An efficient discrete Chebyshev polynomials strategy for tempered time fractional nonlinear Schrödinger problems.

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Fractional Calculus

Background:

• Tempered fractional derivatives generalize classical fractional derivatives (Caputo, Riemann-Liouville) with an added parameter (λ).
• These derivatives are crucial for modeling phenomena with memory and non-local characteristics.

Purpose of the Study:

• To define time fractional nonlinear Schrödinger equations using the Caputo tempered fractional derivative.
• To develop a numerical method employing orthonormal discrete Chebyshev polynomials (ODCPs) for solving these equations.

Main Methods:

• Derivation of operational matrices for ordinary and tempered fractional derivatives of ODCPs.
• Representation of solutions using ODCPs and collocation strategy to form nonlinear algebraic systems.
• Solving the algebraic systems to obtain coefficients and the final solution.

Main Results:

• Numerical examples demonstrate the high accuracy of the proposed method.
• The developed approach effectively solves fractional nonlinear Schrödinger equations.

Conclusions:

• The Caputo tempered fractional derivative and ODCPs provide an effective numerical strategy.
• The method yields accurate solutions for time fractional nonlinear Schrödinger equations and coupled systems.
• The study validates the efficiency and accuracy of the proposed algorithms.