Partial Differential Equations

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Paperback | 20 July 1987

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532 Pages

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22.86 x 15.24 x 3.02

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This book is a rigorous introduction to the abstract theory of partial differential equations. The main prerequisite is familiarity with basic functional analysis: more advanced topics such as Fredholm operators, the Schauder fixed point theorem and Bochner integrals are introduced when needed, and the book begins by introducing the necessary material from the theory of distributions and Sobolev spaces. Using such techniques, the author presents different methods available for solving elliptic, parabolic and hyperbolic equations. He also considers the difference process for the practical solution of a partial differential equation, emphasising that it is possible to solve them numerically by simple methods. Many examples and exercises are provided throughout, and care is taken to explain difficult points. Advanced undergraduates and graduate students will appreciate this self-contained and practical introduction.

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Integral Calculus & Equations

Preface
Sobolev Spaces
Notation, basic properties, distributions
Geometric assumptions for the domain
Definitions and density properties for the Sobolev-Slobodeckii spaces
The transformation theorem and Sobolev spaces on differentiable manifolds
Definition of Sobolev spaces by the Fourier transformation and extension theorems
Continuous embeddings and Sobolev's lemma
Compact embeddings
The trace operator
Weak sequential compactness and approximation of derivatives by difference quotients
Elliptic Differential Operators
Linear differential operators
The Lopatinskil-Sapiro condition and examples
Fredholm operators
The main theorem and some theorems on the index of elliptic boundary value problems
Green's formulae
The adjoint boundary value problem and the connection with the image space of the original operator
Examples
Strongly Elliptic Differential Operators and the Method of Variations
Gelfand triples, the Law-Milgram, V-elliptic and V-coercive operators
Agmon's condition
Agmon's theorem: conditions for the V-coercion of strongly elliptic differential operators
Regularity of the solutions of strongly elliptic equations
The solution theorem for strongly elliptic equations and examples
The Schauder fixed point theorem and a non-linear problem
Elliptic boundary value problemss for unbounded regions
Parabolic Differential Operators
The Bochner integral
Distributions with values in a Hilbert space H and the space W
The existence and uniqueness of the solution of a parabolic differential equation
The regularity of solutions of the parabolic differential equation
Examples
Hyperbolic Differential Operators
Existence and uniqueness of the solution
Regularity of the solutions of the hyperbolic differential equation
Difference Processes for the Calculation of the Solution of the Partial Differential Equation
Functional analytic concepts for difference processes
Difference processes for elliptic differential equations and for the wave equation
Evolution equations
References
Function and distribution spaces
Index
Table of Contents provided by Publisher. All Rights Reserved.

Partial Differential Equations

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Paperback

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