Cubic Trigonometric B-Spline VO-L1 Collocation for Variable-Order Caputo Diffusion

Articles in Press

Document Type : Research article

Authors

1 Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran

2 School of Mathematics and Computer Science, Damghan University, Damghan, Iran

Abstract

Variable-order (VO) fractional diffusion equations provide a flexible framework for modeling transport phenomena with time-dependent memory effects‎. ‎However‎, ‎the nonlocality of fractional operators‎, ‎combined with the variable-order structure‎, ‎makes the development of stable and accurate numerical schemes considerably more challenging‎. ‎In this paper‎, ‎a fully discrete numerical method is developed for VO time-fractional diffusion equations involving the Caputo fractional derivative‎. ‎The spatial discretization is based on cubic trigonometric B-spline collocation‎, ‎which provides a smooth approximation to the spatial solution‎. ‎For the temporal discretization‎, ‎an adapted VO--L1 scheme is employed to approximate the VO Caputo derivative‎. ‎The variable-order function $\lambda(t)$ incorporates the time-dependent memory effects into the model‎, ‎and the global temporal convergence rate is of order $2-\lambda_{\max}$‎, ‎where $\lambda_{\max}=\max_{0\le t\le T_F}\lambda(t)$‎. ‎By incorporating the Dirichlet boundary conditions into the collocation formulation‎, ‎the fully discrete system reduces at each time level to a tridiagonal linear algebraic system‎, ‎which can be solved efficiently‎. ‎A matrix-based stability analysis is carried out under suitable assumptions on the coefficient matrix‎. ‎Moreover‎, ‎a convergence analysis is provided by combining the consistency of the temporal and spatial discretizations with the stability of the overall scheme‎. ‎Under appropriate regularity assumptions‎, ‎the proposed method satisfies the global maximum-norm error estimate $\|\mathbf{e}^{n}\|_{\infty} \leq C\bigl(h_x^2+h_t^{\,2-\lambda_{\max}}\bigr)$‎, ‎where $C$ is independent of $h_x$ and $h_t$‎. ‎Several numerical experiments are conducted to validate the theoretical findings and assess the accuracy and efficiency of the proposed approach‎. ‎The results support the theoretical analysis and demonstrate the effectiveness of the cubic trigonometric B-spline VO--L1 collocation scheme for solving VO fractional diffusion problems with time-dependent memory effects‎.

Keywords

Main Subjects

Analytical and Numerical Solutions for Nonlinear Equations

Articles in Press, Accepted Manuscript
Available Online from 03 October 2026

Files

History

  • Receive Date: 31 August 2026
  • Revise Date: 23 September 2026
  • Accept Date: 02 October 2026
  • Publish Date: 03 October 2026

Share

How to cite

Statistics