{"product_id":"introduction-to-computation-and-modeling-for-differential-equations-9781119018445","title":"Introduction to Computation and Modeling for","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e\u003cb\u003eUses mathematical, numerical, and programming tools to solve differential equations for physical phenomena and engineering problems\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eIntroduction to Computation and Modeling for Differential Equations, Second Edition \u003c\/i\u003efeatures the essential principles and applications of problem solving across disciplines such as engineering, physics, and chemistry. The \u003ci\u003eSecond Edition \u003c\/i\u003eintegrates the science of solving differential equations with mathematical, numerical, and programming tools, specifically with methods involving ordinary differential equations; numerical methods for initial value problems (IVPs); numerical methods for boundary value problems (BVPs); partial differential equations (PDEs); numerical methods for parabolic, elliptic, and hyperbolic PDEs; mathematical modeling with differential equations; numerical solutions; and finite difference and finite element methods.\u003c\/p\u003e \u003cp\u003eThe author features a unique Five-M approach: Modeling, Mathematics, Methods, MATL\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003c\/p\u003e\u003cp\u003ePreface xi\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Introduction 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 What is a Differential Equation? 1\u003c\/p\u003e \u003cp\u003e1.2 Examples of an Ordinary and a Partial Differential Equation, 2\u003c\/p\u003e \u003cp\u003e1.3 Numerical Analysis, a Necessity for Scientific Computing, 5\u003c\/p\u003e \u003cp\u003e1.4 Outline of the Contents of this Book, 8\u003c\/p\u003e \u003cp\u003eBibliography, 10\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Ordinary Differential Equations 11\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Problem Classification, 11\u003c\/p\u003e \u003cp\u003e2.2 Linear Systems of ODEs with Constant Coefficients, 16\u003c\/p\u003e \u003cp\u003e2.3 Some Stability Concepts for ODEs, 19\u003c\/p\u003e \u003cp\u003e2.3.1 Stability for a Solution Trajectory of an ODE System, 20\u003c\/p\u003e \u003cp\u003e2.3.2 Stability for Critical Points of ODE Systems, 23\u003c\/p\u003e \u003cp\u003e2.4 Some ODE models in Science and Engineering, 26\u003c\/p\u003e \u003cp\u003e2.4.1 Newton’s Second Law, 26\u003c\/p\u003e \u003cp\u003e2.4.2 Hamilton’s Equations, 27\u003c\/p\u003e \u003cp\u003e2.4.3 Electrical Networks, 27\u003c\/p\u003e \u003cp\u003e2.4.4 Chemical Kinetics, 28\u003c\/p\u003e \u003cp\u003e2.4.5 Control Theory, 29\u003c\/p\u003e \u003cp\u003e2.4.6 Compartment Models, 29\u003c\/p\u003e \u003cp\u003e2.5 Some Examples from Applications, 30\u003c\/p\u003e \u003cp\u003eBibliography, 36\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Numerical Methods for Initial Value Problems 37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 Graphical Representation of Solutions, 38\u003c\/p\u003e \u003cp\u003e3.2 Basic Principles of Numerical Approximation of ODEs, 40\u003c\/p\u003e \u003cp\u003e3.3 Numerical Solution of IVPs with Euler’s method, 41\u003c\/p\u003e \u003cp\u003e3.3.1 Euler’s Explicit Method: Accuracy, 43\u003c\/p\u003e \u003cp\u003e3.3.2 Euler’s Explicit Method: Improving the Accuracy, 46\u003c\/p\u003e \u003cp\u003e3.3.3 Euler’s Explicit Method: Stability, 48\u003c\/p\u003e \u003cp\u003e3.3.4 Euler’s Implicit Method, 53\u003c\/p\u003e \u003cp\u003e3.3.5 The Trapezoidal Method, 55\u003c\/p\u003e \u003cp\u003e3.4 Higher Order Methods for the IVP, 56\u003c\/p\u003e \u003cp\u003e3.4.1 Runge–Kutta Methods, 56\u003c\/p\u003e \u003cp\u003e3.4.2 Linear Multistep Methods, 60\u003c\/p\u003e \u003cp\u003e3.5 Special Methods for Special Problems, 62\u003c\/p\u003e \u003cp\u003e3.5.1 Preserving Linear and Quadratic Invariants, 62\u003c\/p\u003e \u003cp\u003e3.5.2 Preserving Positivity of the Numerical Solution, 64\u003c\/p\u003e \u003cp\u003e3.5.3 Methods for Newton’s Equations of Motion, 64\u003c\/p\u003e \u003cp\u003e3.6 The Variational Equation and Parameter Fitting in IVPs, 66\u003c\/p\u003e \u003cp\u003eBibliography, 69\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Numerical Methods for Boundary Value Problems 71\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Applications, 73\u003c\/p\u003e \u003cp\u003e4.2 Difference Methods for BVPs, 78\u003c\/p\u003e \u003cp\u003e4.2.1 A Model Problem for BVPs, Dirichlet’s BCs, 79\u003c\/p\u003e \u003cp\u003e4.2.2 A Model Problem for BVPs, Mixed BCs, 83\u003c\/p\u003e \u003cp\u003e4.2.3 Accuracy, 86\u003c\/p\u003e \u003cp\u003e4.2.4 Spurious Solutions, 87\u003c\/p\u003e \u003cp\u003e4.2.5 Linear Two-Point BVPs, 89\u003c\/p\u003e \u003cp\u003e4.2.6 Nonlinear Two-Point BVPs, 91\u003c\/p\u003e \u003cp\u003e4.2.7 The Shooting Method, 92\u003c\/p\u003e \u003cp\u003e4.3 Ansatz Methods for BVPs, 94\u003c\/p\u003e \u003cp\u003e4.3.1 Starting with the ODE Formulation, 95\u003c\/p\u003e \u003cp\u003e4.3.2 Starting with the Weak Formulation, 96\u003c\/p\u003e \u003cp\u003e4.3.3 The Finite Element Method, 100\u003c\/p\u003e \u003cp\u003eBibliography, 103\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Partial Differential Equations 105\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Classical PDE Problems, 106\u003c\/p\u003e \u003cp\u003e5.2 Differential Operators Used for PDEs, 110\u003c\/p\u003e \u003cp\u003e5.3 Some PDEs in Science and Engineering, 114\u003c\/p\u003e \u003cp\u003e5.3.1 Navier–Stokes Equations for Incompressible Flow, 114\u003c\/p\u003e \u003cp\u003e5.3.2 Euler’s Equations for Compressible Flow, 115\u003c\/p\u003e \u003cp\u003e5.3.3 The Convection–Diffusion–Reaction Equations, 116\u003c\/p\u003e \u003cp\u003e5.3.4 The Heat Equation, 117\u003c\/p\u003e \u003cp\u003e5.3.5 The Diffusion Equation, 117\u003c\/p\u003e \u003cp\u003e5.3.6 Maxwell’s Equations for the Electromagnetic Field, 117\u003c\/p\u003e \u003cp\u003e5.3.7 Acoustic Waves, 118\u003c\/p\u003e \u003cp\u003e5.3.8 Schrödinger’s Equation in Quantum Mechanics, 119\u003c\/p\u003e \u003cp\u003e5.3.9 Navier’s Equations in Structural Mechanics, 119\u003c\/p\u003e \u003cp\u003e5.3.10 Black–Scholes Equation in Financial Mathematics, 120\u003c\/p\u003e \u003cp\u003e5.4 Initial and Boundary Conditions for PDEs, 121\u003c\/p\u003e \u003cp\u003e5.5 Numerical Solution of PDEs, Some General Comments, 121\u003c\/p\u003e \u003cp\u003eBibliography, 122\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Numerical Methods for Parabolic Partial Differential Equations 123\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Applications, 125\u003c\/p\u003e \u003cp\u003e6.2 An Introductory Example of Discretization, 127\u003c\/p\u003e \u003cp\u003e6.3 The Method of Lines for Parabolic