Differentiable functions on modules and the equation $grad(w)=Mgrad(v)$

TL;DR

This paper extends the concept of differentiability to modules over finite-dimensional algebras, provides explicit formulas for such functions, and solves related partial differential equations with boundary conditions.

Contribution

It introduces a new framework for $A$-differentiable functions on modules and derives explicit formulas, including solutions to specific PDEs involving matrix equations.

Findings

Explicit formulas for $A$-differentiable functions on modules.

Complete solutions to the equation $grad(w)=Mgrad(v)$.

Existence and uniqueness results for boundary value problems of generalized Laplace equations.

Abstract

Let $A$ be a finite-dimensional, commutative algebra over $\mathbb{R}$ or $\mathbb{C}$. The notion of $A$-differentiable functions on $A$ is extended to the notion of $A$-differentiable functions on a finitely generated $A$-module $B$. Let $U$ be an open, bounded and convex subset of $B$. When $A$ is singly generated and $B$ is arbitrary or $A$ is arbitrary and $B$ is a free module, an explicit formula for an $A$-differentiable functions on $U$, of a prescribed class of differentiability, is given in terms of real or complex differentiable functions. It appears, even in case of real algebras, that certain components of $A$-differentiable function are of higher differentiability than the function itself. Let $M$ be a constant, square matrix. Using the aforementioned formula we find the complete solution of the equation $grad(w)=Mgrad(v)$. The boundary value problem for generalized Laplace equations $M\nabla^2 v=\nabla^2v M^{\intercal}$ is formulated and it is proved that for the given boundary data there exists an unique solution, for which a formula is provided.