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

Abstract

Introduction: An interesting type of fractional derivatives that has received widespread attention in recent years is the tempered fractional derivatives. These fractional derivatives are a generalization of the well-known fractional derivatives, such as Caputo and Riemann-Liouville. In fact, these derivatives are obtained by multiplying the expressed fractional derivatives by an exponential factor. These fractional derivatives have an additional parameter called λ such that in the case of λ=0, the classical Caputo or Riemann-Liouville fractional derivative is obtained.

Objectives: Employing the Caputo tempered fractional derivative to define time fractional nonlinear Schrödinger equation and a coupled system of nonlinear Schrödinger equations. Applying the orthonormal discrete Chebyshev polynomials (ODCPs) to solve these problems. For this purposes, the operational matrices of ordinary and tempered fractional derivatives of the ODCPs are obtained.

Methods: By representing the problem's solutions in terms of the ODCPs (with some unknown coefficients) and exploiting the expressed operational matrices, along with the collocation strategy, two systems of nonlinear algebraic equations are derived. By solving these systems, the expressed coefficients, and subsequently the solution of the main fractional problems are obtained.

Results: Some numerical examples are investigated to acknowledge the high accuracy of the designed approaches.

Conclusion: The tempered fractional derivative in the Caputo form is utilized to define the time fractional nonlinear Schrödinger equation and a coupled system of nonlinear Schrödinger equations. The ODCPs are used to design a numerical strategy for these problems. To this purpose, some operational matrices for these polynomials are obtained. In the designed procedures, the problem's solution are obtained by solving an algebraic system of equations. These systems are obtained by approximating the solution with the ODCPs and employing the expressed matrix relationships, along with the collocation technique. Some examples are presented to check the validity of the developed algorithms. The reported results acknowledged the high accuracy of the designed schemes.

Keywords: Operational matrices; Orthonormal discrete Chebyshev polynomials; Tempered time fractional coupled nonlinear Schrödinger equations; Tempered time fractional nonlinear Schrödinger.