Partial Differential Equations: Modeling, Analysis, Computation

Contents: Preface. 1. Differential and difference equations. 2. Characterization and classification. 3. Fourier theory. 4. Distributions and fundamental solutions. 5. Approximation by finite differences. 6. The Equations of continuum mechanics and electromagnetics. 7. The art of modeling. 8. The analysis of elliptic equations. 9. Numerical methods for elliptic equations. 10. Analysis of parabolic equations. 11. Numerical methods for parabolic equations. 12. Analysis of hyperbolic equations. 13. Numerical methods for scalar hyperbolic equations. 14. Numerical methods for hyperbolic systems. 15. Perturbation methods. 16. Modeling, analyzing, and simulating problems from practice. Appendices: Useful definitions and properties. Bibliography. Index. Partial differential equations (PDEs) are used to describe a large variety of physical phenomena, from fluid flow to electromagnetic fields, and are indispensable to such disparate fields as aircraft simulation and computer graphics. While most existing texts on PDEs deal with either analytical or numerical aspects of PDEs, this innovative and comprehensive textbook features a unique approach that integrates analysis and numerical solution methods and includes a third component modeling to address real-life problems. The authors believe that modeling can be learned only by doing; hence a separate chapter containing 16 user-friendly case studies of elliptic, parabolic, and hyperbolic equations is included and numerous exercises are included in all other chapters. The book enables readers to deepen their understanding of a topic ubiquitous in mathematics and science and to tackle practical problems. The advent of fast computers and the development of numerical methods have enabled the modern engineer to use a large variety of packages to find numerical approximations to solutions of PDEs. Problems are usually standard and a thorough knowledge of a well-chosen subset of analytical and numerical tools and methodologies is necessary when dealing with real-life problems. When one is dealing with PDEs in practice, it becomes clear that both numerical and analytical treatments of the problem are needed. This comprehensive book is intended for graduate students in applied mathematics, engineering, and physics and may be of interest to advanced undergraduate students. Mathematicians, scientists, and engineers also will find the book useful. (jacket).…