Partial Differential Equations with Numerical Methods

A Two-Point Boundary Value Problem.- Elliptic Equations.- Finite Difference Methods for Elliptic Equations.- Finite Element Methods for Elliptic Equations.- The Elliptic Eigenvalue Problem.- Initial-Value Problems for Ordinary Differential Equations.- Parabolic Equations.- Finite Difference Methods for Parabolic Problems.- The Finite Element Method for a Parabolic Problem.- Hyperbolic Equations.- Finite Difference Methods for Hyperbolic Equations.- The Finite Element Method for Hyperbolic Equations.- Some Other Classes of Numerical Methods.

Very well-written Elementary introduction Includes supplementary material: sn.pub/extras

Klappentext

The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices.

Zusammenfassung

The main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.