New Meshless Method Solves Fractional Cable Equations

Researchers have proposed a semi-analytical meshless collocation method designed to solve the distributed-order time-fractional cable equation. The study was published in the journal Boundary Value Problems.

To address the complexity of the distributed-order terms, the proposed method first approximates these terms using Gauss–Legendre quadrature. This step effectively reduces the continuous distribution of fractional orders into a finite weighted sum of standard fractional derivatives.

For the temporal discretization of the resulting Riemann–Liouville derivatives, the study employs a second-order weighted and shifted Grünwald scheme. The authors established the stability and convergence of the semi-discrete formulation through an energy-based analysis.

The spatial discretization utilizes the backward substitution method as a semi-analytical meshless collocation technique. In this approach, boundary conditions are treated separately from the governing equation. The method begins by constructing an approximation directly from the prescribed boundary data. A corrective function is then introduced to enforce the governing equation while preserving homogeneous boundary conditions. The final numerical solution is obtained by combining this boundary approximation with the corrective component.

Numerical results presented in the paper validate the theoretical findings and demonstrate the accuracy of the proposed method. The study confirms that the semi-analytical meshless collocation technique effectively handles the distributed-order time-fractional cable equation.

The article is available as an open access publication. It is part of the early sharing initiative, providing faster access to peer-reviewed, accepted research.

Subject of Research: Numerical methods for fractional partial differential equations

Article Title: A semi-analytical meshless collocation technique for the distributed-order time-fractional cable equation

Article References: Nikan, O. (2026). A semi-analytical meshless collocation technique for the distributed-order time-fractional cable equation. Boundary Value Problems. https://doi.org/10.1186/s13661-026-02360-3

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DOI: 10.1186/s13661-026-02360-3

Keywords: fractional calculus, meshless methods, cable equation, numerical analysis, collocation technique, distributed-order, time-fractional, boundary value problems, semi-analytical, meshless, collocation, technique