Pre-twisted algebrizable differential equations

We introduce the \emph{$\varphi\mathbb{A}$-differentiability} for functions $f:U\subset \mathbb R^{k}\to\mathbb A$ where $\mathbb A$ is the linear space $\mathbb R^{n}$ endowed with an algebra product which is unital, associative, commutative, $U$ is an open set, and $\varphi:U\subset \mathbb R^{k}\to\mathbb A$ is a differentiable function in the usual sense. We also introduce the corresponding generalized Cauchy-Riemann equations ($\varphi\mathbb{A}$-CREs), the Cauchy-integral theorem, and the \emph{$\varphi\mathbb A$-differential equations}. The four-dimensional vector fields associated with triangular billiards are $\varphi\mathbb{A}$-differentiable. The \emph{$\varphi\mathbb A$-differential equations} can be used for constructing exact solutions of partial differential equations like the three-dimensional heat equation.