Hybrid Wavelet Methods for Nonlinear Multi-Term Caputo Variable-Order Partial Differential Equations

Abstract

In recent years, variable-order fractional partial differential equations have attracted growing interest due to their enhanced ability to model complex physical phenomena with memory and spatial heterogeneity. However, existing numerical methods often struggle with the computational challenges posed by such equations, especially in nonlinear, multi-term formulations. This study introduces two hybrid numerical methods-the Linear-Sine and Cosine (L1-CAS) and fast-CAS schemes-for solving linear and nonlinear multi-term Caputo variable-order (CVO) fractional partial differential equations. These methods combine CAS wavelet-based spatial discretization with L1 and fast algorithms in the time domain. A key feature of the approach is its ability to efficiently handle fully coupled spacetime variable-order derivatives and nonlinearities through a second-order interpolation technique. In addition, we derive CAS wavelet operational matrices for variable-order integration and for boundary value problems, forming the foundation of the spatial discretization. Numerical experiments confirm the accuracy, stability, and computational efficiency of the proposed methods.