Integration of PDEs by differential geometric means

TL;DR

This paper introduces a geometric approach using Vessiot theory and exterior calculus to solve various PDEs by identifying integrable sub-distributions and applying integrating factors, effectively handling both linear and nonlinear equations.

Contribution

It develops an algorithm to find the largest integrable sub-distributions of PDEs using differential geometric methods, advancing solution techniques for complex equations.

Findings

Successfully applied to a broad class of linear and nonlinear PDEs

Demonstrates effectiveness of geometric methods in PDE integration

Provides a systematic approach for solving evolution equations

Abstract

We use Vessiot theory and exterior calculus to solve partial differential equations(PDEs) of the type uyy = F(x, y,u,ux,uy,uxx,uxy) and associated evolution equations. These equations are represented by the Vessiot distribution of vector fields. We develop and apply an algorithm to find the largest integrable sub-distributions and hence solutions of the PDEs. We then apply the integrating factor technique [19] to integrate this integrable Vessiot sub-distribution. The method is successfully applied to a large class of linear and non-linear PDEs.