Some properties of band matrix and its application to the numerical solution one-dimensional Bratu's problem

A Class of new methods based on a septic non-polynomial spline function for the numerical solution one-dimensional Bratu's problem are presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Associated boundary formulas are developed. Convergence analysis of these methods is discussed. Numerical results are given to illustrate the efficiency of methods.