A generalized Haar wavelet-homotopy method for solving steady and unsteady differential and partial differential equations

This paper introduces the Haar wavelet homotopy collocation method (HWHCM), a novel approach for numerical approximation of steady and unsteady problems governed by linear or nonlinear ordinary or partial differential equations. The proposed method combines the principles of homotopy analysis and the generalized Haar orthogonal wavelet, which can be considered as an improvement of the wavelet homotopy analysis method (WHAM) for the reduction of calculation complexity and the storage usage. Additionally, the HWHCM method enhances the traditional Haar wavelet collocation method (HWCM) by incorporating homotopy iterative techniques, resulting in improved convergence, stability, and computational accuracy. It offers flexibility in adjusting homotopy parameters and resolution levels, allowing for adaptive balance between accuracy and efficiency tailored to specific problem requirements. The effectiveness and accuracy of the HWHCM are evaluated using rigorous criteria such as relative variance and maximum error. Through its successful application to initial value problems, Poisson equation, Burgers equation, sine-Gordon equation and Schrödinger equation, the numerical results support the significant advantages and validity of the HWHCM, confirming its superior accuracy.