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Partial Differential Equations in Clifford Analysis



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ISBN 9780582317499
Published January 6, 1999 by Chapman and Hall/CRC
160 Pages

 
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Book Description

Clifford analysis represents one of the most remarkable fields of modern mathematics. With the recent finding that almost all classical linear partial differential equations of mathematical physics can be set in the context of Clifford analysis-and that they can be obtained without applying any physical laws-it appears that Clifford analysis itself can suggest new equations or new generalizations of classical equations that may have some physical content.
Partial Differential Equations in Clifford Analysis considers-in a multidimensional space-elliptic, hyperbolic, and parabolic operators related to Helmholtz, Klein-Gordon, Maxwell, Dirac, and heat equations. The author addresses two kinds of parabolic operators, both related to the second-order parabolic equations whose principal parts are the Laplacian and d'Alembertian: an elliptic-type parabolic operator and a hyperbolic-type parabolic operator. She obtains explicit integral representations of solutions to various boundary and initial value problems and their properties and solves some two-dimensional and non-local problems.
Written for the specialist but accessible to non-specialists as well, Partial Differential Equations in Clifford Analysis presents new results, reformulations, refinements, and extensions of familiar material in a manner that allows the reader to feel and touch every formula and problem. Mathematicians and physicists interested in boundary and initial value problems, partial differential equations, and Clifford analysis will find this monograph a refreshing and insightful study that helps fill a void in the literature and in our knowledge.

Table of Contents

Introduction
Principles of Clifford Algebra and Analysis
The Basic Notions and Definitions
Matrix Representations of Clifford Algebras for any Dimension
Differential Operators and Classification
Lorentz Transformations in the Elliptic and Hyperbolic Cases
Elliptic Partial Differential Equations
Introduction
Cauchy Kernel and Representations for h-regular Functions in R(n)
Extension Theorems and The Riemann-Schwarz Principle of Reflection
The Poincaré-Bertrand Transformation Formula
Generalized Riesz System
The Basic L-Theory of the Fourier Integral Transformation
Boundary Value Problems for Regular Functions with Values in R(n)
Boundary Value Problems for h-Regular Functions with Values in R(n), n³1
The Beltrami Equation in R(n)
Hyperbolic Partial Differential Equations
Introduction
Generalized Maxwell and Dirac Equations
The Hyperbolic Beltrami Equation
Initial Value Problems for the Klein-Gordon Equation
Cauchy's Initial Value Problem and its Modification for the Regular and h-Regular Functions with Values in R(n,n-1) and in R(n,n-2), n³3
Parabolic Partial Differential Equations
Introduction
Parabolic Regular System of the First Kind
Initial Value Problems for Parabolic Equations of the First Kind
Parabolic Regular Equations of the Second Kind and Initial Value Problems
Effective Solutions for Some Non-Local Problems
Introduction
Wiener-Hopf and Dual Integral Equations of Convolution Type
Generalized Wiener-Hopf Integral Equation with Two Kernels Depending on the Difference and Sum of the Arguments
Dual Integral Equations with Kernels Depending on the Difference and Sum of Arguments
Non-Local Problems for Holomorphic Functions and Applications in Elasticity Theory
Non-Local Problems for Generalized Holomorphic Functions and the Generalized Beltrami Equation
Non-Local Problems for Polyharmonic Functions
Epilogue
Bibliography

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