Rational hybrid block method for solving Bratu-type boundary value problems

Abstract

This study introduces a novel rational hybrid block method for solving Bratu-type boundary value problems, offering significant improvements in efficiency and accuracy. The method enhances the traditional block hybrid approach by incorporating rational approximations of grid points, which effectively reduce local truncation errors and improve numerical stability. Unlike existing methods, the proposed technique provides higher precision with fewer computational resources, making it particularly advantageous for problems requiring fine resolution. Extensive numerical experimentation on selected Bratu-type problems demonstrates superior performance in terms of convergence and accuracy, especially when using carefully optimized parameters. Moreover, the method’s robustness and adaptability make it well-suited for handling challenging problems, such as those involving bifurcations or steep gradients. These advantages position the method as a powerful tool for solving complex boundary value problems with broad applications in engineering and the physical sciences.