Introduction to Numerical Methods for Time Dependent Differential Equations.

Intro -- Half Title page -- Title page -- Copyright page -- Dedication -- Preface -- Acknowledgements -- Part I: Ordinary Differential Equations and Their Approximations -- Chapter 1: First-Order Scalar Equations -- 1.1 Constant coefficient linear equations -- 1.2 Variable coefficient linear equations -- 1.3 Perturbations and the concept of stability -- 1.4 Nonlinear equations: the possibility of blow-up -- 1.5 Principle of linearization -- Chapter 2: Method of Euler -- 2.1 Explicit Euler method -- 2.2 Stability of the explicit Euler method -- 2.3 Accuracy and truncation error -- 2.4 Discrete Duhamel's principle and global error -- 2.5 General one-step methods -- 2.6 How to test the correctness of a program -- 2.7 Extrapolation -- Chapter 3: Higher-Order Methods -- 3.1 Second-order Taylor method -- 3.2 Improved Euler's method -- 3.3 Accuracy of the solution computed -- 3.4 Runge-Kutta methods -- 3.5 Regions of stability -- 3.6 Accuracy and truncation error -- 3.7 Difference approximations for unstable problems -- Chapter 4: Implicit Euler Method -- 4.1 Stiff equations -- 4.2 Implicit Euler method -- 4.3 Simple variable-step-size strategy -- Chapter 5: Two-Step and Multistep Methods -- 5.1 Multistep methods -- 5.2 Leapfrog method -- 5.3 Adams methods -- 5.4 Stability of multistep methods -- Chapter 6: Systems of Differential Equations -- Part II: Partial Differential Equations and Their Approximations -- Chapter 7: Fourier Series and Interpolation -- 7.1 Fourier expansion -- 7.2 L2-norm and scalar product -- 7.3 Fourier interpolation -- Chapter 8: 1-Periodic Solutions of time Dependent Partial Differential Equations with Constant Coefficients -- 8.1 Examples of equations with simple wave solutions.

8.2 Discussion of well posed problems for time dependent partial differential equations with constant coefficients and with 1-periodic boundary conditions -- Chapter 9: Approximations of 1-Periodic Solutions of Partial Differential Equations -- 9.1 Approximations of space derivatives -- 9.2 Differentiation of Periodic Functions -- 9.3 Method of lines -- 9.4 Time Discretizations and Stability Analysis -- Chapter 10: Linear Initial Boundary Value Problems -- 10.1 Well-Posed Initial Boundary Value Problems -- 10.2 Method of lines -- Chapter 11: Nonlinear Problems -- 11.1 Initial value problems for ordinary differential equations -- 11.2 Existence theorems for nonlinear partial differential equations -- 11.3 Nonlinear example: Burgers' equation -- Appendix A: Auxiliary Material -- A.1 Some useful Taylor series -- A.2 "O" notation -- A.3 Solution expansion -- Appendix B: Solutions to Exercises -- References -- Index.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.