Exploring the Efficacy of the Weighted Average Method for Solving Nonlinear Partial Differential Equations: A Study on the Burger-Fisher Equation

This study explores the application of the weighted average method for solving the Burger-Fisher equation, a nonlinear partial differential equation (PDE) of significant interest in various scientific disciplines. Nonlinear PDEs, such as the Burger-Fisher equation, are fundamental in describing complex physical, biological, and engineering phenomena but pose challenges for both analytical and numerical solutions. The weighted average method, known for its ability to converge rapidly to exact solutions, offers a promising approach for tackling such equations. By discretizing both spatial and temporal derivatives using a combination of forward, backward, and central differences, the method approximates solutions with high accuracy and stability. Conducting convergence and stability analyses, this study elucidates the computational requirements and performance characteristics of the weighted average method. Utilizing mathematical software like MATLAB and MAPLE, the method's implementation involves solving a tridiagonal matrix system at each time step. Comparison between numerical solutions obtained using the method and exact solutions demonstrates the method's accuracy, with negligible errors observed. Visual representations further illustrate the close agreement between the numerical and exact solutions, validating the method's reliability for practical applications. The study's findings underscore the practical utility of the weighted average method in solving the Burger-Fisher equation and similar nonlinear PDEs. Its ability to accurately approximate solutions while maintaining stability highlights its efficacy as a computational tool for addressing complex mathematical problems across diverse scientific and engineering fields. The study contributes to advancing the understanding and application of numerical methods for nonlinear PDEs, offering valuable insights for researchers and practitioners seeking precise and reliable solutions to complex mathematical models. Overall, the study emphasizes the importance of fine-tuning numerical parameters and leveraging computational resources to achieve optimal accuracy when utilizing the weighted average method for solving nonlinear PDEs.