Pre-twisted calculus and differential equations

Abstract

In this paper we introduce the φA-differentiability for functions f:U⊂R^k→R^n, where U is an open set, A is the linear space R^n endowed with a unital associative commutative algebra product, and φ:U⊂R^k→A is a differentiable function in the usual sense. We call it pre-twisted differentiability. With respect to the φA-differentiability we introduce: (a) a type Cauchy–Riemann equations, which serve as φA-differentiability criteria, (b) a Cauchy-integral theorem, and (c) φA-differential equations, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the φA-differentiable functions define a complete solutions for the PDEs of the form Auxx+Buxy+Cuyy=0, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of φA-differential equations define solutions for linear and nonlinear PDE systems.