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Solving linear and nonlinear Caputo fractional differential equations with a quantum pseudo-spectral approach

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  • Abbasbandy, Saeid

Abstract

Linear and nonlinear Caputo time-fractional differential equations play a fundamental role in pure and applied mathematics as well as theoretical physics. This article develops a hybrid methodology that combines quantum computing paradigms with spectral methods to solve such equations, employing shifted fractional Chebyshev polynomials as basis functions. The simultaneous treatment of linear and nonlinear fractional equations requires careful selection of both basis functions and collocation points. This choice proves essential for avoiding the chain rule complication inherent in Caputo’s derivative formulation. Crucially, the chosen basis functions generate a triangular operational matrix, thereby improving both the accuracy and computational efficiency of the pseudo-spectral approach. Within our computational framework, the solution at the terminal time is encoded as a final quantum state. We demonstrate the method’s efficacy through numerical experiments and comparative analysis with existing approaches.

Suggested Citation

  • Abbasbandy, Saeid, 2026. "Solving linear and nonlinear Caputo fractional differential equations with a quantum pseudo-spectral approach," Applied Mathematics and Computation, Elsevier, vol. 511(C).
    • Handle: RePEc:eee:apmaco:v:511:y:2026:i:c:s0096300325004515
    DOI: 10.1016/j.amc.2025.129726
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    References listed on IDEAS

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    1. Muhammad Imran Liaqat & Adnan Khan & Hafiz Muhammad Anjum & Gregory Abe-I-Kpeng & Emad E. Mahmoud & S. A. Edalatpanah, 2024. "A Novel Efficient Approach for Solving Nonlinear Caputo Fractional Differential Equations," Advances in Mathematical Physics, Hindawi, vol. 2024, pages 1-22, November.
    2. Muhammad Asif Zahoor Raja & Junaid Ali Khan & Ijaz Mansoor Qureshi, 2011. "Solution of Fractional Order System of Bagley-Torvik Equation Using Evolutionary Computational Intelligence," Mathematical Problems in Engineering, Hindawi, vol. 2011, pages 1-18, March.
    3. Qinwu Xu & Zhoushun Zheng, 2019. "Spectral Collocation Method for Fractional Differential/Integral Equations with Generalized Fractional Operator," International Journal of Differential Equations, Hindawi, vol. 2019, pages 1-14, January.
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