Differential calculus and equations Books

Book SynopsisPresents a systematic and unified study of geometric nonlinear functional analysis. This book presents a study of uniformly continuous and Lipschitz functions between Banach spaces, which leads naturally also to the classification of Banach spaces and of their important subsets (mainly spheres) in the uniform and Lipschitz categories.Table of ContentsIntroduction Retractions, extensions and selections Retractions, extensions and selections (special topics) Fixed points Differentiation of convex functions The Radon-Nikodym property Negligible sets and Gateaux differentiability Lipschitz classification of Banach spaces Uniform embeddings into Hilbert space Uniform classification of spheres Uniform classification of Banach spaces Nonlinear quotient maps Oscillation of uniformly continuous functions on unit spheres of finite-dimensional subspaces Oscillation of uniformly continuous functions on unit spheres of infinite-dimensional subspaces Perturbations of local isometries Perturbations of global isometries Twisted sums Group structure on Banach spaces Appendices Bibliography Index.

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Book SynopsisOrdinary Differential Equations covers the fundamentals of the theory of ordinary differential equations (ODEs), including an extensive discussion of the integration of differential inequalities, on which this theory relies heavily. In addition to these results, the text illustrates techniques involving simple topological arguments, fixed point theorems, and basic facts of functional analysis. Unlike many texts, which supply only the standard simplified theorems, Ordinary Differential Equations presents the basic theory of ODEs in a general way, making it a valuable reference. This SIAM reissue of the 1982 second edition covers invariant manifolds, perturbations, and dichotomies, making the text relevant to current studies of geometrical theory of differential equations and dynamical systems.

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Book SynopsisDifferent linear mathematical methods can be used to solve these models analytically, such as the Inverse Scattering Transformation (IST), Adomian Decomposition Method, Variational Iteration Method (VIM), Homotopy Analysis Method (HAM) and Homotopy Perturbation Method (HPM).Table of ContentsNonlinear Water Waves and Nonlinear Evolution Equations with ApplicationsInverse Scattering Transform and the Theory of SolitonsKorteweg-de Vries Equation (KdV), Different Analytical Methods for Solving theKorteweg-de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of theSemi-analytical Methods for Solving the KdV and mKdV EquationsKorteweg-de Vries Equation (KdV), Some Numerical Methods for Solving theNonlinear Internal WavesPartial Differential Equations that Lead to SolitonsShallow Water Waves and Solitary WavesSoliton PerturbationSolitons and CompactonsSolitons: Historical and Physical IntroductionSolitons InteractionsSolitons, Introduction toSolitons, Tsunamis and Oceanographical Applications ofWater Waves and the Korteweg-de Vries EquationSoliton Solutions for Some Nonlinear Water Wave Dynamical ModelsAnalytical Soliton Solutions for Some Nonlinear Dynamical Water Waves ModelsSoliton Propagation in Solids: Advances and ApplicationsApplications of lump and interaction soliton solutions to the model of liquid crystals and nerve fibersPeriodic cross-kink, rogue-waves, and lump interaction soliton solutions with kink and periodic waves for fractional Bogoyavlenskii equationDouble Tchebyshev spectral tau algorithm for solving KdV equation, with soliton application

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Book SynopsisPresents a self-contained introduction to continuum mechanics that illustrates how many of the important partial differential equations of applied mathematics arise from continuum modeling principles Written as an accessible introduction, Continuum Mechanics: The Birthplace of Mathematical Models provides a comprehensive foundation for mathematical models used in fluid mechanics, solid mechanics, and heat transfer. The book features derivations of commonly used differential equations based on the fundamental continuum mechanical concepts encountered in various fields, such as engineering, physics, and geophysics. The book begins with geometric, algebraic, and analytical foundations before introducing topics in kinematics. The book then addresses balance laws, constitutive relations, and constitutive theory. Finally, the book presents an approach to multiconstituent continua based on mixture theory to illustrate how phenomena, such as diffusion and porous-Table of ContentsPreface v 1 Geometric Setting 1 1.1 Vectors and Euclidean Point Space 2 1.1.1 Vectors 2 1.1.2 Euclidean Point Space 6 1.1.3 Summary 8 1.2 Tensors 8 1.2.1 First-Order Tensors and Vectors 8 1.2.2 Second-Order Tensors 11 1.2.3 Cross Products, Triple Products, and Determinants 15 1.2.4 Orthogonal Tensors 20 1.2.5 Invariants of a Tensor 21 1.2.6 Derivatives of Tensor-Valued Functions 24 1.2.7 Summary 27 2 Kinematics I: The Calculus of Motion 29 2.1 Bodies, Motions, and Deformations 29 2.1.1 Deformation 32 2.1.2 Examples of Motions 33 2.1.3 Summary 36 2.2 Derivatives of Motion 36 2.2.1 Time Derivatives 37 2.2.2 Derivatives with Respect to Position 38 2.2.3 The Deformation Gradient 40 2.2.4 Summary 42 2.3 Pathlines, Streamlines, and Streaklines 43 2.3.1 Three Types of Arc 43 2.3.2 An Example 45 2.3.3 Summary 49 2.4 Integrals Under Motion 49 2.4.1 Arc, Surface, and Volume Integrals 49 2.4.2 Reynolds Transport Theorem 55 2.4.3 Summary 57 3 Kinematics II: Strain and its Rates 59 3.1 Strain 59 3.1.1 Symmetric Tensors 60 3.1.2 Polar Decomposition and the Deformation Gradient 64 3.1.3 Examples 66 3.1.4 Cauchy–Green and Strain Tensors 68 3.1.5 Strain Invariants 70 3.1.6 Summary 71 3.2 Infinitesimal Strain 72 3.2.1 The Infinitesimal Strain Tensor 72 3.2.2 Summary 75 3.3 Strain Rates 75 3.3.1 Stretching and Spin Tensors 76 3.3.2 Skew Tensors, Spin, and Vorticity 79 3.3.3 Summary 84 3.4 Vorticity and Circulation 84 3.4.1 Circulation 84 3.4.2 Summary 88 3.5 Observer Transformations 89 3.5.1 Changes in Frame of Reference 89 3.5.2 Summary 95 4 Balance Laws 97 4.1 Mass Balance 98 4.1.1 Local Forms of Mass Balance 99 4.1.2 Summary 102 4.2 Momentum Balance 102 4.2.1 Analysis of Stress 104 4.2.2 Inertial Frames of Reference 110 4.2.3 Momentum Balance in Referential Coordinates 113 4.2.4 Summary 114 4.3 Angular Momentum Balance 115 4.3.1 Symmetry of the Stress Tensor 117 4.3.2 Summary 118 4.4 Energy Balance 119 4.4.1 Thermal Energy Balance 122 4.4.2 Summary 124 4.5 Entropy Inequality 124 4.5.1 Motivation 125 4.5.2 Clausius–Duhem Inequality 126 4.5.3 Summary 127 4.6 Jump Conditions 127 4.6.1 Singular Surfaces 129 4.6.2 Localization 132 4.6.3 Summary 135 5 Constitutive Relations: Examples of Mathematical Models 137 5.1 Heat Transfer 138 5.1.1 Properties of the Heat Equation 140 5.1.2 Summary 142 5.2 Potential Theory 143 5.2.1 Motivation 143 5.2.2 Boundary Conditions 144 5.2.3 Uniqueness of Solutions to the Poisson Equation 146 5.2.4 Maximum Principle 147 5.2.5 Mean Value Property 150 5.2.6 Summary 151 5.3 Fluid Mechanics 152 5.3.1 Ideal Fluids 152 5.3.2 An Ideal Fluid in a Rotating Frame of Reference 154 5.3.3 Acoustics 155 5.3.4 Incompressible Newtonian Fluids 158 5.3.5 Stokes Flow 159 5.3.6 Summary 163 5.4 Solid Mechanics 164 5.4.1 Static Displacements 164 5.4.2 Elastic Waves 167 5.4.3 Summary 170 6 Constitutive Theory 173 6.1 Conceptual Setting 174 6.1.1 The Need to Close the System 174 6.1.2 Summary 176 6.2 Determinism and Equipresence 177 6.2.1 Determinism 177 6.2.2 Equipresence 177 6.2.3 Summary 178 6.3 Objectivity 179 6.3.1 Reducing Functional Dependencies 180 6.3.2 Summary 182 6.4 SYMMETRY 183 6.4.1 Changes in Reference Configuration 183 6.4.2 Symmetry Groups 186 6.4.3 Classification of Materials 189 6.4.4 Implications for Thermoviscous Fluids 193 6.4.5 Summary 193 6.5 Admissibility 194 6.5.1 Implications of the Entropy Inequality 195 6.5.2 Analysis of Equilibrium 197 6.5.3 Linear, Isotropic, Thermoelastic Solids 199 6.5.4 Summary 202 7 Multiconstituent Continua 203 7.1 Constituents 204 7.1.1 Configurations and Motions 204 7.1.2 Volume Fractions and Densities 206 7.1.3 Summary 208 7.2 Multiconstituent Balance Laws 209 7.2.1 Multiconstituent Mass Balance 210 7.2.2 Multiconstituent Momentum Balance 212 7.2.3 Multiconstituent Angular Momentum Balance 214 7.2.4 Multiconstituent Energy Balance 215 7.2.5 Multiconstituent Entropy Inequality 216 7.2.6 Isothermal, Nonreacting Multiphase Mixtures 217 7.2.7 Summary 219 7.3 Fluid Flow in a Porous Solid 220 7.3.1 Modeling Assumptions for Porous Media 221 7.3.2 Balance Laws for the Fluid and Solid Phases 223 7.3.3 Equilibrium Constraints 225 7.3.4 Linear Extensions From Equilibrium 226 7.3.5 Commentary 228 7.3.6 Potential Formulation of Darcy’s Law 229 7.3.7 Summary 233 7.4 Diffusion in a Binary Fluid Mixture 234 7.4.1 Modeling Assumptions for Binary Diffusion 235 7.4.2 Balance Laws for the Two Species 235 7.4.3 Constitutive Relationships for Diffusion 236 7.4.4 Modeling Solute Transport 239 7.4.5 Summary 242 A Guide to Notation 243 A.1 General Conventions 243 A.2 Letters Reserved for Dedicated Uses 244 A.3 Special Symbols 245 B Vector Integral Theorems 247 B.1 Stokes’s Theorem 248 B.2 The Divergence Theorem 249 B.3 The Change-of-variables Theorem 252 C Hints and Solutions to Exercises 253 References 265 Index 269

