Approximate Solutions of Bratu-Type Boundary Value Problems

Abstract

The Bratu's equation is a strongly nonlinear second-order ordinary differential equation that arises in electrospinning process and models temperature distribution within a flame in combustion theory. Bratu-type equations are used to simulate the ignition of flammable gases and flame propagation. In this paper, two methods are proposed to obtain highly accurate and reliable approximate solutions of Bratu-type boundary value problems. The first technique is a power series method which is based on the generalised Cauchy product that simplifies the difficulty associated with the nonlinear terms. Subsequently, explicit recurrence relations for the expansion coefficients of the series solutions are obtained. The second approach uses a twelfth-order second derivative backward differentiation formula that is implemented as a boundary value method. This numerical method is referred to as second derivative backward differentiation boundary value method. Three examples are given to illustrate the effectiveness, reliability, and accuracy of the proposed methods. The results obtained from both methods are in excellent agreement with the known exact solution. Comparison of the approximate and exact solutions shows that the proposed methods are reliable and accurate in solving a class of strongly nonlinear boundary value problems of Bratu-type.