Abstract
In the present work, a numerical technique for solving a general form of nonlinear fractional order integro-differential equations (GNFIDEs) with linear functional arguments using Chebyshev series is presented. The recommended equation with its linear functional argument produces a general form of delay, proportional delay, and advanced non-linear arbitrary order Fredholm–Volterra integro-differential equations. Spectral collocation method is extended to study this problem as a matrix discretization scheme, where the fractional derivatives are characterized in the Caputo sense. The collocation method transforms the given equation and conditions to an algebraic nonlinear system of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. The introduced operational matrix of derivatives includes arbitrary order derivatives and the operational matrix of ordinary derivative as a special case. To the best of authors’ knowledge, there is no other work discussing this point. Numerical test examples are given, and the achieved results show that the recommended method is very effective and convenient.
Highlights
Nonlinear differential (DEs) and integro-differential equations (IDEs) have a great importance in modeling of many phenomena in physics and engineering [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
Monotone iterative technique was introduced with Riemann– Liouville fractional derivative to deal with fractional integro-differential equations (FIDEs) with advanced arguments [61], while the collocation method with Bessel polynomials treated linear Fredholm integro-differentialdifference equations [62]
Tau method with the Chebyshev polynomials was employed to deal with linear fractional differential equations with linear functional arguments [63]; the Chebyshev collocation method was extended to fractional differential equations with delay [64]
Summary
Nonlinear differential (DEs) and integro-differential equations (IDEs) have a great importance in modeling of many phenomena in physics and engineering [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. In the last decade or so, several studies have been carried out to develop numerical schemes to deal with fractional integro-differential equations (FIDEs) of both linear and nonlinear type. Differential equations of advanced argument had fewer contributions in mathematics research compared to delay differential equations, which had a great development in the last decade [59, 60]. Monotone iterative technique was introduced with Riemann– Liouville fractional derivative to deal with FIDEs with advanced arguments [61], while the collocation method with Bessel polynomials treated linear Fredholm integro-differentialdifference equations [62]. Tau method with the Chebyshev polynomials was employed to deal with linear fractional differential equations with linear functional arguments [63]; the Chebyshev collocation method was extended to fractional differential equations with delay [64]. All reported works considered a generalization of equations with functional argument with integer order derivative or with fractional derivative in the linear case
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