On the Role of Taylor Expansion with Exponential Nonlinearity in BVP

I.K. Youssef , H.S. Mansour

The finite difference method is employed to introduce a consistent nonlinear system of algebraic equations corresponding to the nonlinear boundary value problem (BVP). Taylor expansion is used as a linearization technique to introduce a linear algebraic system of equations approximating the nonlinear system. Solutions of the linearized system are taken as the initiation for the Newton’s Raphson iteration when solving the nonlinear system. Application to Bratu’s problem and similar problems with damping effects have illustrated the efficiency of the treatment. Two numerical examples with their graphical representation are given. The calculated results have illustrated the correctness of the treatment.