| Description |
1 online resource (xiv, 280 pages) : illustrations |
| Series |
De Gruyter textbook |
|
De Gruyter textbook
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| Contents |
Preface; Notation; I Classical Theory; 1 Preliminaries; 1.1 Basic Differential Operators; 1.2 Linear and Quasilinear Partial Differential Equations; 1.3 Solutions of Some Particular Equations; 1.4 Boundary Value Problems; 1.4.1 Boundary Value Problems for Poisson's Equation; 1.4.2 Boundary Value Problems for the Heat Equation; 1.4.3 Boundary Value Problems for the Wave Equation; 2 Partial Differential Equations and Mathematical Modeling; 2.1 Conservation Laws: Continuity Equations; 2.2 Reaction-Diffusion Systems; 2.3 The One-Dimensional Wave Equation |
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2.4 Other Equations in Mathematical Physics3 Elliptic Boundary Value Problems; 3.1 Green's Formulas; 3.2 The Fundamental Solution of Laplace's Equation; 3.3 Mean Value Theorems for Harmonic Functions; 3.4 The Maximum Principle; 3.5 Uniqueness and Continuous Dependence on Data for the Dirichlet Problem; 3.6 Green's Function of the Dirichlet Problem; 3.7 Poisson's Formula; 3.8 Dirichlet's Principle; 3.9 The Generalized Solution of the Dirichlet Problem; 3.10 Abstract Fourier Series; 3.11 The Eigenvalues and Eigenfunctions of the Dirichlet Problem |
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3.12 The Case of Elliptic Equations in Divergence Form3.13 The Generalized Solution of the Neumann Problem; 3.14 Complements; 3.14.1 Harnack's Inequality; 3.14.2 Hopf's Maximum Principle; 3.14.3 The Newtonian Potential; 3.14.4 Perron's Method; 3.14.5 Layer Potentials; 3.14.6 Fredholm's Method of Integral Equations; 3.15 Problems; 4 Mixed Problems for Evolution Equations; 4.1 The Maximum Principle for the Heat Equation; 4.2 Vector-Valued Functions; 4.3 The Cauchy-Dirichlet Problem for the Heat Equation; 4.4 The Cauchy-Dirichlet Problem for the Wave Equation; 4.5 Problems |
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5 The Cauchy Problem for Evolution Equations5.1 The Fourier Transform; 5.1.1 The Fourier Transform on L1 (Rn); 5.1.2 Fourier Transform and Convolution; 5.1.3 The Fourier Transform on the Schwartz Space S (R"); 5.2 The Cauchy Problem for the Heat Equation; 5.3 The Cauchy Problem for the Wave Equation; 5.4 Nonhomogeneous Equations: Duhamel's Principle; 5.5 Problems; II Modern Theory; 6 Distributions; 6.1 The Fundamental Spaces of the Theory of Distributions; 6.2 Distributions: Examples; Operations with Distributions; 6.2.1 Regular Distributions; 6.2.2 The Dirac Distribution |
| Summary |
This textbook provides a brief and lucid introduction to the theory of linear partial differential equations. It clearly explains the transition from classical to generalized solutions and the natural way in which Sobolev spaces appear as completions of spaces of continuously differentiable functions. The solution operators associated to non-homogeneous equations are used to make transition to the theory of nonlinear PDEs. Organized on three parts, this material is suitable for three one-semester courses, a beginning one in the frame of classical analysis, a more advanced course in modern theo |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Differential equations, Linear -- Textbooks
|
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Differential equations, Partial -- Textbooks
|
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MATHEMATICS -- Differential Equations -- Partial.
|
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Differential equations, Linear
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Differential equations, Partial
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| Genre/Form |
Textbooks
|
| Form |
Electronic book
|
| ISBN |
3110269058 |
|
9783110269055 |
|
9781299719774 |
|
1299719775 |
|