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Book SynopsisA new edition of this classic work, comprehensively revised to present exciting new developments in this important subject The study of numerical methods for solving ordinary differential equations is constantly developing and regenerating, and this third edition of a popular classic volume, written by one of the world's leading experts in the field, presents an account of the subject which reflects both its historical and well-established place in computational science and its vital role as a cornerstone of modern applied mathematics. In addition to serving as a broad and comprehensive study of numerical methods for initial value problems, this book contains a special emphasis on Runge-Kutta methods by the mathematician who transformed the subject into its modern form dating from his classic 1963 and 1972 papers. A second feature is general linear methods which have now matured and grown from being a framework for a unified theory of a wide range of diverse numTable of ContentsForeword xiii Preface to the first edition xv Preface to the second edition xix Preface to the third edition xxi 1 Differential and Difference Equations 1 10 Differential Equation Problems 1 100 Introduction to differential equations 1 101 The Kepler problem 4 102 A problem arising from the method of lines 7 103 The simple pendulum 11 104 A chemical kinetics problem 14 105 The Van der Pol equation and limit cycles 16 106 The Lotka–Volterra problem and periodic orbits 18 107 The Euler equations of rigid body rotation 20 11 Differential Equation Theory 22 110 Existence and uniqueness of solutions 22 111 Linear systems of differential equations 24 112 Stiff differential equations 26 12 Further Evolutionary Problems 28 120 Many-body gravitational problems 28 121 Delay problems and discontinuous solutions 30 122 Problems evolving on a sphere 33 123 Further Hamiltonian problems 35 124 Further differential-algebraic problems 36 13 Difference Equation Problems 38 130 Introduction to difference equations 38 131 A linear problem 39 132 The Fibonacci difference equation 40 133 Three quadratic problems 40 134 Iterative solutions of a polynomial equation 41 135 The arithmetic-geometric mean 43 14 Difference Equation Theory 44 140 Linear difference equations 44 141 Constant coefficients 45 142 Powers of matrices 46 15 Location of Polynomial Zeros 50 150 Introduction 50 151 Left half-plane results 50 152 Unit disc results 52 Concluding remarks 53 2 Numerical Differential Equation Methods 55 20 The Euler Method 55 200 Introduction to the Euler method 55 201 Some numerical experiments 58 202 Calculations with stepsize control 61 203 Calculations with mildly stiff problems 65 204 Calculations with the implicit Euler method 68 21 Analysis of the Euler Method 70 210 Formulation of the Euler method 70 211 Local truncation error 71 212 Global truncation error 72 213 Convergence of the Euler method 73 214 Order of convergence 74 215 Asymptotic error formula 78 216 Stability characteristics 79 217 Local truncation error estimation 84 218 Rounding error 85 22 Generalizations of the Euler Method 90 220 Introduction 90 221 More computations in a step 90 222 Greater dependence on previous values 92 223 Use of higher derivatives 92 224 Multistep–multistage–multiderivative methods 94 225 Implicit methods 95 226 Local error estimates 96 23 Runge–Kutta Methods 97 230 Historical introduction 97 231 Second order methods 98 232 The coefficient tableau 98 233 Third order methods 99 234 Introduction to order conditions 100 235 Fourth order methods 101 236 Higher orders 103 237 Implicit Runge–Kutta methods 103 238 Stability characteristics 104 239 Numerical examples 108 24 Linear MultistepMethods 111 240 Historical introduction 111 241 Adams methods 111 242 General form of linear multistep methods 113 243 Consistency, stability and convergence 113 244 Predictor–corrector Adams methods 115 245 The Milne device 117 246 Starting methods 118 247 Numerical examples 119 25 Taylor Series Methods 120 250 Introduction to Taylor series methods 120 251 Manipulation of power series 121 252 An example of a Taylor series solution 122 253 Other methods using higher derivatives 123 254 The use of f derivatives 126 255 Further numerical examples 126 26 MultivalueMulitistage Methods 128 260 Historical introduction 128 261 Pseudo Runge–Kutta methods 128 262 Two-step Runge–Kutta methods 129 263 Generalized linear multistep methods 130 264 General linear methods 131 265 Numerical examples 133 27 Introduction to Implementation 135 270 Choice of method 135 271 Variable stepsize 136 272 Interpolation 138 273 Experiments with the Kepler problem 138 274 Experiments with a discontinuous problem 139 Concluding remarks 142 3 Runge–KuttaMethods 143 30 Preliminaries 143 300 Trees and rooted trees 143 301 Trees, forests and notations for trees 146 302 Centrality and centres 147 303 Enumeration of trees and unrooted trees 150 304 Functions on trees 153 305 Some combinatorial questions 155 306 Labelled trees and directed graphs 156 307 Differentiation 159 308 Taylor’s theorem 161 31 Order Conditions 163 310 Elementary differentials 163 311 The Taylor expansion of the exact solution 166 312 Elementary weights 168 313 The Taylor expansion of the approximate solution 171 314 Independence of the elementary differentials 174 315 Conditions for order 174 316 Order conditions for scalar problems 175 317 Independence of elementary weights 178 318 Local truncation error 180 319 Global truncation error 181 32 Low Order ExplicitMethods 185 320 Methods of orders less than 4 185 321 Simplifying assumptions 186 322 Methods of order 4 189 323 New methods from old 195 324 Order barriers 200 325 Methods of order 5 204 326 Methods of order 6 206 327 Methods of order greater than 6 209 33 Runge–Kutta Methods with Error Estimates 211 330 Introduction 211 331 Richardson error estimates 211 332 Methods with built-in estimates 214 333 A class of error-estimating methods 215 334 The methods of Fehlberg 221 335 The methods of Verner 223 336 The methods of Dormand and Prince 223 34 Implicit Runge–Kutta Methods 226 340 Introduction 226 341 Solvability of implicit equations 227 342 Methods based on Gaussian quadrature 228 343 Reflected methods 233 344 Methods based on Radau and Lobatto quadrature 236 35 Stability of Implicit Runge–Kutta Methods 243 350 A-stability, A(α)-stability and L-stability 243 351 Criteria for A-stability 244 352 Padé approximations to the exponential function 245 353 A-stability of Gauss and related methods 252 354 Order stars 253 355 Order arrows and the Ehle barrier 256 356 AN-stability 259 357 Non-linear stability 262 358 BN-stability of collocation methods 265 359 The V and W transformations 267 36 Implementable Implicit Runge–Kutta Methods 272 360 Implementation of implicit Runge–Kutta methods 272 361 Diagonally implicit Runge–Kutta methods 273 362 The importance of high stage order 274 363 Singly implicit methods 278 364 Generalizations of singly implicit methods 283 365 Effective order and DESIRE methods 285 37 Implementation Issues 288 370 Introduction 288 371 Optimal sequences 288 372 Acceptance and rejection of steps 290 373 Error per step versus error per unit step 291 374 Control-theoretic considerations 292 375 Solving the implicit equations 293 38 Algebraic Properties of Runge–Kutta Methods 296 380 Motivation 296 381 Equivalence classes of Runge–Kutta methods 297 382 The group of Runge–Kutta tableaux 299 383 The Runge–Kutta group 302 384 A homomorphism between two groups 308 385 A generalization of G1 309 386 Some special elements of G 311 387 Some subgroups and quotient groups 314 388 An algebraic interpretation of effective order 316 39 Symplectic Runge–Kutta Methods 323 390 Maintaining quadratic invariants 323 391 Hamiltonian mechanics and symplectic maps 324 392 Applications to variational problems 325 393 Examples of symplectic methods 326 394 Order conditions 327 395 Experiments with symplectic methods 328 4 Linear Multistep Methods 333 40 Preliminaries 333 400 Fundamentals 333 401 Starting methods 334 402 Convergence 335 403 Stability 336 404 Consistency 336 405 Necessity of conditions for convergence 338 406 Sufficiency of conditions for convergence 339 41 The Order of Linear Multistep Methods 344 410 Criteria for order 344 411 Derivation of methods 346 412 Backward difference methods 347 42 Errors and Error Growth 348 420 Introduction 348 421 Further remarks on error growth 350 422 The underlying one-step method 352 423 Weakly stable methods 354 424 Variable stepsize 355 43 Stability Characteristics 357 430 Introduction 357 431 Stability regions 359 432 Examples of the boundary locus method 360 433 An example of the Schur criterion 363 434 Stability of predictor–corrector methods 364 44 Order and Stability Barriers 367 440 Survey of barrier results 367 441 Maximum order for a convergent k-step method 368 442 Order stars for linear multistep methods 371 443 Order arrows for linear multistep methods 373 45 One-leg Methods and G-stability 375 450 The one-leg counterpart to a linear multistep method 375 451 The concept of G-stability 376 452 Transformations relating one-leg and linear multistep methods 379 453 Effective order interpretation 380 454 Concluding remarks on G-stability 380 46 Implementation Issues 381 460 Survey of implementation considerations 381 461 Representation of data 382 462 Variable stepsize for Nordsieck methods 385 463 Local error estimation 386 Concluding remarks 387 5 General Linear Methods 389 50 RepresentingMethods in General Linear Form 389 500 Multivalue–multistage methods 389 501 Transformations of methods 391 502 Runge–Kutta methods as general linear methods 392 503 Linear multistep methods as general linear methods 393 504 Some known unconventional methods 396 505 Some recently discovered general linear methods 398 51 Consistency, Stability and Convergence 400 510 Definitions of consistency and stability 400 511 Covariance of methods 401 512 Definition of convergence 403 513 The necessity of stability 404 514 The necessity of consistency 404 515 Stability and consistency imply convergence 406 52 The Stability of General Linear Methods 412 520 Introduction 412 521 Methods with maximal stability order 413 522 Outline proof of the Butcher–Chipman conjecture 417 523 Non-linear stability 419 524 Reducible linear multistep methods and G-stability 422 53 The Order of General Linear Methods 423 530 Possible definitions of order 423 531 Local and global truncation errors 425 532 Algebraic analysis of order 426 533 An example of the algebraic approach to order 428 534 The underlying one-step method 429 54 Methods with Runge–Kutta stability 431 540 Design criteria for general linear methods 431 541 The types of DIMSIM methods 432 542 Runge–Kutta stability 435 543 Almost Runge–Kutta methods 438 544 Third order, three-stage ARK methods 441 545 Fourth order, four-stage ARK methods 443 546 A fifth order, five-stage method 446 547 ARK methods for stiff problems 446 55 Methods with Inherent Runge–Kutta Stability 448 550 Doubly companion matrices 448 551 Inherent Runge–Kutta stability 450 552 Conditions for zero spectral radius 452 553 Derivation of methods with IRK stability 454 554 Methods with property F 457 555 Some non-stiff methods 458 556 Some stiff methods 459 557 Scale and modify for stability 460 558 Scale and modify for error estimation 462 56 G-symplectic methods 464 560 Introduction 464 561 The control of parasitism 467 562 Order conditions 471 563 Two fourth order methods 474 564 Starters and finishers for sample methods 476 565 Simulations 480 566 Cohesiveness 481 567 The role of symmetry 487 568 Efficient starting 492 Concluding remarks 497 References 499 Index 509

