An In-Depth Analysis of Wavelet Transform Performance in Solving the Burgers–Fisher Equation: A Foundation for Improving Accuracy and Efficiency in Numerical Computation

Abstract

The Burgers–Fisher equation is a nonlinear partial differential equation with a hybrid hyperbolic–parabolic nature and arises in many areas of applied and physical sciences, including fluid mechanics, gas dynamics, and reaction–diffusion processes. In this study, the effectiveness of the Haar wavelet transform and the Daubechies (DB8) wavelet transform is investigated for obtaining numerical solutions of the Burgers–Fisher equation. Numerical experiments demonstrate that both wavelet-based approaches provide efficient approximations and satisfactory convergence behavior. However, the Daubechies (DB8) transform achieves significantly higher accuracy and better numerical stability than the Haar transform. The obtained results show that the DB8 wavelet produces errors several orders of magnitude smaller than those generated by the Haar wavelet, yielding excellent agreement with the exact solution. This improvement is attributed to the smoother basis functions and higher vanishing moments of the DB8 wavelet, which allow a more effective representation of the nonlinear characteristics of the Burgers–Fisher equation. Therefore, the proposed wavelet-based approach, particularly the DB8 transform, is shown to be a reliable and accurate technique for solving nonlinear partial differential equations.