An introduction to modeling and computation for differential equations

This book provides a unified view of numerical analysis, mathematical modeling in applications, and programming, which is known as scientific computing or computational science. The integrated science of solving a problem with mathematical, numerical, and programming tools makes this book quite unique among related books since it covers a wide array of topics from mathematical modeling to implementing a working computer program. The book describes: how models areset up; how they are preprocessed mathematically with scaling, classification, and approximation; and how a problem is solved numerically with suitable numerical methods. All results are shown appropriately with visualization. The examples in the book are taken from scientific and engineering applications, such as mechanics, fluid dynamics, solid mechanics, chemical engineering, electromagnetic filed theory, control theory, etc. The numerical methods are demonstrated on simple model problems and programmed in MATLAB. The author also highlights the ideas behind some well-known methods, such as finite differences and finite elements. Both MATLAB and the interactive scientific computing program Comsol Multiphysics are used to solve real-world problems found throughout thebook. INDICE: Preface. 1. Introduction. 1.1 What is a Differential Equation? 1.2Examples of an ordinary and a partial differential equation. 1.3 Numerical analysis, a necessity for scientific computing. 1.4 Outline of the contents of this book. 2. Ordinary differential equations. 2.1 Problem classification. 2.2 Linear systems of ODEs with constant coefficients. 2.3 Some stability conceptsfor ODEs. 2.3.1 Stability for a solution trajectory of an ODE-system. 2.3.2 Stability for critical points of ODE-systems. 2.4 Some ODE-models in science and engineering. 2.4.1 Newtons second law. 2.4.2 Hamiltons equations. 2.4.3 Electrical networks. 2.4.4 Chemical kinetics. 2.4.5 Control theory. 2.4.6 Compartment models. 2.5 Some examples from applications. 3. Numerical methods for IVPs. 3.1 Graphical representation of solutions. 3.2 Basic principles of numericalapproximation of ODEs. 3.3 Numerical solution of IVPs with Eulers method. 3.3.1 Eulers explicit method: accuracy. 3.3.2 Eulers explicit method: improving the accuracy. 3.3.3 Eulers explicit method: stability. 3.3.4 Eulers implicit method. 3.3.5 The trapezoidal method. 3.4 Higher order methods for the IVP. 3.4.1 Runge-Kutta methods. 3.4.2 Linear multistep methods. 3.5 The variational equation and parameter fitting in IVPs. 3.6 References. 4. Numerical methods for BVPs. 4.1 Applications. 4.2 Difference methods for BVPs. 4.2.1 A model problemfor BVPs. 4.2.2 Accuracy. 4.2.3 Spurious oscillations. 4.2.4 Linear two-pointboundary value problems. 4.2.5 Nonlinear two-point boundary value problems. 4.2.6 The shooting method. 4.3 Ansatz methods for BVPs. 5. Partial differentialequations. 5.1 Classical PDE-problems. 5.2 Differential operators used for PDEs. 5.3 Some PDEs in science and engineering. 5.3.1 Navier-Stokes equations influid dynamics. 5.3.2 The convection-diffusion-reaction equations. 5.3.3 The heat equation. 5.3.4 The diffusion equation. 5.3.5 Maxwells equations for the electromagnetic field. 5.3.6 Acoustic waves. 5.3.7 Schrödingers equation in quantum mechanics. 5.3.8 Naviers equations in structural mechanics. 5.3.9 Black-Scholes equation in financial mathematics. 5.4 Initial and boundary conditionsfor PDEs. 5.5 Numerical solution of PDEs, some general comments. 6. Numericalmethods for parabolic PDEs. 6.1 Applications. 6.2 An introductory example of discretization. 6.3 The Method of Lines for parabolic PDEs. 6.3.1 Solving the test problem with MoL. 6.3.2 Various types of boundary conditions. 6.3.3 An example of a mixed BC. 6.4 Generalizations of the heat equation. 6.4.1 The heat equation with variable conductivity. 6.4.2 The convection-diffusion-reaction PDE. 6.4.3 The general nonlinear parabolic PDE. 6.5 Ansatz methods for the model equation. 7. Numerical methods for elliptic PDEs. 7.1 Applications. 7.2 The Finite Difference Methods. 7.3 Discretization of a problem with different BCs.7.4 The Finite Element Method. 8. Numerical methods for hyperbolic PDEs. 8.1 Applications. 8.2 Numerical solution of hyperbolic PDEs. 8.3 Introduction to numerical stability for hyperbolic PDEs. 9. Mathematical modeling with differential equations. 9.1 Nature laws. 9.2 Constitutive equations. 9.2.1 Equations in heat conduction problems. 9.2.2 Equations in mass diffusion problems. 9.2.3 Equations in mechanical moment diffusion problems. 9.2.4 Equations in elastic solid mechanics problems. 9.2.5 Equations in chemical reaction engineering problems. 9.2.6 Equations in electrical engineering problems. 9.3 Conservative equations. 9.3.1 Some examples of lumped models. 9.3.2 Some examples of distributed models. 9.4 Scaling of differential equations to dimensionless form. A. Appendix. A.1 Newtons method for systems of nonlinear algebraic equations. A.1.1Quadratic systems. A.1.2 Overdetermined systems. A.2 Some facts about linear difference equations. A.3 Derivation of difference approximations. A.4 The interpretations of div and curl. A.5 Numerical solution of algebraic systems of equations. A.5.1 Direct methods. A.5.2 Iterative methods for linear systems of equations. A.6 Some results for fourier transforms. B. Software for scientificcomputing. B.1 Matlab. B.2 Comsol Multiphysics. C. Computer exercises to support the chapters. References. Index.

• ISBN: 978-0-470-27085-1
• Editorial: John Wiley & Sons
• Encuadernacion: Cartoné
• Páginas: 276
• Fecha Publicación: 31/07/2008
• Nº Volúmenes: 1
• Idioma: Inglés