Open Access

ARTICLE

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

Junseo Lee¹, Bongsoo Jang¹, Umer Saeed^1,2,*

1 Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan, 44919, Republic of Korea
2 NUST Institute of Civil Engineering, School of Civil and Environmental Engineering, National University of Sciences and Technology (NUST), Sector H-12, Islamabad, 44000, Pakistan

* Corresponding Author: Umer Saeed. Email:

(This article belongs to the Special Issue: Analytical and Numerical Solution of the Fractional Differential Equation)

Computer Modeling in Engineering & Sciences 2025, 144(2), 2165-2189. https://doi.org/10.32604/cmes.2025.069023

Received 12 June 2025; Accepted 30 July 2025; Issue published 31 August 2025

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 space-time 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.

---

Keywords

---