Book contents
- Frontmatter
- Contents
- Preface
- 1 An Introduction to the Method of Lines
- 2 A One-Dimensional, Linear Partial Differential Equation
- 3 Green's Function Analysis
- 4 Two Nonlinear, Variable-Coefficient, Inhomogeneous Partial Differential Equations
- 5 Euler, Navier Stokes, and Burgers Equations
- 6 The Cubic Schrödinger Equation
- 7 The Korteweg–deVries Equation
- 8 The Linear Wave Equation
- 9 Maxwell's Equations
- 10 Elliptic Partial Differential Equations: Laplace's Equation
- 11 Three-Dimensional Partial Differential Equation
- 12 Partial Differential Equation with a Mixed Partial Derivative
- 13 Simultaneous, Nonlinear, Two-Dimensional Partial Differential Equations in Cylindrical Coordinates
- 14 Diffusion Equation in Spherical Coordinates
- Appendix 1 Partial Differential Equations from Conservation Principles: The Anisotropic Diffusion Equation
- Appendix 2 Order Conditions for Finite-Difference Approximations
- Appendix 3 Analytical Solution of Nonlinear, Traveling Wave Partial Differential Equations
- Appendix 4 Implementation of Time-Varying Boundary Conditions
- Appendix 5 The Differentiation in Space Subroutines Library
- Appendix 6 Animating Simulation Results
- Index
- Plate section
Preface
Published online by Cambridge University Press: 08 October 2009
- Frontmatter
- Contents
- Preface
- 1 An Introduction to the Method of Lines
- 2 A One-Dimensional, Linear Partial Differential Equation
- 3 Green's Function Analysis
- 4 Two Nonlinear, Variable-Coefficient, Inhomogeneous Partial Differential Equations
- 5 Euler, Navier Stokes, and Burgers Equations
- 6 The Cubic Schrödinger Equation
- 7 The Korteweg–deVries Equation
- 8 The Linear Wave Equation
- 9 Maxwell's Equations
- 10 Elliptic Partial Differential Equations: Laplace's Equation
- 11 Three-Dimensional Partial Differential Equation
- 12 Partial Differential Equation with a Mixed Partial Derivative
- 13 Simultaneous, Nonlinear, Two-Dimensional Partial Differential Equations in Cylindrical Coordinates
- 14 Diffusion Equation in Spherical Coordinates
- Appendix 1 Partial Differential Equations from Conservation Principles: The Anisotropic Diffusion Equation
- Appendix 2 Order Conditions for Finite-Difference Approximations
- Appendix 3 Analytical Solution of Nonlinear, Traveling Wave Partial Differential Equations
- Appendix 4 Implementation of Time-Varying Boundary Conditions
- Appendix 5 The Differentiation in Space Subroutines Library
- Appendix 6 Animating Simulation Results
- Index
- Plate section
Summary
In the analysis and the quest for an understanding of a physical system, generally, the formulation and use of a mathematical model that is thought to describe the system is an essential step. That is, a mathematical model is formulated (as a system of equations) that is thought to quantitatively define the interrelationships between phenomena that define the characteristics of the physical system. The mathematical model is usually tested against observations of the physical system, and if the agreement is considered acceptable, the model is then taken as a representation of the physical system, at least until improvements in the observations lead to refinements and extensions of the model. Often the model serves as a guide to new observations. Ideally, this process of refinement of the observations and model leads to improvements of the model and thus enhanced understanding of the physical system.
However, this process of comparing observations with a proposed model is not possible until the model equations are solved to give a solution that is then the basis for the comparison with observations. The solution of the model equations is often a challenge. Typically in science and engineering this involves the integration of systems of ordinary and partial differential equations (ODE/PDEs). The intent of this volume is to assist scientists and engineers in the process of solving differential equation models by explaining some numerical, computer-based methods that have generally been proved to be effective for the solution of a spectrum of ODE/PDE system problems.
- Type
- Chapter
- Information
- A Compendium of Partial Differential Equation ModelsMethod of Lines Analysis with Matlab, pp. ix - xivPublisher: Cambridge University PressPrint publication year: 2009