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Book SynopsisGENERALIZED ORDINARY DIFFERENTIAL EQUATIONS IN ABSTRACT SPACES AND APPLICATIONS Explore a unified view of differential equations through the use of the generalized ODE from leading academics in mathematics Generalized Ordinary Differential Equations in Abstract Spaces and Applications delivers a comprehensive treatment of new results of the theory of Generalized ODEs in abstract spaces. The book covers applications to other types of differential equations, including Measure Functional Differential Equations (measure FDEs). It presents a uniform collection of qualitative results of Generalized ODEs and offers readers an introduction to several theories, including ordinary differential equations, impulsive differential equations, functional differential equations, dynamical equations on time scales, and more. Throughout the book, the focus is on qualitative theory and on corresponding results for other types of differential equations, as well as the connection between Generalized OrdinarTable of ContentsList of Contributors xi Foreword xiii Preface xvii 1 Preliminaries 1Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, Jaqueline G. Mesquita, and Eduard Toon 1.1 Regulated Functions 2 1.1.1 Basic Properties 2 1.1.2 Equiregulated Sets 7 1.1.3 Uniform Convergence 9 1.1.4 Relatively Compact Sets 11 1.2 Functions of Bounded B-Variation 14 1.3 Kurzweil and Henstock Vector Integrals 19 1.3.1 Definitions 20 1.3.2 Basic Properties 25 1.3.3 Integration by Parts and Substitution Formulas 29 1.3.4 The Fundamental Theorem of Calculus 36 1.3.5 A Convergence Theorem 44 Appendix 1.A: The McShane Integral 44 2 The Kurzweil Integral 53Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Jaqueline G. Mesquita 2.1 The Main Background 54 2.1.1 Definition and Compatibility 54 2.1.2 Special Integrals 56 2.2 Basic Properties 57 2.3 Notes on Kapitza Pendulum 67 3 Measure Functional Differential Equations 71Everaldo M. Bonotto, Márcia Federson, Miguel V. S. Frasson, Rogelio Grau, and Jaqueline G. Mesquita 3.1 Measure FDEs 74 3.2 Impulsive Measure FDEs 76 3.3 Functional Dynamic Equations on Time Scales 86 3.3.1 Fundamentals of Time Scales 87 3.3.2 The Perron Δ-integral 89 3.3.3 Perron Δ-integrals and Perron–Stieltjes integrals 90 3.3.4 MDEs and Dynamic Equations on Time Scales 98 3.3.5 Relations with Measure FDEs 99 3.3.6 Impulsive Functional Dynamic Equations on Time Scales 104 3.4 Averaging Methods 106 3.4.1 Periodic Averaging 107 3.4.2 Nonperiodic Averaging 118 3.5 Continuous Dependence on Time Scales 135 4 Generalized Ordinary Differential Equations 145Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita 4.1 Fundamental Properties 146 4.2 Relations with Measure Differential Equations 153 4.3 Relations with Measure FDEs 160 5 Basic Properties of Solutions 173Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon 5.1 Local Existence and Uniqueness of Solutions 174 5.1.1 Applications to Other Equations 178 5.2 Prolongation and Maximal Solutions 181 5.2.1 Applications to MDEs 191 5.2.2 Applications to Dynamic Equations on Time Scales 197 6 Linear Generalized Ordinary Differential Equations 205Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Miguel V. S. Frasson 6.1 The Fundamental Operator 207 6.2 A Variation-of-Constants Formula 209 6.3 Linear Measure FDEs 216 6.4 A Nonlinear Variation-of-Constants Formula for Measure FDEs 220 7 Continuous Dependence on Parameters 225Suzete M. Afonso, Everaldo M. Bonotto, Márcia Federson, and Jaqueline G. Mesquita 7.1 Basic Theory for Generalized ODEs 226 7.2 Applications to Measure FDEs 236 8 StabilityTheory 241Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Luciene P. Gimenes (in memorian), Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon 8.1 Variational Stability for Generalized ODEs 244 8.1.1 Direct Method of Lyapunov 246 8.1.2 Converse Lyapunov Theorems 247 8.2 Lyapunov Stability for Generalized ODEs 256 8.2.1 Direct Method of Lyapunov 257 8.3 Lyapunov Stability for MDEs 261 8.3.1 Direct Method of Lyapunov 263 8.4 Lyapunov Stability for Dynamic Equations on Time Scales 265 8.4.1 Direct Method of Lyapunov 267 8.5 Regular Stability for Generalized ODEs 272 8.5.1 Direct Method of Lyapunov 275 8.5.2 Converse Lyapunov Theorem 282 9 Periodicity 295Marielle Ap. Silva, Everaldo M. Bonotto, Rodolfo Collegari, Márcia Federson, and Maria Carolina Mesquita 9.1 Periodic Solutions and Floquet’s Theorem 297 9.1.1 Linear Differential Systems with Impulses 303 9.2 (θ,T)-Periodic Solutions 307 9.2.1 An Application to MDEs 313 10 Averaging Principles 317Márcia Federson and Jaqueline G. Mesquita 10.1 Periodic Averaging Principles 320 10.1.1 An Application to IDEs 325 10.2 Nonperiodic Averaging Principles 330 11 Boundedness of Solutions 341Suzete M. Afonso, Fernanda Andrade da Silva, Everaldo M. Bonotto, Márcia Federson, Rogelio Grau, Jaqueline G. Mesquita, and Eduard Toon 11.1 Bounded Solutions and Lyapunov Functionals 342 11.2 An Application to MDEs 352 11.2.1 An Example 356 12 Control Theory 361Fernanda Andrade da Silva, Márcia Federson, and Eduard Toon 12.1 Controllability and Observability 362 12.2 Applications to ODEs 365 13 Dichotomies 369Everaldo M. Bonotto and Márcia Federson 13.1 Basic Theory for Generalized ODEs 370 13.2 Boundedness and Dichotomies 381 13.3 Applications to MDEs 391 13.4 Applications to IDEs 400 14 Topological Dynamics 407Suzete M. Afonso, Marielle Ap. Silva, Everaldo M. Bonotto, and Márcia Federson 14.1 The Compactness of the Class F0(Ω,h) 408 14.2 Existence of a Local Semidynamical System 411 14.3 Existence of an Impulsive Semidynamical System 418 14.4 LaSalle’s Invariance Principle 423 14.5 Recursive Properties 425 15 Applications to Functional Differential Equations of Neutral Type 429Fernando G. Andrade, Miguel V. S. Frasson, and Patricia H. Tacuri 15.1 Drops of History 429 15.2 FDEs of Neutral Type with Finite Delay 435 References 455 List of Symbols 471 Index 473

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Book SynopsisMany physical problems that are usually solved by differential equation methods can be solved more effectively by integral equation methods.Trade ReviewA nice introductory text... Presents the basics of linear integral equations theory in a very comprehensive way... [The] richness of examples and applications makes the book extremely useful for teachers and also researchers. —Applications of Mathematics (Review of the Second Edition) This second edition of this highly useful book continues the emphasis on applications and presents a variety of techniques with extensive examples...The book is ideal as a text for a beginning graduate course. Its excellent treatment of boundary value problems and an up-to-date bibliography make the book equally useful for researchers in many applied fields.—MathSciNet ​(Review of the Second Edition)Table of ContentsIntroduction.- Integral Equations with Separable Kernels.- Method Of Successive Approximations.- Classical Fredholm Theory.- Applications of Ordinary Differential Equations.- Applications of Partial Differential Equations.- Symmetric Kernels.- Singular Integral Equations.- Integral Transformation Methods.- Applications to Mixed Boundary Value Problems.- Integral Equations Perturbation Methods.- Appendix.- Bibliography.- Index.​

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Book SynopsisContains the proceedings of two AMS Special Sessions Recent Developments on Analysis and Computation for Inverse Problems for PDEs', held in March 2021, and Recent Advances in Inverse Problems for Partial Differential Equations', held in October 2021.Table of Contents U. G. Abdulla and S. Seif, Discretization and convergence of the EIT optimal control problem in Sobolev spaces with dominating mixed smoothness T. T. Le, Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method I. Harris, Regularization of the factorization method with applications to inverse scattering T. Le, D.-L. Nguyen, V. Nguyen, and T. Truong, Sampling type method combined with deep learning for inverse scattering with one incident wave D.-L. Nguyen and T. Truong, Fast numerical solutions to direct and inverse scattering for bi-anisotropic periodic Maxwell's equation L. H. Nguyen and H. T.T. Vu, Reconstructing a space-dependent source term via the quasi-reversibility method Q. Tran, Convergence analysis of Nedelec finite element approximations for a stationary Maxwell's system M. V. Klibanov, K. V. Golubnichiy, and A. V. Nikitin, Quasi-reversibility method and neural network machine learning for forecasting of stock option prices V. A. Khoa, M. V. Klibanov, W. G. Powell, and L. C. Nguyen, Numerical reconstruction for 3D nonlinear SAR imaging via a version of the convexification method V. A. Khoa, M. T. N. Truong, I. Hogan, and R. Williams, Initial state reconstruction on graphs L. Besabe and D. Onofrei, Active control of scalar Helmholtz fields in the presence of known impenetrable obstacles

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Book SynopsisThe subjects explored in the book are dimensional analysis and scaling, dynamical systems, perturbation methods, and calculus of variations. These are immense subjects of wide applicability and a fertile ground for critical thinking and quantitative reasoning, in which every student of mathematics should have some experience.Table of Contents Dimensional analysis Scaling One-dimensional dynamics Two-dimensional dynamics Perturbation methods Calculus of variations Bibliography Index

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Book SynopsisPrimarily a book on partial differential equations with two definite slants: toward inverse problems and to the inclusion of fractional derivatives. The book also has an extensive historical section and the material that can be called fractional calculus' and ordinary differential equations with fractional derivatives.Table of Contents Preamble Genesis of fractional models Special functions and tools Fractional calculus Fractional ordinary differential equations Mathematical theory of subdiffusion Analysis of fractionally damped wave equations Methods for solving inverse problems Fundamental inverse problems for fractional order models Inverse problems for fractional diffusion Inverse problems for fractionally damped wave equations Outlook beyond Abel Mathematical preliminaries Bibliography Index

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Book SynopsisTable of Contents Strings, drums, and the Laplacian The spectral theorems Variational principles and applications Nodal geometry of eigenfunctions Eigenvalue inequalities Heat equation, spectral invariants, and isospectrality The Steklov problem and the Dirichlet-to-Neumann map A short tutorial on numerical spectral geometry Background definitions and notation Image credits Bibliography Index

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Book SynopsisExplores the interplay between modeling, analysis, discretization, matrix computation, and model reduction. The authors link PDE analysis, functional analysis, and calculus of variations with matrix iterative computation using Krylov subspace methods and address the challenges that arise during formulation of the mathematical model through to efficient numerical solution of the algebraic problem.

