Legendre Ritz-Least squares method for solving a class of Bratu-types

A numerical method for solving initial value problems of Bratu–types is presented. Initial value problems of Bratu-type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. In this manuscript a new method is introduced for solving initial value problems of Bratu-type. By transforming the Bratu’s problem into an optimization problem and using polynomial basis functions, we obtain the system of algebraic equation. Then, we solve the system of nonlinear algebraic equation and obtain the coefficients of polynomial expansion. Also, we show the convergence of the method. Some numerical examples are presented which illustrate the theoretical results and the performance of the method.

Keywords:

Optimization problem , Bratu’s problem , Initial value problems