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  • Open Access

    ARTICLE

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

    Junseo Lee1, Bongsoo Jang1, Umer Saeed1,2,*

    CMES-Computer Modeling in Engineering & Sciences, Vol.144, No.2, pp. 2165-2189, 2025, DOI:10.32604/cmes.2025.069023 - 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 More >

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  • Open Access

    ARTICLE

    Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials

    Kamal Shah1,2, Hafsa Naz2, Thabet Abdeljawad1,3,*, Aziz Khan1, Manar A. Alqudah4

    CMES-Computer Modeling in Engineering & Sciences, Vol.134, No.2, pp. 941-955, 2023, DOI:10.32604/cmes.2022.021483 - 31 August 2022

    Abstract In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices. Further, operational matrices are constructed using variable order differentiation and integration. We are finding the operational matrices of variable order differentiation and integration by omitting the discretization of data. With the help of aforesaid matrices, considered FDEs are converted to algebraic equations of Sylvester type. Finally, the algebraic equations we get are More >

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