Ordinary Complex Differential Equations with Applications in Science and Engineering

TL;DR

This paper explores complex differential equations, establishing their fundamental properties, introducing a novel solution method using Homotopy Perturbation, and demonstrating applications in engineering and complex geometry.

Contribution

It is the first to apply Homotopy Perturbation Method to solve linear complex differential equations and provides comprehensive analytical solutions using Laurent series expansions.

Findings

Proved existence and uniqueness of solutions for complex differential equations.

First implementation of Homotopy Perturbation Method in the complex plane.

Applied methods to engineering problems like airfoil design and Schwarz-Christoffel transformation.

Abstract

In this work, we spotted the light on one of the really important concepts and turned it into a mathematical branch instead of separate equations studied individually in different specializations of science. The existence and uniqueness of solutions for complex differential equations have been proved with many mathematical generalized tools. Homotopy Perturbation Method has been used and implemented as a method for solving linear complex differential equations with which is the first time such a method used to solve an equation in the complex plane. An analytical method of solutions has been investigated deeply with complex series solutions have generalized also using Laurent series expansions. Some really important application in engineering and complex geometry has been implemented with deep understand- ing, such as Airfoil application for airplane and spaceships wings and the fans of jet engines, and the Schwarz-Christoffel transformation.