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Book SynopsisWeb-like waves, often observed on the surface of shallow water, are examples of nonlinear waves. They are generated by nonlinear interactions among several obliquely propagating solitary waves, also known as solitons. In this book, modern mathematical tools—algebraic geometry, algebraic combinatorics, and representation theory, among others—are used to analyze these two-dimensional wave patterns. The author’s primary goal is to explain some details of the classification problem of the soliton solutions of the KP equation (or KP solitons) and their applications to shallow water waves.This book is intended for researchers and graduate students.

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Book SynopsisThis book addresses an important class of mathematical problems (the Riemann problem) for first-order hyperbolic partial differential equations (PDEs), which arise when modeling wave propagation in applications such as fluid dynamics, traffic flow, acoustics, and elasticity.It covers the fundamental ideas related to classical Riemann solutions, including their special structure and the types of waves that arise, as well as the ideas behind fast approximate solvers for the Riemann problem.The emphasis is on the general ideas, but each chapter delves into a particular application. The book is available in electronic form as a collection of Jupyter notebooks that contain executable computer code and interactive figures and animations.

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Book SynopsisThe Mathematical Elasticity set contains three self-contained volumes that together provide the only modern treatise on elasticity. They introduce contemporary research on three-dimensional elasticity, the theory of plates, and the theory of shells. Each volume contains proofs, detailed surveys of all mathematical prerequisites, and many problems for teaching and self-study. An extended preface and extensive bibliography have been added to each volume to highlight the progress that has been made since the original publication.The first book, Three-Dimensional Elasticity, covers the modeling and mathematical analysis of nonlinear three-dimensional elasticity. In volume two, Theory of Plates, asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear plate and shallow shell theories. The objective of Theory of Shells, the final volume, is to show how asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear shell theories: membrane, generalized membrane, and flexural.These classic textbooks are for advanced undergraduates, first-year graduate students, and researchers in pure or applied mathematics or continuum mechanics. They are appropriate for courses in mathematical elasticity, theory of plates and shells, continuum mechanics, computational mechanics, and applied mathematics in general.

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Book SynopsisIn recent years, there has been an explosion of interest in network-based modeling in many branches of science. This book synthesizes some of the common features of many such models, providing a general framework analogous to the modern theory of nonlinear dynamical systems. How networks lead to behavior not typical in a general dynamical system and how the architecture and symmetry of the network influence this behavior are the book's main themes.Dynamics and Bifurcation in Networks: Theory and Applications of Coupled Differential Equations is the first book to describe the formalism for network dynamics developed over the past 20 years. In it, the authors introduce a definition of a network and the associated class of "admissible" ordinary differential equations, in terms of a directed graph whose nodes represent component dynamical systems and whose arrows represent couplings between these systems. They also develop connections between network architecture and the typical dynamics and bifurcations of these equations and discuss applications of this formalism to various areas of science, including gene regulatory networks, animal locomotion, decision-making, homeostasis, binocular rivalry, and visual illusions.This book will be of interest to scientific researchers in any area that uses network models, which includes many parts of biology, physics, chemistry, computer science, electrical and electronic engineering, psychology, and sociology.

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Book SynopsisExtremum Seeking through Delays and PDEs, the first book on the topic, expands the scope of applicability of the extremum seeking method, from static and finite-dimensional systems to infinite-dimensional systems. Readers will find: Numerous algorithms for model-free real-time optimization are developed and their convergence guaranteed. Extensions from single-player optimization to noncooperative games, under delays and PDEs, are provided. The delays and pdes are compensated in the control designs using the PDE backstepping approach, and stability is ensured using infinite-dimensional versions of averaging theory. Accessible and powerful tools for analysis. This book is intended for control engineers in all disciplines (electrical, mechanical, aerospace, chemical), mathematicians, physicists, biologists, and economists. It is appropriate for graduate students, researchers, and industrial users.

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Book SynopsisDivided into two self-contained parts, this textbook is an introduction to modern real analysis. More than 350 exercises and 100 examples are integrated into the text to help clarify the theoretical considerations and the practical applications to differential geometry, Fourier series, differential equations, and other subjects. The first section of Classical Analysis of Real-Valued Functions covers the theorems of existence of supremum and infimum of bounded sets on the real line and the Lagrange formula for differentiable functions. Applications of these results are crucial for classical mathematical analysis, andmany are threaded through the text. In the second part of the book, the implicit function theorem plays a central role, while the Gauss–Ostrogradskii formula, surface integration, Heine–Borel lemma, the Ascoli–Arzelà theorem, and the one-dimensional indefinite Lebesgue integral are also covered. This book is intended for students in the first and second years of classical universities majoring in pure and applied mathematics, but students of engineering disciplines will also gain important and helpful insights. It is appropriate for courses in mathematical analysis, functional analysis, real analysis, and calculus and can be used for self-study as well.

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Book SynopsisBasic Theory of Fractional Differential Equations is a contemporary collection of 16 articles that explores modern methods and applications of FDEs. It covers the extended Jacobi elliptic function expansion method, numerical approximation techniques like -step continuous BDFs for FIVPs, stability theories, and various fractional derivatives. The book finds applications in diverse fields, making it a valuable tool for solving real-world problems in physics, engineering, finance, and biology.Table of Contents Chapter 1 Introduction Chapter 2 Exact Solutions for Some Fractional Differential Equations Chapter 3 Compact and Noncompact Solutions to Generalized Sturm–Liouville and Langevin Equation with Caputo–Hadamard Fractional Derivative Chapter 4 Solution of Fractional Partial Differential Equations Using Fractional Power Series Method Chapter 5 Novel Stability Results for Caputo Fractional Differential Equations Chapter 6 Block Backward Differentiation Formulas for Fractional Differential Equations Chapter 7 Nonlinear Fractional Differential Equations with Nonlocal Fractional Integro-Differential Boundary Conditions Chapter 8 A New Fractional Jacobi Elliptic Equation Method for Solving Fractional Partial Differential Equations Chapter 9 Existence of Solutions for Nonlinear Singular Fractional Differential Equations with Fractional Derivative Condition Chapter 10 On the Nonlinear Fractional Differential Equations with Caputo Sequential Fractional Derivative Chapter 11 On Fractional Order Hybrid Differential Equations Chapter 12 Fuzzy Conformable Fractional Differential Equations Chapter 13 On Hilfer-Type Fractional Impulsive Differential Equations Chapter 14 The Numerical Investigation of Fractional-Order Zakharov–Kuznetsov Equations Chapter 15 Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative Chapter 16 Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives Chapter 17 Stability of a Nonlinear Fractional Langevin System with Nonsingular Exponential Kernel and Delay Control

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Book SynopsisThe purpose of this book is to introduce and study numerical methods basic and advanced ones for scientific computing. This last refers to the implementation of appropriate approaches to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or of engineering (mechanics of structures, mechanics of fluids, treatment signal, etc.). Each chapter of this book recalls the essence of the different methods resolution and presents several applications in the field of engineering as well as programs developed under Matlab software.Table of ContentsPreface ix Part 1. Solving Equations 1 Chapter 1. Solving Nonlinear Equations 3 1.1 Introduction 3 1.2 Separating the roots 3 1.3 Approximating a separated root 4 1.3.1 Bisection method (or dichotomy method) 4 1.3.2 Fixed-point method 6 1.3.3 First convergence criterion 7 1.3.4 Iterative stopping criteria.8 1.3.5 Second convergence criterion (local criterion) 9 1.3.6 Newton’s method (or the method of tangents) 10 1.3.7 Secant method 12 1.3.8 Regula falsi method (or false position method) 17 1.4 Order of an iterative process.19 1.5 Using Matlab 19 1.5.1 Finding the roots of polynomials 19 1.5.2 Bisection method 21 1.5.3 Newton’s method 22 Chapter 2. Numerically Solving Differential Equations 25 2.1 Introduction 25 2.2 Cauchy problem and discretization 27 2.3 Euler’s method 30 2.3.1 Interpretation 30 2.3.2 Convergence 30 2.4 One-step Runge–Kutta method 31 2.4.1 Second-order Runge–Kutta method 32 2.4.2 Fourth-order Runge–Kutta method 33 2.5 Multi-step Adams methods 36 2.5.1 Open Adams methods 36 2.5.2 Closed Adams formulas 39 2.6 Predictor–Corrector method.41 2.7 Using Matlab 43 Part 2. Solving PDEs 47 Chapter 3. Finite Difference Methods 49 3.1 Introduction 49 3.2 Presentation of the finite difference method 51 3.2.1 Convergence, consistency and stability 53 3.2.2 Courant–Friedrichs–Lewy condition 56 3.2.3 Von Neumann stability analysis 57 3.3 Hyperbolic equations 58 3.3.1 Key results 59 3.3.2 Numerical schemes for solving the transport equation 63 3.3.3 Wave equation 66 3.3.4 Burgers equation 68 3.4 Elliptic equations 72 3.4.1 Poisson equation 72 3.5 Parabolic equations 74 3.5.1 Heat equation 74 3.6 Using Matlab 76 Chapter 4. Finite Element Method 83 4.1 Introduction 83 4.2 One-dimensional finite element methods 83 4.3 Two-dimensional finite element methods 88 4.4 General procedure of the method 93 4.5 Finite element method for computing elastic structures 93 4.5.1 Linear elasticity 93 4.5.2 Variational formulation of the linear elasticity problem 97 4.5.3 Planar linear elasticity problems 99 4.5.4 Applying the finite element method to planar problems 101 4.5.5 Axisymmetric problems.105 4.5.6 Three-dimensional problems 107 4.6 Using Matlab 107 4.6.1 Solving Poisson’s equation 108 4.6.2 Solving the heat equation.111 4.6.3 Computing structures 112 Chapter 5. Finite Volume Methods 117 5.1 Introduction 117 5.2 Finite volume method (FVM) 118 5.2.1 Conservation properties of the method 118 5.2.2 The stages of the method.119 5.2.3 Convergence 120 5.2.4 Consistency 120 5.2.5 Stability 120 5.3 Advection schemes 121 5.3.1 Two-dimensional FVM. 126 5.3.2 Convection-diffusion equation 129 5.3.3 Central differencing scheme 131 5.3.4 Upwind (decentered) scheme 133 5.3.5 Hybrid scheme 136 5.3.6 Power-law scheme 136 5.3.7 QUICK scheme 137 5.3.8 Higher-order schemes 139 5.3.9 Unsteady one-dimensional convection-diffusion Equation 140 5.3.10 Explicit scheme 142 5.3.11 Crank–Nicolson scheme.142 5.3.12 Implicit scheme 143 5.4 Using Matlab 144 Chapter 6. Meshless Methods. 147 6.1 Introduction 147 6.2 Limitations of the FEM and motivation of meshless methods 148 6.3 Examples of meshless methods148 6.3.1 Advantages of meshless methods 149 6.3.2 Disadvantages of meshless methods150 6.3.3 Comparison of the finite element method and meshless methods 151 6.4 Basis of meshless methods 151 6.4.1 Approximations 151 6.4.2 Kernel (weight) functions.152 6.4.3 Completeness 152 6.4.4 Partition of unity 152 6.5 Meshless method (EFG) 153 6.5.1 Theory 153 6.5.2 Moving Least-Squares Approximation 153 6.6 Application of the meshless method to elasticity 163 6.6.1 Formulation of static linear elasticity 163 6.6.2 Imposing essential boundary conditions 165 6.7 Numerical examples 170 6.7.1 Fixed-free beam 170 6.7.2 Compressed block 171 6.8 Using Matlab 173 Part 3. Appendices 179 Appendix 1181 Appendix 2189 Bibliography 195 Index 199

