Partial Differential Equations in Clifford Analysis

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CRC Press, Jan 6, 1999 - Mathematics - 160 pages
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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.
 

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Contents

Matrix representations of Clifford algebras for any dimension
9
Differential operators and classification
14
Lorentz transformations in the elliptic and hyperbolic cases
22
Elliptic partial differential equations
26
Cauchy kernel and representations for hregular functions in Rn
30
Extension theorems and the RiemannSchwarz principle of reflection
38
The PoincaréBertrand transformation formula
41
Generalized Riesz system
45
The Beltrami equation in Rn
72
Hyperbolic partial differential equations
78
Generalized Maxwell and Dirac equations
81
The hyperbolic Beltrami equation
86
Initial value problems for the KleinGordon equation
90
Cauchys initial value problem and its modification for the regular and hregular functions with values in Rínn1 and
97
Rnn2 n 3
101
Parabolic partial differential equations
103

The basic Ltheory of the Fourier integral transformation
49
Boundary value problems for regular functions with values in Rn
61
Boundary value problems for hregular functions with values in Rn n 1
69
Effective solutions of some nonlocal problems
118
Epilogue
146
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Page 151 - Solution of initial value problems in classes of generalized analytic functions, Teubner, Leipzig and Springer- Verlag, 1989.
Page 151 - Titchmarsh, E. Introduction to the Theory of Fourier Integrals. Clarendon Press, Oxford, 1962. [Tr] Treves, F.

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