A novel Bernstein operational matrix approach for tempered fractional differential equations: Convergence and stability analysis
Document Type
Article
Source of Publication
Aims Mathematics
Publication Date
1-1-2026
Abstract
Tempered fractional differential equations (TFDEs) incorporate exponential decay into fractional operators to account for truncated memory and semi-long-range dependence in a variety of applications, including anomalous diffusion, viscoelasticity, transport phenomena, geophysical processes, and financial dynamics. In this work, a tempered fractional Bernstein method (TFBM) was proposed for the numerical solution of TFDEs involving Caputo-type derivatives. The proposed formulation combined a Bernstein polynomial approximation with an analytic representation of the Caputo–tempered fractional derivative through operational matrices. On this basis, two collocation-based variants were developed, namely, a Chebyshev-type method (TFBM-C) and a Legendre-type method (TFBM-L). For the linear setting, a convergence analysis established norm convergence of the numerical solution to the exact solution as the polynomial degree increased under standard stability and consistency assumptions. Stability was investigated under perturbations in the forcing term as well as under combined perturbations in the system matrix and righthand side, and explicit normwise error bounds were derived using classical matrix perturbation theory. Numerical experiments involving linear and nonlinear TFDEs, weakly singular solutions, multi-term operators, and benchmark test problems demonstrated that the proposed methods achieve higher accuracy than finite-difference and shifted Legendre operational matrix schemes while maintaining low computational cost.
DOI Link
ISSN
Publisher
American Institute of Mathematical Sciences (AIMS)
Volume
11
Issue
4
First Page
10311
Last Page
10341
Disciplines
Mathematics
Keywords
Bernstein operational matrices, Caputo fractional derivative, convergence analysis, error estimate, stability analysis, tempered fractional differential equations
Scopus ID
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Al Hallak, Jalal; Alshbool, Mohammed; Hashim, Ishak; Ismail, Eddie Shahril; and Momani, Shaher, "A novel Bernstein operational matrix approach for tempered fractional differential equations: Convergence and stability analysis" (2026). All Works. 7989.
https://zuscholars.zu.ac.ae/works/7989
Indexed in Scopus
yes
Open Access
yes
Open Access Type
Gold: This publication is openly available in an open access journal/series