{"product_id":"introduction-to-numerical-methods-for-time-dependent-differential-equations-hardback-9781118838952","title":"Introduction to Numerical Methods for Time Dependent Differential Equations (Hardback) 9781118838952","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003eIntroduction to Numerical Methods for Time Dependent Differential Equations\u003c\/font\u003e\u003cbr\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003c\/p\u003e\n\u003cp\u003e\u003cfont size=\"4\"\u003eHeinz-Otto Kreiss (Author), Omar Eduardo Ortiz (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9781118838952, Wiley\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eHardback, published 13 May 2014\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e192 pages\u003cbr\u003e24.3 x 16 x 1.8 cm, 0.435 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003e\u003cp\u003e\u003cb\u003eIntroduces both the fundamentals of time dependent differential equations and their numerical solutions\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e\u003ci\u003eIntroduction to Numerical Methods for Time Dependent Differential Equations \u003c\/i\u003edelves into the underlying mathematical theory needed to solve time dependent differential equations numerically. Written as a self-contained introduction, the book is divided into two parts to emphasize both ordinary differential equations (ODEs) and partial differential equations (PDEs).\u003c\/p\u003e \u003cp\u003eBeginning with ODEs and their approximations, the authors provide a crucial presentation of fundamental notions, such as the theory of scalar equations, finite difference approximations, and the Explicit Euler method. Next, a discussion on higher order approximations, implicit methods, multistep methods, Fourier interpolation, PDEs in one space dimension as well as their related systems is provided.\u003c\/p\u003e \u003cp\u003e\u003ci\u003eIntroduction to Numerical Methods for Time Dependent Differential Equations \u003c\/i\u003efeatures:\u003c\/p\u003e \u003cul\u003e \u003cli\u003eA step-by-step discussion of the procedures needed to prove the stability of difference approximations\u003c\/li\u003e \u003cli\u003eMultiple exercises throughout with select answers, providing readers with a practical guide to understanding the approximations of differential equations\u003c\/li\u003e \u003cli\u003eA simplified approach in a one space dimension\u003c\/li\u003e \u003cli\u003eAnalytical theory for difference approximations that is particularly useful to clarify procedures\u003c\/li\u003e \u003c\/ul\u003e \u003cp\u003e\u003ci\u003eIntroduction to Numerical Methods for Time Dependent Differential Equations \u003c\/i\u003eis an excellent textbook for upper-undergraduate courses in applied mathematics, engineering, and physics as well as a useful reference for physical scientists, engineers, numerical analysts, and mathematical modelers who use numerical experiments to test designs or predict and investigate phenomena from many disciplines.\u003c\/p\u003e\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\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\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Mathematics [\u003ca title=\"See our other books on Mathematics\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Mathematics%20%5BPB%5D%22\"\u003ePB\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Wiley","offers":[{"title":"Brand 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