A robust framework for solving PDEs: Biorthogonal spline wavelet methods

This paper presents a new numerical approach for solving the partial differential equations (PDEs), focusing on the Diffusion equation. The method combines a collocation approach with wavelet techniques to achieve high accuracy in approximating solutions. A detailed framework for the proposed method, explaining the discretization process at multiple collocation points and the formulation of the resulting system of linear equations is provided. An implementation is conducted to demonstrate the method’s effectiveness in capturing the complex behaviors typical of the model studied. Comparisons with analytical solutions underscore the robustness and precision of the technique, paving the way for its application in diverse fields such as physics, finance, and engineering.