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Singularity-removing Chebyshev collocation methods for nonlinear fractional differential equations with blow-up

Author

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  • Jia, Chengwang
  • Zhang, Hongwei
  • Gao, Guang-hua
  • Wang, Tongke

Abstract

Two singularity-removing Chebyshev collocation methods are proposed for discretizing nonlinear fractional differential equation with singular solution at the initial and terminal points. By reformulating the problem into the Volterra integral equation of the second kind, we obtain the fractional series solution around the origin via successive approximation, which depicts the initial singularity of the unknown solution accurately. By removing the singularity at the origin, we design two Chebyshev collocation methods to solve the equivalent Volterra integral equation and Hadamard finite-part integral equation on a regular interval. The convergence of the scheme for the equivalent Volterra integral form is proved. For blow-up problem, we perform the Padé technique for the obtained series solution to detect the blow-up time. We also explore the blow-up behavior of the solution for three typical types of nonlinear term. By removing the dominant term of the blow-up behavior from the equation, the accuracy of the collocation methods is improved significantly. Finally, numerical examples demonstrate the high efficiency and exponential convergence of the singularity-removing Chebyshev collocation methods.

Suggested Citation

  • Jia, Chengwang & Zhang, Hongwei & Gao, Guang-hua & Wang, Tongke, 2026. "Singularity-removing Chebyshev collocation methods for nonlinear fractional differential equations with blow-up," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 239(C), pages 192-210.
    • Handle: RePEc:eee:matcom:v:239:y:2026:i:c:p:192-210
    DOI: 10.1016/j.matcom.2025.05.014
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    References listed on IDEAS

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    1. Postavaru, Octavian, 2023. "An efficient numerical method based on Fibonacci polynomials to solve fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 406-422.
    2. Manohara, G. & Kumbinarasaiah, S., 2024. "Numerical approximation of fractional SEIR epidemic model of measles and smoking model by using Fibonacci wavelets operational matrix approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 221(C), pages 358-396.
    3. Wang, Yuan-Ming & Xie, Bo, 2023. "A fourth-order fractional Adams-type implicit–explicit method for nonlinear fractional ordinary differential equations with weakly singular solutions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 21-48.
    4. Li, Yuyu & Wang, Tongke & Gao, Guang-hua, 2023. "The asymptotic solutions of two-term linear fractional differential equations via Laplace transform," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 394-412.
    5. Cardone, Angelamaria & Conte, Dajana, 2020. "Stability analysis of spline collocation methods for fractional differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 178(C), pages 501-514.
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    1. Erfanifar, Raziyeh & Hajarian, Masoud, 2026. "A study on the convergence and efficiency of a novel seventh-order iterative method to solve systems of nonlinear equations with electrical engineering applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 241(PB), pages 650-664.

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