Heydari Mohammad Hossein, Baleanu Dumitru
Department of Mathematics, Shiraz University of Technology, Shiraz, Iran.
Department of Computer Science and Mathematics, Lebanese American University, Beirut 13-5053, Lebanon.
J Adv Res. 2025 Sep;75:405-420. doi: 10.1016/j.jare.2024.11.014. Epub 2024 Nov 16.
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.
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.
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.
Some numerical examples are investigated to acknowledge the high accuracy of the designed approaches.
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.
近年来受到广泛关注的一种有趣的分数阶导数类型是 tempered 分数阶导数。这些分数阶导数是著名的分数阶导数（如 Caputo 导数和 Riemann-Liouville 导数）的推广。实际上，这些导数是通过将表示的分数阶导数乘以一个指数因子得到的。这些分数阶导数有一个额外的参数λ，使得在λ = 0的情况下，可以得到经典的 Caputo 或 Riemann-Liouville 分数阶导数。
利用 Caputo tempered 分数阶导数定义时间分数阶非线性薛定谔方程和一个非线性薛定谔方程的耦合系统。应用正交离散切比雪夫多项式（ODCPs）来求解这些问题。为此，得到了 ODCPs 的普通分数阶导数和 tempered 分数阶导数的运算矩阵。
通过用 ODCPs（带有一些未知系数）表示问题的解，并利用所得到的运算矩阵，结合配置策略，导出两个非线性代数方程组。通过求解这些方程组，得到所表示的系数，进而得到主要分数阶问题的解。
研究了一些数值例子以确认所设计方法的高精度。
利用 Caputo 形式的 tempered 分数阶导数定义了时间分数阶非线性薛定谔方程和一个非线性薛定谔方程的耦合系统。ODCPs 被用于为这些问题设计一种数值策略。为此，得到了这些多项式的一些运算矩阵。在所设计的过程中，通过求解一个代数方程组得到问题的解。这些方程组是通过用 ODCPs 逼近解并利用所表示的矩阵关系以及配置技术得到的。给出了一些例子来检验所开发算法的有效性。所报告的结果确认了所设计方案的高精度。
