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A Legendre interpolation method for the nonlinear variable-order fractional advection–diffusion equations

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  • Qu, Hai-Dong
  • Liu, Xuan
  • Yang, Xiao-peng

Abstract

This paper develops a Legendre interpolation method (LIM) for solving initial–boundary value problems of nonlinear variable-order fractional advection–diffusion equations with two-sided derivatives. The proposed approach combines a Crank–Nicolson temporal discretization with spatial approximation using shifted Legendre polynomials. We establish comprehensive stability and convergence analyses for both semi-discrete and fully discrete schemes, proving second-order temporal convergence and exponential spatial convergence. Numerical experiments validate the theoretical findings and demonstrate the method’s high accuracy across various test cases, including nonlinear problems with variable coefficients. The method effectively handles complex variable-order fractional operators while maintaining computational efficiency through optimized matrix structures.

Suggested Citation

  • Qu, Hai-Dong & Liu, Xuan & Yang, Xiao-peng, 2026. "A Legendre interpolation method for the nonlinear variable-order fractional advection–diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 204(C).
    • Handle: RePEc:eee:chsofr:v:204:y:2026:i:c:s0960077925017849
    DOI: 10.1016/j.chaos.2025.117770
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    References listed on IDEAS

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    1. Amr M. S. Mahdy & Mohamed A. Abdou & Doaa Sh. Mohamed, 2023. "Computational Methods for Solving Higher-Order (1+1) Dimensional Mixed-Difference Integro-Differential Equations with Variable Coefficients," Mathematics, MDPI, vol. 11(9), pages 1-25, April.
    2. Ayazi, N. & Mokhtary, P. & Moghaddam, B. Parsa, 2024. "Efficiently solving fractional delay differential equations of variable order via an adjusted spectral element approach," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
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