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Book SynopsisThis book introduces an original fractional calculus methodology (�the infinite state approach�) which is applied to the modeling of fractional order differential equations (FDEs) and systems (FDSs). Its modeling is based on the frequency distributed fractional integrator, while the resulting model corresponds to an integer order and infinite dimension state space representation. This original modeling allows the theoretical concepts of integer order systems to be generalized to fractional systems, with a particular emphasis on a convolution formulation. With this approach, fundamental issues such as system state interpretation and system initialization – long considered to be major theoretical pitfalls – have been solved easily. Although originally introduced for numerical simulation and identification of FDEs, this approach also provides original solutions to many problems such as the initial conditions of fractional derivatives, the uniqueness of FDS transients, formulation of analytical transients, fractional differentiation of functions, state observation and control, definition of fractional energy, and Lyapunov stability analysis of linear and nonlinear fractional order systems. This second volume focuses on the initialization, observation and control of the distributed state, followed by stability analysis of fractional differential systems.Table of ContentsForeword xiii Preface xv Part 1. Initialization, State Observation and Control 1 Chapter 1. Initialization of Fractional Order Systems 3 1.1. Introduction 3 1.2. Initialization of an integer order differential system 4 1.2.1. Introduction 4 1.2.2. Response of a linear system 4 1.2.3. Input/output solution 6 1.2.4. State space solution 7 1.2.5. First-order system example 8 1.3. Initialization of a fractional differential equation 10 1.3.1. Introduction 10 1.3.2. Free response of a simple FDE 10 1.4. Initialization of a fractional differential system 14 1.4.1. Introduction 14 1.4.2. State space representation 14 1.4.3. Input/output formulation 15 1.5. Some initialization examples 17 1.5.1. Introduction 17 1.5.2. Initialization of the fractional integrator 17 1.5.3. Initialization of the Riemann–Liouville derivative 19 1.5.4. Initialization of an elementary FDS 21 1.5.5. Conclusion 33 Chapter 2. Observability and Controllability of FDEs/FDSs 35 2.1. Introduction 35 2.2. A survey of classical approaches to the observability and controllability of fractional differential systems 37 2.2.1. Introduction 37 2.2.2. Definition of observability and controllability 37 2.2.3. Observability and controllability criteria for a linear integer order system 37 2.2.4. Observability and controllability of FDS 39 2.3. Pseudo-observability and pseudo-controllability of an FDS 40 2.3.1. Introduction 40 2.3.2. Elementary approach 41 2.3.3. Cayley–Hamilton approach 45 2.3.4. Gramian approach 49 2.3.5. Gilbert’s approach 52 2.3.6. Conclusion 57 2.3.7. Pseudo-controllability example 58 2.4. Observability and controllability of the distributed state 60 2.4.1. Introduction 60 2.4.2. Observability of the distributed state 62 2.4.3. Controllability of the distributed state 64 2.5. Conclusion 65 Chapter 3. Improved Initialization of Fractional Order Systems 67 3.1. Introduction 67 3.2. Initialization: problem statement 68 3.3. Initialization with a fractional observer 71 3.3.1. Fractional observer definition 71 3.3.2. Stability analysis 72 3.3.3. Convergence analysis 74 3.3.4. Numerical example 1: one-derivative system 76 3.3.5. Numerical example 2: non-commensurate order system 78 3.4. Improved initialization 81 3.4.1. Introduction 81 3.4.2. Non-commensurate order principle 82 3.4.3. Gradient algorithm 84 3.4.4. One-derivative FDE example 87 3.4.5. Two-derivative FDE example 91 A.3. Appendix 95 A.3.1. Convergence of gradient algorithm 95 A.3.2. Stability and limit value of λ 98 Chapter 4. State Control of Fractional Differential Systems 99 4.1. Introduction 99 4.2. Pseudo-state control of an FDS 100 4.2.1. Introduction 100 4.2.2. Numerical simulation example 101 4.3. State control of the elementary FDE 103 4.3.1. Introduction 103 4.3.2. State control of a fractional integrator 104 4.4. State control of an FDS 121 4.4.1. Introduction 121 4.4.2. Principle of state control 122 4.4.3. State control of two integrators in series 124 4.4.4. Numerical example 126 4.4.5. State control of a two-derivative FDE 129 4.4.6. Pseudo-state control of the two-derivative FDE 130 4.5. Conclusion 131 Chapter 5. Fractional Model-based Control of the Diffusive RC Line 133 5.1. Introduction 133 5.2. Identification of the RC line using a fractional model 134 5.2.1. Introduction 134 5.2.2. An identification algorithm dedicated to fractional models 134 5.2.3. Simulation of the diffusive RC line 139 5.2.4. Experimental identification 149 5.3. Reset of the RC line 154 5.3.1. Introduction 154 5.3.2. Natural relaxation 155 5.3.3. Principle of the reset technique 156 5.3.4. Proposed reset procedure 158 5.3.5. Experimental results 159 5.3.6. Comments 164 5.3.7. Conclusion 165 Part 2. Stability of Fractional Differential Equations and Systems 167 Chapter 6. Stability of Linear FDEs Using the Nyquist Criterion 169 6.1. Introduction 169 6.2. Simulation and stability of fractional differential equations 171 6.2.1. Simulation of an FDE 171 6.2.2. Stability of the simulation scheme 172 6.2.3. Stability analysis of FDEs using the Nyquist criterion 174 6.3. Stability of ordinary differential equations 175 6.3.1. Introduction 175 6.3.2. Open-loop transfer function 176 6.3.3. Drawing of H OL (jω) graph in the complex plane 177 6.3.4. Stability of the third-order ODE 178 6.3.5. Conclusion 182 6.4. Stability analysis of FDEs 182 6.4.1. Introduction 182 6.4.2. Drawing of H OL (jω) graph in the complex plane 182 6.4.3. Stability of the one-derivative FDE 184 6.4.4. Stability of the two-derivative FDE 187 6.4.5. Stability of the N-derivative FDE 194 6.4.6. Conclusion 195 6.5. Stability analysis of ODEs with time delays 195 6.5.1. Introduction 195 6.5.2. Definitions 196 6.5.3. Stability analysis 196 6.5.4. Application to an example 198 6.6. Stability analysis of FDEs with time delays 200 6.6.1. Definitions 200 6.6.2. Stability 201 6.6.3. Application to an example 202 Chapter 7. Fractional Energy 205 7.1. Introduction 205 7.2. Pseudo-energy stored in a fractional integrator 206 7.3. Energy stored and dissipated in a fractional integrator 211 7.3.1. Introduction 211 7.3.2. Electrical distributed network 211 7.3.3. Stored energy 214 7.3.4. Power dissipated in the fractional integrator 215 7.3.5. Energy storage 216 7.3.6. Integer order and fractional order integrators 219 7.3.7. Characterization of fractional energy and its dissipation 226 7.3.8. Fractional energy invariance 231 7.4. Closed-loop and open-loop fractional energies 234 7.4.1. Introduction 234 7.4.2. Energy of the closed-loop model 234 7.4.3. Energy of the open-loop model 237 7.4.4. Stored energies with a step input excitation 239 Chapter 8. Lyapunov Stability of Commensurate Order Fractional Systems 247 8.1. Introduction 247 8.2. Lyapunov stability of a one-derivative FDE 249 8.2.1. Problem statement 249 8.2.2. Numerical simulation 251 8.2.3. Physical interpretation 253 8.2.4. Theoretical interpretation 254 8.3. Lyapunov stability of an N-derivative FDE 258 8.3.1. Introduction 258 8.3.2. The integer order case 258 8.3.3. Lyapunov function of N-derivative systems 261 8.3.4. Stability condition 265 8.4. Lyapunov stability of a two-derivative commensurate order FDE 269 8.4.1. Introduction 269 8.4.2. State space model of the open-loop representation 270 8.4.3. State space models of the closed-loop representation 271 8.4.4. Energy and stability of the open-loop representation 272 8.4.5. Energy and stability of the closed-loop representation 274 8.4.6. Definition of a stability test for a > 0 276 8.5. Lyapunov stability of an N-derivative FDE ( N > 2 ) 281 8.5.1. Introduction 281 8.5.2. Problem statement 282 8.5.3. LMI generalization for N = 3 283 8.5.4. Application example 289 A.8. Appendix 290 A.8.1. Lemma 290 A.8.2. Matignon’s criterion 291 Chapter 9. Lyapunov Stability of Non-commensurate Order Fractional Systems 293 9.1. Introduction 293 9.2. Stored energy, dissipation and energy balance in fractional electrical devices 295 9.2.1. Usual capacitor and inductor devices 295 9.2.2. Fractional capacitor and inductor 296 9.2.3. Energy storage and dissipation in fractional devices 299 9.2.4. Reversibility of energy and energy balance 301 9.3. The usual series RLC circuit 302 9.3.1. Introduction 302 9.3.2. Analysis of the series RLC circuit 302 9.3.3. Stability analysis 304 9.4. The series RLC* fractional circuit 306 9.4.1. Introduction 306 9.4.2. Analysis of the series RLC* circuit 306 9.4.3. Experimental stability analysis 307 9.4.4. Theoretical stability analysis 310 9.4.5. Conclusion 314 9.5. The series RLL*C* circuit 315 9.5.1. Circuit modeling 315 9.5.2. Stability analysis 317 9.6. The series RL*C* fractional circuit 320 9.6.1. Introduction 320 9.6.2. Analysis of the series RL*C* circuit 320 9.6.3. Theoretical stability analysis 322 9.7. Stability of a commensurate order FDE: energy balance approach 325 9.7.1. Introduction 325 9.7.2. Analysis of the commensurate order FDE 325 9.7.3. Application to stability 327 9.8. Stability of a commensurate order FDE: physical interpretation of the usual approach 328 9.8.1. Introduction 328 9.8.2. Commensurate order system 329 9.8.3. Lyapunov function of a fractional differential system 329 9.8.4. Stability analysis 331 9.8.5. Conclusion 334 A.9. Appendix 335 A.9.1. The infinite length LG line 335 A.9.2. Energy storage and dissipation in the fractional capacitor 339 A.9.3. Some integrals 341 Chapter 10. An Introduction to the Lyapunov Stability of Nonlinear Fractional Order Systems 343 10.1. Introduction 343 10.2. Indirect Lyapunov method 344 10.2.1. Introduction 344 10.2.2. Linearization 344 10.2.3. Nonlinear system analysis 345 10.2.4. Local stability of a one-derivative nonlinear fractional system 349 10.3. Lyapunov direct method 353 10.3.1. Introduction 353 10.3.2. The variable gradient method 353 10.3.3. Nonlinear system with one derivative 354 10.3.4. Nonlinear system with two fractional derivatives 357 10.4. The Van der Pol oscillator 363 10.4.1. Electrical nonlinear system 363 10.4.2. Van der Pol oscillator 364 10.4.3. Simulation of the nonlinear system 364 10.4.4. Limit cycle 365 10.5. Analysis of local stability 366 10.5.1. Linearization 366 10.5.2. Local stability 367 10.5.3. Validation of stability results 369 10.6. Large signal analysis 371 10.6.1. Introduction 371 10.6.2. Approximation of the first harmonic [MUL 09] 371 10.6.3. Lyapunov function and oscillation frequency 372 10.6.4. Amplitude of the limit cycle 372 10.6.5. Prediction of the limit cycle 374 References 377 Index 395

