Differentiable functions on modules and equation $\mathrm{grad}\,(w)=\mathsf{M}\,\mathrm{grad}\,(v)$

Let $A$ be a finite-dimensional, commutative algebra over $\mathbb{R}$ or $\mathbb{C}$. We extend the notion of $A$-differentiable functions on $A$ and develop a theory of $A$-differentiable functions on finitely generated $A$-modules. Let $U$ be an open, bounded and convex subset of such a module. We give an explicit formula for $A$-differentiable functions on $U$ of prescribed class of differentiability in terms of real or complex differentiable functions, in the case when $A$ is singly generated and the module is arbitrary and in the case when $A$ is arbitrary and the module is free. We prove that certain components of $A$-differentiable function are of higher differentiability than the function itself.
Let $\mathsf{M}$ be a constant, square matrix. Using the formula mentioned above, we find a complete description of solutions of the equation $\mathrm{grad}\,(w)=\mathsf{M}\,\mathrm{grad}\,(v)$.
We formulate the boundary value problem for generalized Laplace equations $\mathsf{M}\,\nabla^2 v=\nabla^2v \mathsf{M}^{\mathsf{T}}$ and prove that for given boundary data there exists a unique solution, for which we provide a formula.