Pre-Twisted Calculus and Differential Equations

24 Pages Posted: 28 Jan 2023

See all articles by Elifalet López-González

Elifalet López-González

affiliation not provided to SSRN

Edgar Alonso Martínez-García

affiliation not provided to SSRN

Rafael Torres-Córdoba

affiliation not provided to SSRN

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

López-González, Elifalet and Martínez-García, Edgar Alonso and Torres-Córdoba, Rafael, Pre-Twisted Calculus and Differential Equations. Available at SSRN: https://ssrn.com/abstract=4340449 or http://dx.doi.org/10.2139/ssrn.4340449

Elifalet López-González (Contact Author)

affiliation not provided to SSRN ( email )

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Edgar Alonso Martínez-García

affiliation not provided to SSRN ( email )

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Rafael Torres-Córdoba

affiliation not provided to SSRN ( email )

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