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Book SynopsisThis monograph presents the most recent progress in bifurcation theory of impulsive dynamical systems with time delays and other functional dependence. It covers not only smooth local bifurcations, but also some non-smooth bifurcation phenomena that are unique to impulsive dynamical systems. The monograph is split into four distinct parts, independently addressing both finite and infinite-dimensional dynamical systems before discussing their applications. The primary contributions are a rigorous nonautonomous dynamical systems framework and analysis of nonlinear systems, stability, and invariant manifold theory. Special attention is paid to the centre manifold and associated reduction principle, as these are essential to the local bifurcation theory. Specifying to periodic systems, the Floquet theory is extended to impulsive functional differential equations, and this permits an exploration of the impulsive analogues of saddle-node, transcritical, pitchfork and Hopf bifurcations.Readers will learn how techniques of classical bifurcation theory extend to impulsive functional differential equations and, as a special case, impulsive differential equations without delays. They will learn about stability for fixed points, periodic orbits and complete bounded trajectories, and how the linearization of the dynamical system allows for a suitable definition of hyperbolicity. They will see how to complete a centre manifold reduction and analyze a bifurcation at a nonhyperbolic steady state.Trade Review“This book is more than a textbook and more than a research monograph. It can be considered as a guiding book to be used in both theoretical and applied research of various level (graduate students, post-graduates, senior).” (Vladimir Răsvan, zbMATH 1467.37001, 2021)Table of ContentsImpulsive functional differential equations.- Preliminaries.- General linear systems.- Linear periodic systems.- Nonlinear systems and stability.- Invariant manifold theory.- Smooth bifurcations.- Finite-dimensional ordinary impulsive differential equations.- Preliminaries.- Linear systems.- Stability for nonlinear systems.- Invariant manifold theory.- Bifurcations.- Special topics and applications.- Continuous approximation.- Non-smooth bifurcations.- Bifurcations in models from mathematical epidemiology and ecology.

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Book SynopsisThis self-contained monograph unifies theorems, applications and problem solving techniques of matrix inequalities. In addition to the frequent use of methods from Functional Analysis, Operator Theory, Global Analysis, Linear Algebra, Approximations Theory, Difference and Functional Equations and more, the reader will also appreciate techniques of classical analysis and algebraic arguments, as well as combinatorial methods. Subjects such as operator Young inequalities, operator inequalities for positive linear maps, operator inequalities involving operator monotone functions, norm inequalities, inequalities for sector matrices are investigated thoroughly throughout this book which provides an account of a broad collection of classic and recent developments. Detailed proofs for all the main theorems and relevant technical lemmas are presented, therefore interested graduate and advanced undergraduate students will find the book particularly accessible. In addition to several areas of theoretical mathematics, Matrix Analysis is applicable to a broad spectrum of disciplines including operations research, mathematical physics, statistics, economics, and engineering disciplines. It is hoped that graduate students as well as researchers in mathematics, engineering, physics, economics and other interdisciplinary areas will find the combination of current and classical results and operator inequalities presented within this monograph particularly useful.Trade Review“The book is written in a readable style and provides several interesting and nice techniques. It is very useful for graduate students and researchers interested in operator and norm inequalities.” (Mohammad Sal Moslehian, Mathematical Reviews, June, 2023)The book contains a bibliography of over 200 items and … the many inequalities presented, usually with full proofs provided. … if you are looking for an inequality in the areas covered, then this should be a useful source.” (John D. Dixon, zbMATH 1477.15001, 2022)Table of Contents1. Elementary linear algebra review.- 2. Interpolating the arithmetic-geometric mean inequality and its operator version.- 3. Operator inequalities for positive linear maps.- 4. Operator inequalities involving operator monotone functions.- 5. Inequalities for sector matrices.- 6. Positive partial transpose matrix inequalities.- References.- Index.

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Book SynopsisThis book is devoted to the development of optimal control theory for finite dimensional systems governed by deterministic and stochastic differential equations driven by vector measures. The book deals with a broad class of controls, including regular controls (vector-valued measurable functions), relaxed controls (measure-valued functions) and controls determined by vector measures, where both fully and partially observed control problems are considered. In the past few decades, there have been remarkable advances in the field of systems and control theory thanks to the unprecedented interaction between mathematics and the physical and engineering sciences. Recently, optimal control theory for dynamic systems driven by vector measures has attracted increasing interest. This book presents this theory for dynamic systems governed by both ordinary and stochastic differential equations, including extensive results on the existence of optimal controls and necessary conditions for optimality. Computational algorithms are developed based on the optimality conditions, with numerical results presented to demonstrate the applicability of the theoretical results developed in the book. This book will be of interest to researchers in optimal control or applied functional analysis interested in applications of vector measures to control theory, stochastic systems driven by vector measures, and related topics. In particular, this self-contained account can be a starting point for further advances in the theory and applications of dynamic systems driven and controlled by vector measures.Trade Review“This book is a masterpiece of mathematical work where the authors joyfully stroll with the reader through the pleasant universe of control theory and its many applications. … I think the authors did a very good job. Their contribution is surely going to be particularly helpful to applied mathematicians. It’s a must read!” (Calvin Tadmon, SIAM Review, Vol. 64 (4), December, 2022)Table of Contents1 Mathematical Preliminaries.- 2 Linear Systems.- 3 Nonlinear Systems.- 4 Optimal Control: Existence Theory.- Optimal Control: Necessary Conditions of Optimality.- 6 Stochastic Systems Controlled by Vector Measures.- 7 Applications to Physical Examples.- Bibliography.- Index.

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Book SynopsisThe volume covers most of the topics addressed and discussed during the Workshop INdAM "Recent advances in kinetic equations and applications", which took place in Rome (Italy), from November 11th to November 15th, 2019. The volume contains results on kinetic equations for reactive and nonreactive mixtures and on collisional and noncollisional Vlasov equations for plasmas. Some contributions are devoted to the study of phase transition phenomena, kinetic problems with nontrivial boundary conditions and hierarchies of models. The book, addressed to researchers interested in the mathematical and numerical study of kinetic equations, provides an overview of recent advances in the field and future research directions.Table of Contents- Sharpening of Decay Rates in Fourier Based Hypocoercivity Methods. - Quantum Drift-Diffusion Equations for a Two-Dimensional Electron Gas with Spin-Orbit Interaction. - A Kinetic BGK Relaxation Model for a Reacting Mixture of Polyatomic Gases. - On Some Recent Progress in the Vlasov–Poisson–Boltzmann System with Diffuse Reflection Boundary. - The Vlasov Equation with Infinite Mass. - Mathematical and Numerical Study of a Dusty Knudsen Gas Mixture: Extension to Non-spherical Dust Particles. - Body-Attitude Alignment: First Order Phase Transition, Link with Rodlike Polymers Through Quaternions, and Stability. - The Half-Space Problem for the Boltzmann Equation with Phase Transition at the Boundary. - Recent Developments on Quasineutral Limits for Vlasov-Type Equations. - A Note on Acoustic Limit for the Boltzmann Equation. - Thermal Boundaries in Kinetic and Hydrodynamic Limits. - Control of Collective Dynamics with Time-Varying Weights. - Kinetic Modelling of Autoimmune Diseases. - A Generalized Slip-Flow Theory for a Slightly Rarefied Gas Flow Induced by Discontinuous Wall Temperature. - A Revisit to the Cercignani–Lampis Model: Langevin Picture and Its Numerical Simulation. - On the Accuracy of Gyrokinetic Equations in Fusion Applications.

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Book SynopsisThe second edition of this successful textbook includes a significantly extended chapter on Climate Change with an analysis of the CO2 budget. It also contains a completely new part on Epidemiology, treating the SEIR-model which describes the behavior and dynamics of epidemics. In particular, COVID-19 with actual data is discussed. This compact introduction to ordinary differential equations and their applications is aimed at anyone who in their studies is confronted voluntarily or involuntarily with this versatile subject. Numerous applications from physics, technology, biomathematics, cosmology, economy and optimization theory are given. Abstract proofs and unnecessary formalism are avoided as far as possible. The focus is on modelling ordinary differential equations of the first and second orders as well as their analytical and numerical solution methods, in which the theory is dealt with briefly before moving on to application examples. In addition, program codes show exemplarily how even more challenging questions can be tackled and represented meaningfully with the help of a computer algebra system. The first chapter deals with the necessary prior knowledge of integral and differential calculus. 103 motivating exercises together with their solutions round off the work. “I am happy to see such a book. It will serve as a support for many students, professors and faculty.” Dr. Alessio Figalli, Professor at the ETH Zürich and Fields medalist 2018Table of ContentsPrerequisites from Calculus.- First Order Differential Equations.- First Order Applications.- Second Order Differential Equations and Systems with Applications.- Numerical MethodsWith Applications.- Climate Change, Epidemiolgy, Signal Processing.

