Introduction to partial differential equations with MATLAB
The text is intended for a first course in partial differential equations at the undergraduate level for students in mathematics, science, and engineering. It is assumed that the student has had the standard three semester calculus sequence including multivariable calculus and a course in ordinary differential equations. Some exposure to matrices and vectors is helpful, but not essential. No prior experience with MATLAB is assumed, although many engineering students will have already seen MATLAB. The material is organized by physical setting and by equation rather than by technique of solution. Each equation is placed in the appropriate physical context with a careful derivation from physical principles. This is followed by a discussion of qualitative properties of the solutions without boundary conditions. Then a nonlinear version of the equation and an appropriate numerical scheme are introduced. Thus Chapter 2 deals with linear first-order equations and the method of characteristics, an example of a scalar nonlinear conservation law, and numerical methods for these equations. Chapter 3 follows the same strategy for the heat equation, while Chapter 4 treats boundary value problems for the heat equation. The wave equation is studied in Chapter 5. Fourier series and the Fourier transform are studied in Chapter 6. Chapter 7 is perhaps novel for this level course. It deals with the method of stationary phase and dispersive equations. The Schrödinger equation is discussed as a dispersive equation. Chapter 8 has fairly standard material on the linear heat and wave equations in higher dimensions. Chapter 9 treats the Laplace and Poisson equations, and includes some examples of nonlinear variational problems. Finally, Chapter 10 provides more numerical methods, suitable for computations in higher dimensions, including an introduction to the finite element method and Galerkin methods. The computer is used extensively in the text with many exercises, although it is possible to use this text without the computer exercises.
scientific article; zbMATH DE number 1102779