Series solutions of partial differential equations using the complex integral method

W Robin

doi:10.12988/imf.2014.48145

Abstract

The complex integral method for solving ordinary differential equations in series [3, 7, 8] is extended to cover the series solution of partial differential equations also. The means of this extension is straightforward, with both ‘ordinary’ and ‘Frobenius’ multiple variable power series being dealt with. Standard examples of the application of the extended method(s) to first-order, second-order, third-order and simultaneous partial differential equations are provided throughout. Examples also include the series solution of a non-linear partial differential equation and the consideration of series solutions with negative powers.

Highlights

In this paper we extend the application of the complex integral method for solving ordinary differential equations (ODE) in series [3, 7, 8] to cover the power series solution of partial differential equations (PDE)
First-Order PDE with Two Independent Variables. In this and the following section, we present a number of examples, including some from the literature involving the differential transform method, which acts as benchmark or comparison for the current method, involving PDE with two independent variables
Herrera’s complex integral method appears to have sufficient flexibility to enable its generalization to the production of series solutions to most types of differential equation

Summary

Introduction

In this paper we extend the application of the complex integral method for solving ordinary differential equations (ODE) in series [3, 7, 8] to cover the power series solution of partial differential equations (PDE) . The means of this extension, from the single independent variable to the several independent variables case, is quite straightforward, as we apply the procedure variable by variable to the particular assumed multiple variable series format in each case, with both the ‘ordinary’ and ‘Frobenius’ multiple variable series being dealt with. The paper is rounded-off with a short conclusions and discussion section, section 7

The Basic Formalism for Two Independent variables

First-Order PDE with Two Independent Variables

Second-Order PDE with Two Independent Variables

Second-Order PDE with Three Independent Variables

Solution of PDE Using Frobenius Series

Discussion and Conclusions