Ebook: Analytic Methods for Partial Differential Equations

The subject of partial differential equations holds an exciting and special position in mathematics. Partial differential equations were not consciously created as a subject but emerged in the 18th century as ordinary differential equations failed to describe the physical principles being studied. The subject was originally developed by the major names of mathematics, in particular, Leonard Euler and Joseph-Louis Lagrange who studied waves on strings; Daniel Bernoulli and Euler who considered potential theory, with later developments by Adrien-Marie Legendre and Pierre-Simon Laplace; and Joseph Fourier's famous work on series expansions for the heat equation. Many of the greatest advances in modern science have been based on discovering the underlying partial differential equation for the process in question. J ames Clerk Maxwell, for example, put electricity and magnetism into a unified theory by estab­ lishing Maxwell's equations for electromagnetic theory, which gave solutions for problems in radio wave propagation, the diffraction of light and X-ray developments. Schrodinger's equation for quantum mechankal processes at the atomic level leads to experimentally verifiable results which have changed the face of atomic physics and chemistry in the 20th century. In fluid mechanics, the Navier-Stokes' equations form a basis for huge number-crunching activities associated with such widely disparate topics as weather forcasting and the design of supersonic aircraft. Inevitably the study of partial differential equations is a large undertaking, and falls into several areas of mathematics.

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The subject of partial differential equations holds an exciting place in mathematics. At one extreme the interest lies in the existence and uniqueness of solutions, and the functional analysis of the proofs of these properties. At the other extreme lies the applied mathematical and engineering quest to find useful solutions, either analytically or numerically, to these important equations which can be used in design and construction. The objective of this book is to actually solve equations rather than discuss the theoretical properties of their solutions. The topics are approached practically, without losing track of the underlying mathematical foundations of the subject. The topics covered include the separation of variables, the characteristic method, D'Alembert's method, integral transforms and Green's functions. Numerous exercises are provided as an integral part of the learning process, with solutions provided in a substantial appendix.

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The subject of partial differential equations holds an exciting place in mathematics. At one extreme the interest lies in the existence and uniqueness of solutions, and the functional analysis of the proofs of these properties. At the other extreme lies the applied mathematical and engineering quest to find useful solutions, either analytically or numerically, to these important equations which can be used in design and construction. The objective of this book is to actually solve equations rather than discuss the theoretical properties of their solutions. The topics are approached practically, without losing track of the underlying mathematical foundations of the subject. The topics covered include the separation of variables, the characteristic method, D'Alembert's method, integral transforms and Green's functions. Numerous exercises are provided as an integral part of the learning process, with solutions provided in a substantial appendix.
Content:
Front Matter....Pages i-xii
Mathematical Preliminaries....Pages 1-48
Separation of the Variables....Pages 49-94
First-order Equations and Hyperbolic Second-order Equations....Pages 95-122
Integral Transforms....Pages 123-162
Green’s Functions....Pages 163-224
Back Matter....Pages 225-299

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The subject of partial differential equations holds an exciting place in mathematics. At one extreme the interest lies in the existence and uniqueness of solutions, and the functional analysis of the proofs of these properties. At the other extreme lies the applied mathematical and engineering quest to find useful solutions, either analytically or numerically, to these important equations which can be used in design and construction. The objective of this book is to actually solve equations rather than discuss the theoretical properties of their solutions. The topics are approached practically, without losing track of the underlying mathematical foundations of the subject. The topics covered include the separation of variables, the characteristic method, D'Alembert's method, integral transforms and Green's functions. Numerous exercises are provided as an integral part of the learning process, with solutions provided in a substantial appendix.
Content:
Front Matter....Pages i-xii
Mathematical Preliminaries....Pages 1-48
Separation of the Variables....Pages 49-94
First-order Equations and Hyperbolic Second-order Equations....Pages 95-122
Integral Transforms....Pages 123-162
Green’s Functions....Pages 163-224
Back Matter....Pages 225-299
....