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

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.