Pre-Twisted Calculus and Differential Equations
24 Pages Posted: 28 Jan 2023
Abstract
In this paper we introduce the \emph{$\varphi\mathbb{A}$-differentiability} for functions $f:U\subset \mathbb R^{k}\to\mathbb R^{n}$, where $U$ is an open set, $\mathbb A$ is the linear space $\mathbb R^{n}$ endowed witha unital associative commutative algebra product, and $\varphi:U\subset \mathbb R^{k}\to\mathbb A$ is a differentiablefunction in the usual sense. We call it \emph{pre-twisted differentiability}. With respect to the $\varphi\mathbb{A}$-differentiability we introduce: a) a type Cauchy-Riemann equations, which serve as $\varphi\mathbb{A}$-differentiability criteria, b) a Cauchy-integral theorem, and c) \emph{$\varphi\mathbb A$-differential equations}, which can be used to solve systems of non-autonomous ODEs. It has recently been shown that the $\varphi\mathbb A$-differentiable functions define a complete solutions for the PDEs of the form $Au_{xx}+Bu_{xy}+Cu_{yy}=0$, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of $\varphi\mathbb A$-differential equations define solutions for PDE systems.
Keywords: Partial Differential Equations, Generalized Cauchy-Riemann equations, Lorch differentiability, Exact solutions forPDEs
Suggested Citation: Suggested Citation