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The book Partial Differential Equations through Examples and Exercises has evolved from the lectures and exercises that the authors have given for more than fifteen years, mostly for mathematics, computer science, physics and chemistry students. By our best knowledge, the book is a first attempt to present the rather complex subject of partial differential equations (PDEs for short) through active reader-participation. Thus this book is a combination of theory and examples. In the theory of PDEs, on one hand, one has an interplay of several mathematical disciplines, including the theories of analytical functions, harmonic analysis, ODEs, topology and last, but not least, functional analysis, while on the other hand there are various methods, tools and approaches. In view of that, the exposition of new notions and methods in our book is "step by step". A minimal amount of expository theory is included at the beginning of each section Preliminaries with maximum emphasis placed on well selected examples and exercises capturing the essence of the material. Actually, we have divided the problems into two classes termed Examples and Exercises (often containing proofs of the statements from Preliminaries). The examples contain complete solutions, and also serve as a model for solving similar problems, given in the exercises. The readers are left to find the solution in the exercises; the answers, and occasionally, some hints, are still given. The book is implicitly divided in two parts, classical and abstract.
Table des matières
1 Introduction.- 1.1 Basic Notions.- 1.2 The Cauchy-Kowalevskaya Theorem.- 1.3 Equations of Mathematical Physics.- 2 First Order PDEs.- 2.1 Quasi-linear PDEs.- 2.2 Pfaff's Equations.- 2.3 Nonlinear First Order PDEs.- 3 Classification of the Second Order PDEs.- 3.1 Two Independent Variables.- 3.2 n Independent Variables.- 3.3 Wave, Potential and Heat Equation.- 4 Hyperbolic Equations.- 4.1 Cauchy Problem for the One-dimensional Wave Equation.- 4.2 Cauchy Problem for the n-dimensional Wave Equation.- 4.3 The Fourier Method of Separation Variables.- 4.4 The Sturm-Liouville Problem.- 4.5 Miscellaneous Problems.- 4.6 The Vibrating String.- 5 Elliptic Equations.- 5.1 Dirichlet Problem.- 5.2 The Maximum Principle.- 5.3 The Green Function.- 5.4 The Harmonic Functions.- 5.5 Gravitational Potential.- 6 Parabolic Equations.- 6.1 Cauchy Problem.- 6.2 Mixed Type Problem.- 6.3 Heat conduction.- 7 Numerical Methods.- 7.0.1 Preliminaries.- 7.0.2 Examples and Exercises.- 8 Lebesgue's Integral, Fourier Transform.- 8.1 Lebesgue's Integral and the L2(Q) Space.- 8.2 Delta Nets.- 8.3 The Surface Integrals.- 8.4 The Fourier Transform.- 9 Generalized Derivative and Sobolev Spaces.- 9.1 Generalized Derivative.- 9.2 Sobolev Spaces.- 10 Some Elements from Functional Analysis.- 10.1 Hilbert Space.- 10.2 The Fredholm Alternatives.- 10.3 Normed Vector Spaces.- 11 Functional Analysis Methods in PDEs.- 11.1 Generalized Dirichlet Problem.- 11.2 The Generalized Mixed Problems.- 11.3 Numerical Solutions.- 11.4 Miscellaneous.- 12 Distributions in the theory of PDEs.- 12.1 Basic Properties.- 12.2 Fundamental Solutions.
