Numerical solution of ordinary differential equations / / Kendall E. Atkinson, Weimin Han, David Stewart

• Record Nr.: UNINA9910141193403321
• Autore: Atkinson Kendall E.
• Pubbl/distr/stampa: Hoboken, New Jersey : , : Wiley, , 2009; ©2009
• ISBN: 1-283-33249-3; 9786613332493; 1-118-16449-0; 1-118-16452-0
• Descrizione fisica: 1 online resource (272 p.)
• Collana: Pure and Applied Mathematics: A Wiley-Interscience Series of Texts, Monographs, and Tracts
• Classificazione: SK 920
• Disciplina: 515.352
• Soggetti: Differential equations - Numerical solutions
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Description based upon print version of record.
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Numerical Solution of Ordinary Differential Equations; CONTENTS; Introduction; 1 Theory of differential equations: An introduction; 1.1 General solvability theory; 1.2 Stability of the initial value problem; 1.3 Direction fields; Problems; 2 Euler's method; 2.1 Definition of Euler's method; 2.2 Error analysis of Euler's method; 2.3 Asymptotic error analysis; 2.3.1 Richardson extrapolation; 2.4 Numerical stability; 2.4.1 Rounding error accumulation; Problems; 3 Systems of differential equations; 3.1 Higher-order differential equations; 3.2 Numerical methods for systems; Problems; 4 The backward Euler method and the trapezoidal method4.1 The backward Euler method; 4.2 The trapezoidal method; Problems; 5 Taylor and Runge-Kutta methods; 5.1 Taylor methods; 5.2 Runge-Kutta methods; 5.2.1 A general framework for explicit Runge-Kutta methods; 5.3 Convergence, stability, and asymptotic error; 5.3.1 Error prediction and control; 5.4 Runge-Kutta-Fehlberg methods; 5.5 MATLAB codes; 5.6 Implicit Runge-Kutta methods; 5.6.1 Two-point collocation methods; Problems; 6 Multistep methods; 6.1 Adams-Bashforth methods; 6.2 Adams-Moulton methods; 6.3 Computer codes; 6.3.1 MATLAB ODE codesProblems; 7 General error analysis for multistep methods; 7.1 Truncation error; 7.2 Convergence; 7.3 A general error analysis; 7.3.1 Stability theory; 7.3.2 Convergence theory; 7.3.3 Relative stability and weak stability; Problems; 8 Stiff differential equations; 8.1 The method of lines for a parabolic equation; 8.1.1 MATLAB programs for the method of lines; 8.2 Backward differentiation formulas; 8.3 Stability regions for multistep methods; 8.4 Additional sources of difficulty; 8.4.1 A-stability and L-stability; 8.4.2 Time-varying problems and stability; 8.5 Solving the finite-difference method8.6 Computer codes; Problems; 9 Implicit RK methods for stiff differential equations; 9.1 Families of implicit Runge-Kutta methods; 9.2 Stability of Runge-Kutta methods; 9.3 Order reduction; 9.4 Runge-Kutta methods for stiff equations in practice; Problems; 10 Differential algebraic equations; 10.1 Initial conditions and drift; 10.2 DAEs as stiff differential equations; 10.3 Numerical issues: higher index problems; 10.4 Backward differentiation methods for DAEs; 10.4.1 Index 1 problems; 10.4.2 Index 2 problems; 10.5 Runge-Kutta methods for DAEs; 10.5.1 Index 1 problems10.5.2 Index 2 problems; 10.6 Index three problems from mechanics; 10.6.1 Runge-Kutta methods for mechanical index 3 systems; 10.7 Higher index DAEs; Problems; 11 Two-point boundary value problems; 11.1 A finite-difference method; 11.1.1 Convergence; 11.1.2 A numerical example; 11.1.3 Boundary conditions involving the derivative; 11.2 Nonlinear two-point boundary value problems; 11.2.1 Finite difference methods; 11.2.2 Shooting methods; 11.2.3 Collocation methods; 11.2.4 Other methods and problems; Problems; 12 Volterra integral equations; 12.1 Solvability theory; 12.1.1 Special equations
• Sommario/riassunto: A concise introduction to numerical methodsand the mathematical framework neededto understand their performance Numerical Solution of Ordinary Differential Equations presents a complete and easy-to-follow introduction to classical topics in the numerical solution of ordinary differential equations. The book's approach not only explains the presented mathematics, but also helps readers understand how these numerical methods are used to solve real-world problems. Unifying perspectives are provided throughout the text, bringing together and categorizing different types of problems in o