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Book SynopsisThis book provides new contributions to the theory of inequalities for integral and sum, and includes four chapters. In the first chapter, linear inequalities via interpolation polynomials and green functions are discussed. New results related to Popoviciu type linear inequalities via extension of the Montgomery identity, the Taylor formula, Abel-Gontscharoff's interpolation polynomials, Hermite interpolation polynomials and the Fink identity with Green’s functions, are presented. The second chapter is dedicated to Ostrowski’s inequality and results with applications to numerical integration and probability theory. The third chapter deals with results involving functions with nondecreasing increments. Real life applications are discussed, as well as and connection of functions with nondecreasing increments together with many important concepts including arithmetic integral mean, wright convex functions, convex functions, nabla-convex functions, Jensen m-convex functions, m-convex functions, m-nabla-convex functions, k-monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace transform and exponentially convex functions, by using the finite difference operator of order m. The fourth chapter is mainly based on Popoviciu and Cebysev-Popoviciu type identities and inequalities. In this last chapter, the authors present results by using delta and nabla operators of higher order.Trade Review“This is an interesting book on the theory of inequalities for integrals and sums, which researchers in this theory should have in their library.” (Gradimir Milovanović, Mathematical Reviews, December, 2023)Table of Contents1 Linear Inequalities via Interpolation Polynomials and Green Functions.- 2 Ostrowski Inequality.- 3 Functions with Nondecreasing Increments.- 4 Popoviciu and Cebysev-Popoviciu Type Identities and Inequalities.

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Book SynopsisThis monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems – as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis – will find this text to be a valuable addition to the mathematical literature.Table of ContentsIntroduction.- Geometric Measure Theory.- Calderon-Zygmund Theory for Boundary Layers in UR Domains.- Boundedness and Invertibility of Layer Potential Operators.- Controlling the BMO Semi-Norm of the Unit Normal.- Boundary Value Problems in Muckenhoupt Weighted Spaces.- Singular Integrals and Boundary Problems in Morrey and Block Spaces.- Singular Integrals and Boundary Problems in Weighted Banach Function Spaces.

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Book SynopsisInput-to-State Stability presents the dominating stability paradigm in nonlinear control theory that revolutionized our view on stabilization of nonlinear systems, design of robust nonlinear observers, and stability of nonlinear interconnected control systems. The applications of input-to-state stability (ISS) are manifold and include mechatronics, aerospace engineering, and systems biology. Although the book concentrates on the ISS theory of finite-dimensional systems, it emphasizes the importance of a more general view of infinite-dimensional ISS theory. This permits the analysis of more general system classes and provides new perspectives on and a better understanding of the classical ISS theory for ordinary differential equations (ODEs). Features of the book include: • a comprehensive overview of the theoretical basis of ISS; • a description of the central applications of ISS in nonlinear control theory; • a detailed discussion of the role of small-gain methods in the stability of nonlinear networks; and • an in-depth comparison of ISS for finite- and infinite-dimensional systems. The book also provides a short overview of the ISS theory for other systems classes (partial differential equations, hybrid, impulsive, and time-delay systems) and surveys the available results for the important stability properties that are related to ISS. The reader should have a basic knowledge of analysis, Lebesgue integration theory, linear algebra, and the theory of ODEs but requires no prior knowledge of dynamical systems or stability theory. The author introduces all the necessary ideas within the book. Input-to-State Stability will interest researchers and graduate students studying nonlinear control from either a mathematical or engineering background. It is intended for active readers and contains numerous exercises of varying difficulty, which are integral to the text, complementing and widening the material developed in the monograph.Table of Contents

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Book SynopsisThis textbook provides an introduction to methods for solving nonlinear partial differential equations (NLPDEs). After the introduction of several PDEs drawn from science and engineering, readers are introduced to techniques to obtain exact solutions of NLPDEs. The chapters include the following topics: Nonlinear PDEs are Everywhere; Differential Substitutions; Point and Contact Transformations; First Integrals; and Functional Separability. Readers are guided through these chapters and are provided with several detailed examples. Each chapter ends with a series of exercises illustrating the material presented in each chapter. This Second Edition includes a new method of generating contact transformations and focuses on a solution method (parametric Legendre transformations) to solve a particular class of two nonlinear PDEs.Table of ContentsNonlinear PDEs are Everywhere.- Differential Substitutions.- Point and Contact Transformations.- First Integrals.- Functional Separability.

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Book SynopsisThe textbook presents a rather unique combination of topics in ODEs, examples and presentation style. Presentation emphasizes the development of practical solution skills by including a very large number of in-text examples and end-of-section exercises.

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Book SynopsisThis monograph addresses two significant related questions in complex geometry: the construction of a Chern character on the Grothendieck group of coherent sheaves of a compact complex manifold with values in its Bott-Chern cohomology, and the proof of a corresponding Riemann-Roch-Grothendieck theorem. One main tool used is the equivalence of categories established by Block between the derived category of bounded complexes with coherent cohomology and the homotopy category of antiholomorphic superconnections. Chern-Weil theoretic techniques are then used to construct forms that represent the Chern character. The main theorem is then established using methods of analysis, by combining local index theory with the hypoelliptic Laplacian.Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck is an important contribution to both the geometric and analytic study of complex manifolds and, as such, it will be a valuable resource for many researchers in geometry, analysis, and mathematical physics. Table of ContentsIntroduction.- Bott-Chern Cohomology and Characteristic Classes.- The Derived Category ${\mathrm{D^{b}_{\mathrm{coh}}}}$.- Preliminaries on Linear Algebra and Differential Geometry.- The Antiholomorphic Superconnections of Block.- An Equivalence of Categories.- Antiholomorphic Superconnections and Generalized Metrics.- Generalized Metrics and Chern Character Forms.- The Case of Embeddings.- Submersions and Elliptic Superconnections.- Elliptic Superconnection Forms and Direct Images.- A Proof of Theorem 10-1 when $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$..- The Hypoelliptic Superconnections.- The Hypoelliptic Superconnection Forms.- The Hypoelliptic Superconnection Forms when $\overline{\partial}^{X}\partial^{X}\omega^{X}=0$.- Exotic Superconnections and Riemann-Roch-Grothendieck.- Subject Index.- Index of Notation.- Bibliography.

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Book SynopsisThis book provides a comprehensive analysis of the existence of weak solutions of unsteady problems with variable exponents. The central motivation is the weak solvability of the unsteady p(.,.)-Navier–Stokes equations describing the motion of an incompressible electro-rheological fluid. Due to the variable dependence of the power-law index p(.,.) in this system, the classical weak existence analysis based on the pseudo-monotone operator theory in the framework of Bochner–Lebesgue spaces is not applicable. As a substitute for Bochner–Lebesgue spaces, variable Bochner–Lebesgue spaces are introduced and analyzed. In the mathematical framework of this substitute, the theory of pseudo-monotone operators is extended to unsteady problems with variable exponents, leading to the weak solvability of the unsteady p(.,.)-Navier–Stokes equations under general assumptions.Aimed primarily at graduate readers, the book develops the material step-by-step, starting with the basics of PDE theory and non-linear functional analysis. The concise introductions at the beginning of each chapter, together with illustrative examples, graphics, detailed derivations of all results and a short summary of the functional analytic prerequisites, will ease newcomers into the subject.Table of Contents- 1. Introduction. - 2. Preliminaries. - Part I Main Part. - 3. Variable Bochner–Lebesgue Spaces. - 4. Solenoidal Variable Bochner–Lebesgue Spaces. - 5. Existence Theory for Lipschitz Domains. - Part II Extensions. - 6. Pressure Reconstruction. - 7. Existence Theory for Irregular Domains. - 8. Existence Theory for p- < 2. - 9. Appendix.

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Book SynopsisThis graduate text presents the elegant and profound theory of continuous parameter Markov processes and many of its applications. The authors focus on developing context and intuition before formalizing the theory of each topic, illustrated with examples.After a review of some background material, the reader is introduced to semigroup theory, including the Hille–Yosida Theorem, used to construct continuous parameter Markov processes. Illustrated with examples, it is a cornerstone of Feller’s seminal theory of the most general one-dimensional diffusions studied in a later chapter. This is followed by two chapters with probabilistic constructions of jump Markov processes, and processes with independent increments, or Lévy processes. The greater part of the book is devoted to Itô’s fascinating theory of stochastic differential equations, and to the study of asymptotic properties of diffusions in all dimensions, such as explosion, transience, recurrence, existence of steady states, and the speed of convergence to equilibrium. A broadly applicable functional central limit theorem for ergodic Markov processes is presented with important examples. Intimate connections between diffusions and linear second order elliptic and parabolic partial differential equations are laid out in two chapters, and are used for computational purposes. Among Special Topics chapters, two study anomalous diffusions: one on skew Brownian motion, and the other on an intriguing multi-phase homogenization of solute transport in porous media.Table of Contents1. A review of Martingaels, stopping times and the Markov property.- 2. Semigroup theory and Markov processes.-3. Regularity of Markov process sample paths.- 4. Continuous parameter jump Markov processes.- 5. Processes with independent increments.- 6. The stochastic integral.- 7. Construction of difficusions as solutions of stochastic differential equations.- 8. Itô's Lemma.- 9. Cameron-Martin-Girsanov theorem.- 10. Support of nonsingular diffusions.- 11. Transience and recurrence of multidimensional diffusions.- 12. Criteria for explosion.- 13. Absorption, reflection and other transformations of Markov processes.- 14. The speed of convergence to equilibrium of discrete parameter Markov processes and Diffusions.- 15. Probabilistic representation of solutions to certain PDEs.- 16. Probabilistic solution of the classical Dirichlet problem.- 17. The functional Central Limit Theorem for ergodic Markov processes.- 18. Asymptotic stability for singular diffusions.- 19. Stochastic integrals with L2-Martingales.- 20. Local time for Brownian motion.- 21. Construction of one dimensional diffusions by Semigroups.- 22. Eigenfunction expansions of transition probabilities for one-dimensional diffusions.- 23. Special Topic: The Martingale Problem.- 24. Special topic: multiphase homogenization for transport in periodic media.- 25. Special topic: skew random walk and skew Brownian motion.- 26. Special topic: piecewise deterministic Markov processes in population biology.- A. The Hille-Yosida theorem and closed graph theorem.- References.- Related textbooks and monographs.

