## Partial Differential Equations in Clifford AnalysisClifford 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

Principles of Clifford algebra and analysis | 4 |

Elliptic partial differential equations | 26 |

Hyperbolic partial differential equations | 78 |

Parabolic partial differential equations | 103 |

Effective solutions of some nonlocal problems | 118 |

38 | 139 |

41 | 145 |

Epilogue | 146 |

### Common terms and phrases

analogous applications assumed basis boundary conditions boundary value problem bounded called Cauchy Cauchy's problem classical Clifford algebra complex variable considered constant constructed continuous convolution corresponding defined definition differential Dirac domain dual integral equations easily elliptic equality equation 0.6 equation 6.1 equivalent exist Find follows formula function W(z fundamental solution given h-regular half-plane harmonic functions heat equation Hence holomorphic functions hyperbolic kernel kind known linearly independent matrices multidimensional non-local problems Note numbers obtain operator parabolic particular piecewise posed principle Proof properties proved quadratures quantum mechanics reduced regular regular functions regular solution representation represented respect satisfying the conditions sense ſº solution of equation solved explicitly space Taking into consideration theorem theory transformation unknown vanishing at infinity vectorial written zero