{"product_id":"introduction-to-numerical-methods-for-time-dependent-differential-equations-9781118838952","title":"Introduction to Numerical Methods for Time","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eIntroduces both the fundamentals of time dependent differential equations and their numerical solutions     Introduction to Numerical Methods for Time Dependent Differential Equations delves into the underlying mathematical theory needed to solve time dependent differential equations numerically.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface xiii\u003c\/p\u003e \u003cp\u003eAcknowledgments xv\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART I ORDINARY DIFFERENTIAL EQUATIONS AND THEIR APPROXIMATIONS\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 First Order Scalar Equations 3\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Constant coefficient linear equations 3\u003c\/p\u003e \u003cp\u003e1.1.1 Duhamel’s principle 8\u003c\/p\u003e \u003cp\u003e1.1.2 Principle of frozen coefficients 10\u003c\/p\u003e \u003cp\u003e1.2 Variable coefficient linear equations 10\u003c\/p\u003e \u003cp\u003e1.2.1 The principle of superposition 10\u003c\/p\u003e \u003cp\u003e1.2.2 Duhamel’s principle for variable coefficients 12\u003c\/p\u003e \u003cp\u003e1.3 Perturbations and the concept of stability 13\u003c\/p\u003e \u003cp\u003e1.4 Nonlinear equations: the possibility of blowup 17\u003c\/p\u003e \u003cp\u003e1.5 The principle of linearization 20\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 The Method of Euler 23\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 The explicit Euler method 23\u003c\/p\u003e \u003cp\u003e2.2 Stability of the explicit Euler method 25\u003c\/p\u003e \u003cp\u003e2.3 Accuracy and truncation error 27\u003c\/p\u003e \u003cp\u003e2.4 Discrete Duhamel’s principle and global error 28\u003c\/p\u003e \u003cp\u003e2.5 General onestep methods. 31\u003c\/p\u003e \u003cp\u003e2.6 How to test the correctness of a program 32\u003c\/p\u003e \u003cp\u003e2.7 Extrapolation 34\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Higher Order Methods 37\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 The secondorder Taylor method 37\u003c\/p\u003e \u003cp\u003e3.2 Improved Euler’s method 39\u003c\/p\u003e \u003cp\u003e3.3 Accuracy of the computed solution 40\u003c\/p\u003e \u003cp\u003e3.4 RungeKutta methods 44\u003c\/p\u003e \u003cp\u003e3.5 Regions of stability 48\u003c\/p\u003e \u003cp\u003e3.6 Accuracy and truncation error 51\u003c\/p\u003e \u003cp\u003e3.7 Difference approximations for unstable problems 52\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 The Implicit Euler Method 55\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Stiff equations 55\u003c\/p\u003e \u003cp\u003e4.2 The implicit Euler method 58\u003c\/p\u003e \u003cp\u003e4.3 A simple variable step size strategy 63\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Two Step and Multistep Methods 67\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Multistep methods 67\u003c\/p\u003e \u003cp\u003e5.2 The leapfrog method 68\u003c\/p\u003e \u003cp\u003e5.3 Adams methods 72\u003c\/p\u003e \u003cp\u003e5.4 Stability of multistep methods 74\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Systems of Differential Equations 77\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003ePART II PARTIAL DIFFERENTIAL EQUATIONS AND THEIR APPROXIMATIONS\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Fourier Series and Interpolation 83\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 Fourier expansion 83\u003c\/p\u003e \u003cp\u003e7.2 The L2norm and scalar product 89\u003c\/p\u003e \u003cp\u003e7.3 Fourier interpolation 92\u003c\/p\u003e \u003cp\u003e7.3.1 Scalar product and norm for 1periodic grid functions 93\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 1periodic Solutions of Time Dependent PDE... 95\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 Examples of equations with simple wave solutions 95\u003c\/p\u003e \u003cp\u003e8.1.1 The oneway wave equation 95\u003c\/p\u003e \u003cp\u003e8.1.2 The heat equation 96\u003c\/p\u003e \u003cp\u003e8.1.3 The wave equation 97\u003c\/p\u003e \u003cp\u003e8.2 Discussion of well posed problems for time dependent PDE... 98\u003c\/p\u003e \u003cp\u003e8.2.1 First order equations 98\u003c\/p\u003e \u003cp\u003e8.2.2 Second order (in space) equations 100\u003c\/p\u003e \u003cp\u003e8.2.3 General equation 101\u003c\/p\u003e \u003cp\u003e8.2.4 Stability against lower order terms and systems of equations 102\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 Approximations of 1periodic Solutions of PDE 105\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Approximations of space derivatives 105\u003c\/p\u003e \u003cp\u003e9.1.1 Smoothness of the Fourier interpolant 108\u003c\/p\u003e \u003cp\u003e9.2 Differentiation of Periodic Functions 109\u003c\/p\u003e \u003cp\u003e9.3 The method of lines 110\u003c\/p\u003e \u003cp\u003e9.3.1 The oneway wave equation 110\u003c\/p\u003e \u003cp\u003e9.3.2 The heat equation 113\u003c\/p\u003e \u003cp\u003e9.3.3 The wave equation 115\u003c\/p\u003e \u003cp\u003e9.4 Time Discretizations and Stability Analysis 116\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Linear InitialBoundary Value Problems 119\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Well Posed InitialBoundary Value Problems 119\u003c\/p\u003e \u003cp\u003e10.1.1 The heat equation on a strip 120\u003c\/p\u003e \u003cp\u003e10.1.2 The oneway wave equation on a strip 122\u003c\/p\u003e \u003cp\u003e10.1.3 The wave equation on a strip 124\u003c\/p\u003e \u003cp\u003e10.2 The method of lines 126\u003c\/p\u003e \u003cp\u003e10.2.1 The heat equation 126\u003c\/p\u003e \u003cp\u003e10.2.2 Finite differences algebra 130\u003c\/p\u003e \u003cp\u003e10.2.3 General parabolic problem 131\u003c\/p\u003e \u003cp\u003e10.2.4 The oneway wave equation 134\u003c\/p\u003e \u003cp\u003e10.2.5 The wave equation 135\u003c\/p\u003e \u003cp\u003e\u003cb\u003e11 Nonlinear Problems 137\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e11.1 Initialvalue problems for ODE 138\u003c\/p\u003e \u003cp\u003e11.2 Existence theorems for nonlinear PDE 141\u003c\/p\u003e \u003cp\u003e11.3 A nonlinear example: Burgers’ equation 145\u003c\/p\u003e \u003cp\u003e\u003cb\u003eA Auxiliary Material 149\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1 Some useful Taylor series 149\u003c\/p\u003e \u003cp\u003eA.2 The “O” notation 150\u003c\/p\u003e \u003cp\u003eA.3 The solution expansion 150\u003c\/p\u003e \u003cp\u003e\u003cb\u003eB Solutions to Exercises 153\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eReferences 171\u003c\/p\u003e \u003cp\u003eIndex 173\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default 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