Abstract
In this article, a numerical technique based on the Chebyshev cardinal functions (CCFs) and the Lagrange multiplier technique for the numerical approximation of the variable-order fractional integrodifferential equations are shown. The variable-order fractional derivative is considered in the sense of regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives. To solve the problem, first, we obtain the operational matrix of the regularized Hilfer-Prabhakar and Hilfer-Prabhakar fractional derivatives of CCFs. Then, this matrix and collocation method are used to reduce the solution of the nonlinear coupled variable-order fractional integrodifferential 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 its accuracy and efficiency.
Highlights
Fractional calculus as a branch of mathematical analysis due to important applications in computational science, engineering [1,2,3,4], chemistry, and many physical equations such as Burger’s equation [5] and Korteweg-de Vries equation [6], and viscous fluid [7] has been broadly significant
Obtaining solution of the linear and nonlinear differential equations has important roles in fractional calculus [8, 9], because the behavior of the equations is determined by their analytical solutions
A differential equation including a variable-order fractional operator is considered in which the equation is an extension of the fractional differential equations of fractional order
Summary
Fractional calculus as a branch of mathematical analysis due to important applications in computational science, engineering [1,2,3,4], chemistry, and many physical equations such as Burger’s equation [5] and Korteweg-de Vries equation [6], and viscous fluid [7] has been broadly significant. Our interest in this type of fractional derivative is related to its application in physical phenomena, for example, linear viscoelastic constitutive equations [30], anomalous dielectric relaxation of Havriliak-Negami function [31, 32], fractional viscoelasticity [33], spherical stellar systems, stochastic processes [34], telegraph equations [34], and anomalous relaxation in dielectrics [35] Since this mathematical system which is given in Equation (1), due to the variableorder 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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