abstract 84

Bulletin of Computational Applied Mathematics (Bull CompAMa) 

84

Parabolic Dirac operators and their fundamental solutions in parametric Clifford algebras

Eusebio Ariza García, Judith Vanegas, Franklin Vargas Jiménez

In this work, we presented a factorization for the non-stationary diffusion operator for anisotropic media $\tilde{{\mathcal L}} = {\partial}_t-div(B\nabla)$, where $B\in{\mathbb{R}}^{n\times n}$  is a symmetric and positive definite matrix. The factorization is given by $\tilde{{\mathcal L}} ={\mathcal D}_{x,t}^2$ where ${\mathcal D}_{x,t}$ is a parabolic Dirac operator defined on a parametric Clifford algebra. For the anisotropic heat diffusion operator and its factor, the parabolic Dirac operator, we find fundamental solutions, and for the last operator, a Cauchy-Pompeiu type integral representation is found. Integral operators derived from this integral representation allow us to pose mixed problems for the ${\mathcal D}_{x,t}$ operator.

Keywords: Dirac operators; parametric Clifford algebra; fundamental solutions.

Cite this paper:

Ariza García E., Vanegas J., Vargas Jiménez F. 

Parabolic Dirac operators and their fundamental solutions in parametric Clifford algebras.

Bull. Comput. Appl. Math. (Bull CompAMa)

Vol. 11, No.1 pp.141-159, 2023.