Efficient Homotopy Technique for the Computation of Bratu Differential Equation

Abstract

In this paper, we adopt a variant of the Homotopy Analysis Method (HAM) for the numerical solutions of the Bratu differential equation. A predictor-corrector approach essentially tracks the solutions of a homotopy path by using small steps together with a tangent vector derived from the system to determine a new point by a refinement of the previous point. An improved Euler step acts as a predictor along the track, followed by a Newton corrector. The procedure which involves the solution of systems of nonlinear differential equations gradually moves along the tangent of a solution path and subsequently arrives at a path of acceptable solutions of the target system. An attractive feature of this approach is the improvement in convergence resulting from the achievement of a global quadratic rate of convergence. Numerical results are compared with closed form solutions taken from literature. From these results, the method can be relied upon for applications to resolve complicated nonlinearities arising from problems in mathematical physics and engineering.