Partial Differential Equations An Accessible Route Through Theory And Applications Solutions
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Partial Differential Equations
Author: András Vasy
language: en
Publisher: American Mathematical Society
Release Date: 2022-07-15
This text on partial differential equations is intended for readers who want to understand the theoretical underpinnings of modern PDEs in settings that are important for the applications without using extensive analytic tools required by most advanced texts. The assumed mathematical background is at the level of multivariable calculus and basic metric space material, but the latter is recalled as relevant as the text progresses. The key goal of this book is to be mathematically complete without overwhelming the reader, and to develop PDE theory in a manner that reflects how researchers would think about the material. A concrete example is that distribution theory and the concept of weak solutions are introduced early because while these ideas take some time for the students to get used to, they are fundamentally easy and, on the other hand, play a central role in the field. Then, Hilbert spaces that are quite important in the later development are introduced via completions which give essentially all the features one wants without the overhead of measure theory. There is additional material provided for readers who would like to learn more than the core material, and there are numerous exercises to help solidify one's understanding. The text should be suitable for advanced undergraduates or for beginning graduate students including those in engineering or the sciences.
Partial Differential Equations
Cover -- Title page -- Contents -- Preface -- Chapter 1. Introduction -- Chapter 2. Where do PDE come from? -- Chapter 3. First order scalar semilinear equations -- Chapter 4. First order scalar quasilinear equations -- Chapter 5. Distributions and weak derivatives -- Chapter 6. Second order constant coefficient PDE: Types and d'Alembert's solution of the wave equation -- Chapter 7. Properties of solutions of second order PDE: Propagation, energy estimates and the maximum principle -- Chapter 8. The Fourier transform: Basic properties, the inversion formula and the heat equation -- Chapter 9. The Fourier transform: Tempered distributions, the wave equation and Laplace's equation -- Chapter 10. PDE and boundaries -- Chapter 11. Duhamel's principle -- Chapter 12. Separation of variables -- Chapter 13. Inner product spaces, symmetric operators, orthogonality -- Chapter 14. Convergence of the Fourier series and the Poisson formula on disks -- Chapter 15. Bessel functions -- Chapter 16. The method of stationary phase -- Chapter 17. Solvability via duality -- Chapter 18. Variational problems -- Bibliography -- Index -- Back Cover
A Course on Partial Differential Equations
Author: Walter Craig
language: en
Publisher: American Mathematical Soc.
Release Date: 2018-12-12
Does entropy really increase no matter what we do? Can light pass through a Big Bang? What is certain about the Heisenberg uncertainty principle? Many laws of physics are formulated in terms of differential equations, and the questions above are about the nature of their solutions. This book puts together the three main aspects of the topic of partial differential equations, namely theory, phenomenology, and applications, from a contemporary point of view. In addition to the three principal examples of the wave equation, the heat equation, and Laplace's equation, the book has chapters on dispersion and the Schrödinger equation, nonlinear hyperbolic conservation laws, and shock waves. The book covers material for an introductory course that is aimed at beginning graduate or advanced undergraduate level students. Readers should be conversant with multivariate calculus and linear algebra. They are also expected to have taken an introductory level course in analysis. Each chapter includes a comprehensive set of exercises, and most chapters have additional projects, which are intended to give students opportunities for more in-depth and open-ended study of solutions of partial differential equations and their properties.