A Meshfree Simulation of a Class of Higher-Order Fractional Differential Equations Using Radial Basis Functions and Caputo Derivative

Javed Khan; Rohul Amin; Arshed Ali; Imtiaz Ahmad; Hadia Atta

Authors

Javed Khan Department of Mathematics, University of Peshawar, Pakistan
Rohul Amin Department of Mathematics, University of Peshawar, Pakistan
Arshed Ali Department of Mathematics, Islamia College Peshawar, Pakistan
Imtiaz Ahmad Institute of Informatics and Computing in Energy (IICE), University Tenaga Nasional (UNITEN), Kajang, Selangor, Malaysia
Hadia Atta Department of Mathematics, Islamia College Peshawar, Pakistan

Keywords:

Radial Basis Functions, Caputo Derivative, Fractional Differential Equations, Meshfree Methods, Finite Difference Method

Abstract

This paper develops an efficient radial basis function (RBF) collocation method for the numerical solution of higher‑order fractional differential equations involving the Caputo derivative of order . The fractional derivative is evaluated using the three-point finite difference method. The proposed approach transforms the governing equation into a linear algebraic system, enabling accurate and computationally efficient numerical simulations. The accuracy and robustness of the method are validated through benchmark problems with fractional orders . Numerical results demonstrate that the method achieves errors ranging from to , depending on the test problem, node density, and choice of RBF. Convergence analysis reveals near‑exponential decay of the error as the number of nodes increases for smooth solutions, with consistent convergence observed for more challenging cases. Comparative studies show that the multiquadric RBF outperforms quintic and spline‑based RBFs, yielding significantly lower errors, often by several orders of magnitude, than classical numerical methods. These findings confirm that the proposed RBF collocation framework is a reliable, accurate, and efficient tool for solving higher‑order fractional differential equations arising in engineering and applied sciences.

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