## 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

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 |

### Common terms and phrases

analogue applications assumed basis boundary conditions boundary value problems bounded called Cauchy Cauchy principal value Cauchy-type integral Cauchy's problem classical Clifford algebra complex variable considered constant constructed continuous corresponding defined definition differential Dirac domain dual easily element elliptic equality equation 0.6 equation 6.1 exist Find follows formula functions with values fundamental solution given h-regular harmonic functions Hence holomorphic functions hyperbolic integral equations invariant inversion kernel known linear linearly independent matrices non-local problems Note numbers obtain obvious operator parabolic particular piecewise posed principle Proof properties proved quadratures quantum mechanics reduced reflection regular functions regular solution representation represented respect satisfying the conditions solution of equation solved explicitly space surface Taking into consideration theorem transformation true two-dimensional uniquely University vanishing at infinity vectorial wave written zero