Abstract
In this article, a numerical method based on the shifted Chebyshev functions for the numerical approximation of the coupled nonlinear variable-order fractional sine-Gordon equations is shown. The variable-order fractional derivative is considered in the sense of Caputo-Prabhakar. To solve the problem, first, we obtain the operational matrix of the Caputo-Prabhakar fractional derivative of shifted Chebyshev polynomials. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional sine-Gordon equations to a system of algebraic equations which is technically simpler for handling. Convergence and error analysis are examined. Finally, some examples are given to test the proposed numerical method to illustrate the accuracy and efficiency of the proposed method.
Highlights
The basic and important applications of differential equations with fractional variable order which are considered as a generalization of ordinary differential equations with integer order in various fields of computational science [1,2,3], engineering [4,5,6], physics [7, 8], and chemistry [9, 10] have attracted significant attention in the literature
One of the important discussions in differential equations is the study of nonlinear partial and nonlinear differential equations that are used in physics and applied mathematics such as the sine-Gordon equation which is a significant nonlinear integrable evolution partial differential system in space-time coordinates [11]
This paper focuses on finding approximate solutions based on the Chebyshev function scheme for fractional differential equations of variable orders which are called coupled nonlinear sine-Gordon equations, and these type of equations of fractional
Summary
The basic and important applications of differential equations with fractional variable order which are considered as a generalization of ordinary differential equations with integer order in various fields of computational science [1,2,3], engineering [4,5,6], physics [7, 8], and chemistry [9, 10] have attracted significant attention in the literature. Bhrawy and Zaky [55] used the shifted Jacobi polynomials to obtain solution variable-order fractional Schrödinger equations, Bhrawy and Zaky [56] studied the Jacobi–Gauss–Lobatto collocation method to obtain solution variable-order fractional Schrödinger equations, Mahmoud et al [57] proposed the Jacobi wavelet collocation method to obtain solution variable-order fractional equations, and Zaky et al [58] proposed the shifted Chebyshev polynomials to obtain solution variable-order fractional equations; in [59], a proper discrete form of fractional Grönwall-type inequality is introduced and other methods such as shifted Jacobi collocation method [60] and shifted Jacobi method [61] Since this mathematical system which is given in Equation (3), due to the variable-order fractional operators and nonlinearity, is very complex, we need to reduce it by using a highly accurate and efficient expansion scheme.
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