The Legendre wavelet method for solving initial value problems of Bratu-type

The aim of this work is to study the Legendre wavelets for the solution of initial value problems of Bratu-type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. The properties of Legendre wavelets together with the Gaussian integration method are used to reduce the problem to the solution of nonlinear algebraic equations. Also a reliable approach for convergence of the Legendre wavelet method when applied to a class of nonlinear Volterra equations is discussed and an error estimation for the proposed method is also introduced. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results have been compared with the exact solution. We finally show the high accuracy and efficiency of the proposed method.