PDEs, 130\u003c\/p\u003e \u003cp\u003e6.3.1 Solving the Test Problem with MoL, 130\u003c\/p\u003e \u003cp\u003e6.3.2 Various Types of Boundary Conditions, 134\u003c\/p\u003e \u003cp\u003e6.3.3 An Example of the Use of MoL for a Mixed Boundary Condition, 135\u003c\/p\u003e \u003cp\u003e6.4 Generalizations of the Heat Equation, 136\u003c\/p\u003e \u003cp\u003e6.4.1 The Heat Equation with Variable Conductivity, 136\u003c\/p\u003e \u003cp\u003e6.4.2 The Convection – Diffusion – Reaction PDE, 138\u003c\/p\u003e \u003cp\u003e6.4.3 The General Nonlinear Parabolic PDE, 138\u003c\/p\u003e \u003cp\u003e6.5 Ansatz Methods for the Model Equation, 139\u003c\/p\u003e \u003cp\u003eBibliography, 140\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Numerical Methods for Elliptic Partial Differential Equations 143\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Applications, 145\u003c\/p\u003e \u003cp\u003e7.2 The Finite Difference Method, 150\u003c\/p\u003e \u003cp\u003e7.3 Discretization of a Problem with Different BCs, 154\u003c\/p\u003e \u003cp\u003e7.4 Ansatz Methods for Elliptic PDEs, 156\u003c\/p\u003e \u003cp\u003e7.4.1 Starting with the PDE Formulation, 156\u003c\/p\u003e \u003cp\u003e7.4.2 Starting with the Weak Formulation, 158\u003c\/p\u003e \u003cp\u003e7.4.3 The Finite Element Method, 159\u003c\/p\u003e \u003cp\u003eBibliography, 164\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Numerical Methods for Hyperbolic PDEs 165\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Applications, 171\u003c\/p\u003e \u003cp\u003e8.2 Numerical Solution of Hyperbolic PDEs, 174\u003c\/p\u003e \u003cp\u003e8.2.1 The Upwind Method (FTBS), 175\u003c\/p\u003e \u003cp\u003e8.2.2 The FTFS Method, 177\u003c\/p\u003e \u003cp\u003e8.2.3 The FTCS Method, 178\u003c\/p\u003e \u003cp\u003e8.2.4 The Lax–Friedrichs Method, 178\u003c\/p\u003e \u003cp\u003e8.2.5 The Leap-Frog Method, 179\u003c\/p\u003e \u003cp\u003e8.2.6 The Lax–Wendroff Method, 179\u003c\/p\u003e \u003cp\u003e8.2.7 Numerical Method for the Wave Equation, 181\u003c\/p\u003e \u003cp\u003e8.3 The Finite Volume Method, 183\u003c\/p\u003e \u003cp\u003e8.4 Some Examples of Stability Analysis for Hyperbolic PDEs, 185\u003c\/p\u003e \u003cp\u003eBibliography, 187\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Mathematical Modeling with Differential Equations 189\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Nature Laws, 190\u003c\/p\u003e \u003cp\u003e9.2 Constitutive Equations, 192\u003c\/p\u003e \u003cp\u003e9.2.1 Equations in Heat Transfer Problems, 192\u003c\/p\u003e \u003cp\u003e9.2.2 Equations in Mass Diffusion Problems, 193\u003c\/p\u003e \u003cp\u003e9.2.3 Equations in Mechanical Moment Diffusion Problems, 193\u003c\/p\u003e \u003cp\u003e9.2.4 Equations in Elastic Solid Mechanics Problems, 194\u003c\/p\u003e \u003cp\u003e9.2.5 Equations in Chemical Reaction Engineering Problems, 194\u003c\/p\u003e \u003cp\u003e9.2.6 Equations in Electrical Engineering Problems, 195\u003c\/p\u003e \u003cp\u003e9.3 Conservative Equations, 195\u003c\/p\u003e \u003cp\u003e9.3.1 Some Examples of Lumped Models, 196\u003c\/p\u003e \u003cp\u003e9.3.2 Some Examples of Distributed Models, 197\u003c\/p\u003e \u003cp\u003e9.4 Scaling of Differential Equations to Dimensionless Form, 201\u003c\/p\u003e \u003cp\u003eBibliography, 204\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Applied Projects on Differential Equations 