Introduction to Partial Differential Equations

This modern take on partial differential equations does not require knowledge beyond vector calculus and linear algebra. The author focuses on the most important classical partial differential equations, including conservation equations and their characteristics, the wave equation, the heat equation, function spaces, and Fourier series, drawing on tools from analysis only as they arise.
Within each section the author creates a narrative that answers the five questions:

1. What is the scientific problem we are trying to understand?

2. How do we model that with PDE?

3. What techniques can we use to analyze the PDE?

4. How do those techniques apply to this equation?

5. What information or insight did we obtain by developing and analyzing the PDE? The text stresses the interplay between modeling and mathematical analysis, providing a thorough source of problems and an inspirationfor the development of methods.

Perfect book for a ?One-semester PDE course Includes a thorough discussion of modeling process for each equation Covers indepth three types of linear PDES: elliptic, parabolic, and hyperbolic? Includes supplementary material: sn.pub/extras

Autorentext

David Borthwick, Ph.D., is a Professor in the Department of Mathematics at Emory University. He received his Ph.D. in physics from Harvard University in 1993 and has taught mathematics at Emory for over 25 years. His research focuses on mathematical physics, spectral theory, and geometric analysis. He is the author of several books, including Spectral Theory in the Springer GTM series.

Inhalt

1. Introduction.- 2. Preliminaries.- 3. Conservation Equations and Characteristics.- 4. The Wave Equation.- 5. Separation of Variables.- 6. The Heat Equation.- 7. Function Spaces.- 8. Fourier Series.- 9. Maximum Principles.- 10. Weak Solutions.- 11. Variational Methods.- 12. Distributions.- 13. The Fourier Transform.- A. Appendix: Analysis Foundations.- References.- Notation Guide.- Index.