Article: An efficient cubic spline method with convergence analysis for nonlinear time-fractional reaction-diffusion problems exhibiting singularities Journal: International Journal of Applied Nonlinear Science (IJANS) 2025 Vol.5 No.2 pp.111 - 131 Abstract: This study presents a high-order numerical method for nonlinear time-fractional reaction-diffusion equations with an initial singularity. The L1 method on a graded mesh is used to approximate the Caputo fractional derivative, while the cubic spline approach is used to discretise the spatial variable. We effectively employ our technique to reduce the consequences of the singularity. The resulting nonlinear system of equations is solved by the Newton linearisation method. We present convergence and stability of the system using Fourier analysis. The approach has been shown to be convergent of order O(N - min{2 - α, rα}, h2) in temporal and spatial directions. Here, h represents the spatial step size, N the number of time grid points, and r the grading parameter, with α ∈ (0, 1) signifying the order of the fractional derivative. In addition, the Caputo derivative kernel function is evaluated by means of the sum-of-exponentials (SOE) approach, resulting in a rapid numerical operation and reduced computational costs. In order to verify our theoretical conclusions and show how the mesh grading affects the convergence order when working with a non-smooth solution to the problem, we finally carry out numerical experiments. Inderscience Publishers - linking academia, business and industry through research

Title: An efficient cubic spline method with convergence analysis for nonlinear time-fractional reaction-diffusion problems exhibiting singularities

Authors: Richa Singh; Priyanka

Addresses: Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh, India ' Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh, India

Abstract: This study presents a high-order numerical method for nonlinear time-fractional reaction-diffusion equations with an initial singularity. The L1 method on a graded mesh is used to approximate the Caputo fractional derivative, while the cubic spline approach is used to discretise the spatial variable. We effectively employ our technique to reduce the consequences of the singularity. The resulting nonlinear system of equations is solved by the Newton linearisation method. We present convergence and stability of the system using Fourier analysis. The approach has been shown to be convergent of order O(N^{- min{2 - α, rα}}, h²) in temporal and spatial directions. Here, h represents the spatial step size, N the number of time grid points, and r the grading parameter, with α ∈ (0, 1) signifying the order of the fractional derivative. In addition, the Caputo derivative kernel function is evaluated by means of the sum-of-exponentials (SOE) approach, resulting in a rapid numerical operation and reduced computational costs. In order to verify our theoretical conclusions and show how the mesh grading affects the convergence order when working with a non-smooth solution to the problem, we finally carry out numerical experiments.

Keywords: cubic spline difference scheme; Caputo derivative; L1 method; graded mesh; convergence analysis.

DOI: 10.1504/IJANS.2025.151257

International Journal of Applied Nonlinear Science, 2025 Vol.5 No.2, pp.111 - 131

Received: 21 Aug 2024
Accepted: 07 Oct 2024
Published online: 19 Jan 2026 *

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Title: An efficient cubic spline method with convergence analysis for nonlinear time-fractional reaction-diffusion problems exhibiting singularities

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