Convergence and Ulam-Hyers-Rassias Stability Analyses of Numerical Solutions of Bratu Type Equations using Picard Method
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The Bratu equation is a basic nonlinear boundary value problem with important applications to fuel ignition, thermal combustion, and nanotechnology. The current article presents a new application of the Picard iterative technique to find accurate approximate solutions for this equation. Firstly, the conditions for existence and uniqueness of the solutions are determined. Furthermore, the article provides an explicit formulation of Picard’s scheme for second-order ordinary differential equations and its particular implementation to the Bratu type problem. The iterative solutions obtained are analyzed thoroughly for convergence and are proved to be Ulam-Hyers-Rassias stable, a very strong type of stability not yet known for these kind of solutions. Numerical tests for three cases ( , 2, and the critical value ) are shown to exemplify outstanding accuracy of the proposed approach. A thorough comparison with known techniques including the Adomian Decomposition Method, Homotopy Perturbation Method, and Variational Iteration Method indicates that the Picard iterative scheme is much better in terms of accuracy since its maximum absolute errors are much smaller
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