205\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eProject 1 Signal propagation in a long electrical conductor, 205\u003c\/p\u003e \u003cp\u003eProject 2 Flow in a cylindrical pipe, 206\u003c\/p\u003e \u003cp\u003eProject 3 Soliton waves, 208\u003c\/p\u003e \u003cp\u003eProject 4 Wave scattering in a waveguide, 209\u003c\/p\u003e \u003cp\u003eProject 5 Metal block with heat sourse and thermometer, 210\u003c\/p\u003e \u003cp\u003eProject 6 Deformation of a circular metal plate, 211\u003c\/p\u003e \u003cp\u003eProject 7 Cooling of a chrystal glass, 212\u003c\/p\u003e \u003cp\u003eProject 8 Rotating fluid in a cylinder, 212\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix A Some Numerical and Mathematical Tools 215\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1 Newton’s Method for Systems of Nonlinear Algebraic Equations, 215\u003c\/p\u003e \u003cp\u003eA.1.1 Quadratic Systems, 215\u003c\/p\u003e \u003cp\u003eA.1.2 Overdetermined Systems, 218\u003c\/p\u003e \u003cp\u003eA.2 Some Facts about Linear Difference Equations, 219\u003c\/p\u003e \u003cp\u003eA.3 Derivation of Difference Approximations, 223\u003c\/p\u003e \u003cp\u003eBibliography, 225\u003c\/p\u003e \u003cp\u003eA.4 The Interpretations of Grad, Div, and Curl, 225\u003c\/p\u003e \u003cp\u003eA.5 Numerical Solution of Algebraic Systems of Equations, 229\u003c\/p\u003e \u003cp\u003eA.5.1 Direct Methods, 229\u003c\/p\u003e \u003cp\u003eA.5.2 Iterative Methods for Linear Systems of Equations, 233\u003c\/p\u003e \u003cp\u003eA.6 Some Results for Fourier Transforms, 237\u003c\/p\u003e \u003cp\u003eBibliography, 239\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix B Software for Scientific Computing 241\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB.1 MATLAB, 242\u003c\/p\u003e \u003cp\u003eB.1.1 Chapter 3: IVPs, 242\u003c\/p\u003e \u003cp\u003eB.1.2 Chapter 4: BVPs, 244\u003c\/p\u003e \u003cp\u003eB.1.3 Chapter 6: Parabolic PDEs, 245\u003c\/p\u003e \u003cp\u003eB.1.4 Chapter 7: Elliptic PDEs, 246\u003c\/p\u003e \u003cp\u003eB.1.5 Chapter 8: Hyperbolic PDEs, 246\u003c\/p\u003e \u003cp\u003eB.2 COMSOL MULTIPHYSICS, 247\u003c\/p\u003e \u003cp\u003eBibliography and Resources, 249\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix C Computer Exercises to Support the Chapters 251\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eC.1 Computer Lab 1 Supporting Chapter 2, 251\u003c\/p\u003e \u003cp\u003eC.1.1 ODE Systems of LCC Type and Stability, 251\u003c\/p\u003e \u003cp\u003eC.2 Computer Lab 2 Supporting Chapter 3, 254\u003c\/p\u003e \u003cp\u003eC.2.1 Numerical Solution of Initial Value Problems, 254\u003c\/p\u003e \u003cp\u003eC.3 Computer Lab 3 Supporting Chapter 4, 257\u003c\/p\u003e \u003cp\u003eC.3.1 Numerical Solution of a Boundary Value Problem, 257\u003c\/p\u003e \u003cp\u003eC.4 Computer Lab 4 Supporting Chapter 6, 258\u003c\/p\u003e \u003cp\u003eC.4.1 Partial Differential Equation of Parabolic Type, 258\u003c\/p\u003e \u003cp\u003eC.5 Computer Lab 5 Supporting Chapter 7, 261\u003c\/p\u003e \u003cp\u003eC.5.1 Numerical Solution of Elliptic PDE Problems, 261\u003c\/p\u003e \u003cp\u003eC.6 Computer Lab 6 Supporting Chapter 8, 263\u003c\/p\u003e \u003cp\u003eC.6.1 Numerical Experiments with the Hyperbolic Model PDE\u003c\/p\u003e \u003cp\u003eProblem, 263\u003c\/p\u003e \u003cp\u003eIndex 265\u003c\/p\u003e","brand":"John Wiley \u0026 Sons 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