Theory and Applications of Partial Differential EquationsThis book is a product of the experience of the authors in teaching partial differential equations to students of mathematics, physics, and engineering over a period of 20 years. Our goal in writing it has been to introduce the subject with precise and rigorous analysis on the one hand, and interesting and significant applications on the other. The starting level of the book is at the first-year graduate level in a U.S. university. Previous experience with partial differential equations is not required, but the use of classical analysis to find solutions of specific problems is not emphasized. From that perspective our treatment is decidedly theoretical. We have avoided abstraction and full generality in many situations, however. Our plan has been to introduce fundamental ideas in relatively simple situations and to show their impact on relevant applications. The student is then, we feel, well prepared to fight through more specialized treatises. There are parts of the exposition that require Lebesgue integration, distributions and Fourier transforms, and Sobolev spaces. We have included a long appendix, Chapter 8, giving precise statements of all results used. This may be thought of as an introduction to these topics. The reader who is not familiar with these subjects may refer to parts of Chapter 8 as needed or become somewhat familiar with them as prerequisite and treat Chapter 8 as Chapter O. |
Contents
Introduction to Partial Differential Equations | 1 |
Reflection Problem | 24 |
Wave Equation in Two and Three Dimensions | 32 |
Copyright | |
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Other editions - View all
Theory and Applications of Partial Differential Equations Piero Bassanini,Alan R. Elcrat Limited preview - 2013 |
Theory and Applications of Partial Differential Equations Piero Bassanini,Alan R. Elcrat No preview available - 2013 |
Theory and Applications of Partial Differential Equations Piero Bassanini,Alan R. Elcrat No preview available - 2014 |
Common terms and phrases
arbitrary B₁ B₂ Banach space boundary conditions boundary values bounded C²(N Cauchy problem Chapter choose compact conservation law consider constant continuous converges convex curve defined denote depends derivatives Dirichlet problem discontinuity domain eigenvalues elliptic entropy entropy condition Exercise exists finite fixed follows Fourier genuinely nonlinear given grad harmonic function heat equation hence Hilbert space Hint Igrad implies inequality initial data integral k₁ Lemma linear Lipschitz Lipschitz continuous maximum principle neighborhood norm obtain operator potential proof of Theorem Proposition Prove rarefaction fan result Riemann problem satisfies scalar Section sequence shock smooth Sobolev spaces Suppose t₁ Theorem 2.1 transformation u₁ u₂ uniformly unique v₁ value problem variables vector w₁ weak solution z₁ zero ΘΩ