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Book SynopsisThis volume collects papers based on plenary and invited talks given at the 50th Barrett Memorial Lectures on Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models that was organized by the University of Tennessee, Knoxville and held virtually in May 2021. The three-day meeting brought together experts from the computational, scientific, engineering, and mathematical communities who work with nonlocal models. These proceedings collect contributions and give a survey of the state of the art in computational practices, mathematical analysis, applications of nonlocal models, and explorations of new application domains. The volume benefits from the mixture of contributions by computational scientists, mathematicians, and application specialists. The content is suitable for graduate students as well as specialists working with nonlocal models and covers topics on fractional PDEs, regularity theory for kinetic equations, approximation theory for fractional diffusion, analysis of nonlocal diffusion model as a bridge between local and fractional PDEs, and more.Table of ContentsCTRW approximations for fractional equations with variable order (Kolokoltsov).- Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation (Borthagaray).- Regularity estimates and open problems in kinetic equations (Silvestre).- An optimization-based strategy for peridynamic-FEM coupling and for the prescription of nonlocal boundary conditions (Littlewood).- Nonlocal diffusion models with consistent local and fractional limits (Du).- A one-dimensional symmetric force-based blending method for atomistic-to-continuum coupling (Li).- A note on estimates of level sets and their role in demonstrating regularity of solutions to nonlocal double phase equations (Mengesha).- An overview of Almost Minimizers of Bernoulli-Type Functionals (Garcia).

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Book SynopsisThis book covers problems involving a variety of fractional differential equations, as well as some involving the generalized Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives. The authors highlight the existence, uniqueness, and stability results for various classes of fractional differential equations based on the most recent research in the area. The book discusses the classic and novel fixed point theorems related to the measure of noncompactness in Banach spaces and explains how to utilize them as tools. The authors build each chapter upon the previous one, helping readers to develop their understanding of the topic. The book includes illustrated results, analysis, and suggestions for further study.Table of ContentsIntroduction.- Preliminary Background.- Hybrid Fractional Differential Equations.- Fractional Differential Equations with Retardation and Anticipation.- Impulsive Fractional Differential Equations with Retardation and Anticipation.- Coupled Systems for Fractional Differential Equations.

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Book SynopsisIntroduction.- The Equations of Maxwell.- Laplace's Equation.- Fourier Series, Bessel Functions, and Mathematical Physics.- The Fourier Transform, Heat Conduction, and the Wave Equation.- The ThreeDimensional Wave Equation.- An Introduction to Nonlinear Partial Differential Equations.- Raman Scattering and Numerical Methods.- ReactionDiffusion Equations.- The HartmanGrobman Theorem.

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Book SynopsisThis book, originating from a seminar held at Oberwolfach in 2022, introduces to state-of-the-art methods and results in the study of free boundary problems which are arising from compressible as well as from incompressible Euler's equations in general.

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Book Synopsis- 1. Preliminaries.- 2. Eigenvalue Problem and Integro-Differential Inequalities.- 3. Best Possible Estimates of Solutions to the Interface Problem for Linear Elliptic DivergenceSecond Order Equations in a Conical Domain.- 4. Interface Problem for the Laplace Operator with N Different Media.- 5. Interface Problem for Weak Quasi-Linear Elliptic Equations in a Conical Domain.- 6. Interface Problem for Strong Quasi-Linear Elliptic Equations in a Conical Domain.- 7. Best Possible Estimates of Solutions to the Interface Problem for a Quasi-Linear Elliptic Divergence Second Order Equation in a Domain with a Boundary Edge.- 8. Interface Oblique Derivative Problem for Perturbed p(x)-Laplacian Equation in a Bounded n- Dimensional Cone.- 9. Existence of Bounded Weak Solutions.

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Book SynopsisPreface.- Review of Mechanics of Materials.- Cartesian Tensors.- Kinematics.- Nonlinear Theory for Large Deformation.- Linear Theory for Small Deformation.- Saint-Venant’s Problem.- Some Simple Problems.- Antiplane Problems.- Plane Strain and Plane Stress.- Waves and Vibrations.- Appendices.

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Book SynopsisWhat is an ordinary differential equation?.- First-order differential equations.- Existence and uniqueness theorems.- Linear equations of higher order.- Systems of differential equations.- The qualitative theory and the phase plane.- Solution of differential equations by power series.- The Laplace transform.- Appendix: The orbits of the planets.- Appendix: Historical notes.

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Book Synopsis- The Fractional-Order Selkov-Schnakenberg Reaction-Diffusion model: Stability and Numerical simulations (Iqbal H. Jebril, Issam Bendib, Adel Ouannas, Salah Boulaaras, Iqbal M. Batiha and Shaher Momani).- Finite-Time Stability Analysis of Reaction-Diffusion Systems with Fractional-Order Dynamics: A Study Using the Selkov-Schnakenberg Model (Issam Bendib, Adel Ouannas, Shaher Momani and Chaouki Aouiti).- Global Stability Analysis of Fractional Selkov-Schnakenberg Reaction-Diffusion Systems (Iqbal H. Jebril, Issam Bendib, Adel Ouannas, Salah Boulaaras, Iqbal M. Batiha and Shaher Momani).- Dynamics in Finite-Time of the Fractional-Order FitzHugh-Nagumo model: stability, synchronization, and simulations (Issam Bendib, Adel Ouannas, Mohammed Al Horani and Mohamed Dalah).- On Fractional Variable-Order Neural Networks under Atangana-Baleanu-Caputo Derivative (Ma’mon Abu Hammad, Amel Hioual, Adel Ouannas, Shaher Momani and Zohir Dibi).- Blow up solutions for a variant of the Cahn-Hilliard equation describing growth of cancerous cells (Hussein Fakih, Salam Abou Baraki, Ragheb Mghames and Yahia Awad).- A New Fractional Discrete Memristive Map with Incommensurate Order and Hidden Dynamics (Imane Zouak, Adel Ouannas and Amina-Aicha Khennaoui).- Qualitative Analysis and Hopf bifurcation for a fractional order ratio-dependent prey-predator model (Canan Celik and Kübra Degerli).- Hidden Chaos in new Fractional Sigmoidal-Based Quadratic Memristive Map (Louiza Diabi and Adel Ouannas).- Stability Investigation of Nonlinear Fractional Difference Equations with Incommensurate Orders (Noureddine Djenina, Adel Ouannas, Shaher Momani and Giuseppe Grassi).- Resolvent operator approach for solving some fractional abstract Volterra-Fredholm integro-differential equations with deviated argument (Fatiha Boutaous).- Sequential Bayesian A-Optimal Sampling Locations for Fractional Partial Differential Equations (Ryad Ghanam and Edward L. Boone).- On the Solutions of Two-Point Nonlinear Fractional Differential Equations with Multiple Fractional Boundary Conditions (Yahia Awad, Hussein Fakih, Karim Amin and Ragheb Mghames).- Some new Chebyshev and Grüss-type fractional inequalities obtained by a generalized fractional integral operator (Mustafa Gürbüz and Çagri Asak).- Regularity of Solutions for a Class of Neutral Fractional Stochastic Differential Equations (Jihen Sallay).- The Lindley q-Distribution: Development, Properties, and Statistical Applications (Bouzida Imed and Zitouni Mouna).

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Book SynopsisChapter 1. Introduction.- Chapter 2. Quasi-interpolation.- Chapter 3. Approximation of integral operators.- Chapter 4. Some other cubature problems.- Chapter 5. Approximate solution of non-stationary problems.- Chapter 6. Integral operators over hyper-rectangular domains.

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Book SynopsisIntroduction.- Exact and approximate solutions.- Solidification of a thin liquid layer.- Variable property Stefan problem.- Variable interface conditions.- Non-Fourier Stefan problems.- Hints to Exercises.- Index.

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Book SynopsisThe first part of the book provides an introduction to key tools and techniques in dispersive equations: Strichartz estimates, bilinear estimates, modulation and adapted function spaces, with an application to the generalized Korteweg-de Vries equation and the Kadomtsev-Petviashvili equation. The energy-critical nonlinear Schrödinger equation, global solutions to the defocusing problem, and scattering are the focus of the second part. Using this concrete example, it walks the reader through the induction on energy technique, which has become the essential methodology for tackling large data critical problems. This includes refined/inverse Strichartz estimates, the existence and almost periodicity of minimal blow up solutions, and the development of long-time Strichartz inequalities. The third part describes wave and Schrödinger maps. Starting by building heuristics about multilinear estimates, it provides a detailed outline of this very active area of geometric/dispersive PDE. It focuses on concepts and ideas and should provide graduate students with a stepping stone to this exciting direction of research.​Table of ContentsLocal existence of solutions to the initial value problem for dispersive equations.- The energy critical nonlinear Schrödinger equation.- Wave maps and Schrödinger maps.

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Book SynopsisModeling, in particular with partial differential equations, plays an ever growing role in the applied sciences. Hence its mathematical understanding is an important issue for today's research. This book provides an introduction to three different topics in partial differential equations arising from applications. The subject of the first course by Michel Chipot (Zurich) is equilibrium positions of several disks rolling on a wire. In particular, existence and uniqueness of and the exact position for an equilibrium are discussed. The second course by Josselin Garnier (Toulouse) deals with problems arising from acoustics and geophysics where waves propagate in complicated media, the properties of which can only be described statistically. It turns out that if the different scales presented in the problem can be separated, there exists a deterministic result. The third course by Otared Kavian (Versailles St.-Quentin) is devoted to so-called inverse problems where one or several parameters of a partial differential equation need to be determined by using, for instance, measurements on the boundary of the domain. The question that arises naturally is what information is necessary to determine the unknown parameters. This question is answered in different settings. The text is addressed to students and researchers with a basic background in partial differential equations.

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Book SynopsisThis textbook provides a self-contained and elementary introduction to the modern theory of pseudodifferential operators and their applications to partial differential equations. In the first chapters, the necessary material on Fourier transformation and distribution theory is presented. Subsequently the basic calculus of pseudodifferential operators on the n-dimensional Euclidean space is developed. In order to present the deep results on regularity questions for partial differential equations, an introduction to the theory of singular integral operators is given - which is of interest for its own. Moreover, to get a wide range of applications, one chapter is devoted to the modern theory of Besov and Bessel potential spaces. In order to demonstrate some fundamental approaches and the power of the theory, several applications to wellposedness and regularity question for elliptic and parabolic equations are presented throughout the book. The basic notation of functional analysis needed in the book is introduced and summarized in the appendix. The text is comprehensible for students of mathematics and physics with a basic education in analysis.

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