Integration of PDEs by differential geometric means

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 Sherring and Prince (1992 Trans. Am. Math. Soc. 433 453) to integrate this integrable Vessiot sub-distribution. The method is successfully applied to a large class of linear and nonlinear PDEs.