Partial Differential Equations through Examples and Exercises - Arpad Takaci

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Résumé :
This book examines the complicated subject of Partial Differential Equations (PDEs). It involves the reader throughout by presenting theory, examples and exercises together. br/ Both the classical and abstract aspects of the theory are dealt with, so that, for example, classical and generalised solutions in Sobolev and distribution spaces are treated. Most of the work is devoted to second or higher order PDEs; part of the distribution theory is included, covering Dirac's delta distribution delta function. br/ Many practical tools are offered for solving important problems with the basic three PDEs, namely the wave equation, the Laplace equation, the heat equation and their generalisations. br/ The majority of the problems are mathematical in character, though often physical interpretations are given. br/ emAudience:/em This volume is intended for undergraduate and graduate students in mathematics, physics technology and economics interested in PDEs for modelling complex systems.

Sommaire:
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.