Polynomial solution of singular differential equations using Weighted Sobolev gradients

摘要

In this work, we proposed a new scheme based on Sobolev gradient approach for finding an approximate polynomial solution of certain first and second order ordinary differential equations. Continuous function instead of discretized differential operator is used to avoid numerical issues posed by the size of grid. For example, a simple first order equation is solved using different polynomial basis functions to illustrate the effectiveness of the algorithm. Then the theory of weighted Sobolev gradients is used for the singular Legendre's equation. Numerical experiments indicate that the new algorithm is more efficient than the previous algorithms discussed in the literature on the subject.