Initial value problems in Clifford-type analysis

TL;DR

This paper addresses initial value problems involving Clifford-type algebra-valued functions, providing conditions under which associated operators can be used to solve these problems using the technique of associated spaces.

Contribution

It introduces sufficient conditions for operators in Clifford analysis to form an associated pair, enabling the solution of initial value problems with Clifford algebra-valued functions.

Findings

Established conditions for associated operators in Clifford analysis.

Demonstrated the use of associated spaces to solve initial value problems.

Extended the framework of Clifford-type analysis for differential equations.

Abstract

We consider an initial value problem of type $$ \frac{\partial u}{\partial t}={\cal F}(t,x,u,\partial_j u), \quad u(0,x)=\phi(x), $$ where $t$ is the time, $x \in \mathbb{R}^n $ and $u_0$ is a Clifford type algebra-valued function satisfying ${\bf D}u=\displaystyle\sum_{j=0}^{n}\lambda_j(x)e_j\partial_ju = 0$, $\lambda_j(x)\in \mathbb{R} $ for all $j$. We will solve this problem using the technique of associated spaces. In order to do that, we give sufficient conditions on the coefficients of the operators ${\cal F}$ and ${\bf D}$, where ${\cal F}(u)= \displaystyle\sum_{i=0}^{n}A^{(i)}(x)\displaystyle\partial_iu$ for $A^{(i)}(x) \in \mathbb{R}$ or $A^{(i)}(x)$ belonging to a Clifford-type algebra, such that these operators are an associated pair.