A novel Bernstein operational matrix approach for tempered fractional differential equations: Convergence and stability analysis

Jalal Al Hallak; Mohammed Alshbool; Ishak Hashim; Eddie Shahril Ismail; Shaher Momani; Jalal Al Hallak; Mohammed Alshbool; Ishak Hashim; Eddie Shahril Ismail; Shaher Momani

doi:10.3934/math.2026426

AIMS Mathematics

2026, Volume 11, Issue 4: 10311-10341. doi: 10.3934/math.2026426

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A novel Bernstein operational matrix approach for tempered fractional differential equations: Convergence and stability analysis

1.
Department of Mathematical Sciences, Universiti Kebangsaan Malaysia, Bangi, 43600, Selangor, Malaysia
2.
Department of Mathematics and Statistics, Zayed University, Abu Dhabi, United Arab Emirates
3.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, P.O. Box 346, United Arab Emirates
4.
Department of Mathematics, Faculty of Science, University of Jordan, Amman, 11942, Jordan

Received: 17 January 2026 Revised: 31 March 2026 Accepted: 08 April 2026 Published: 15 April 2026
MSC : 65L05, 26A33

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 norm-wise 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.
- tempered fractional differential equations,
- Bernstein operational matrices,
- Caputo fractional derivative,
- convergence analysis,
- stability analysis,
- error estimate
Citation: Jalal Al Hallak, Mohammed Alshbool, Ishak Hashim, Eddie Shahril Ismail, Shaher Momani. A novel Bernstein operational matrix approach for tempered fractional differential equations: Convergence and stability analysis[J]. AIMS Mathematics, 2026, 11(4): 10311-10341. doi: 10.3934/math.2026426

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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 norm-wise 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.

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