Abstract
The main purpose of this study is to present an approximation method based on the Laguerre polynomials for fractional linear Volterra integro-differential equations. This method transforms the integro-differential equation to a system of linear algebraic equations by using the collocation points. In addition, the matrix relation for Caputo fractional derivatives of Laguerre polynomials is also obtained. Besides, some examples are presented to illustrate the accuracy of the method and the results are discussed.
Highlights
The fractional calculus represents a powerful tool in applied mathematics to study numerous problems from different fields of science and engineering such as mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology, and bioengineering [1]
The methods that are used to find the solutions of the fractional Volterra integro-differential equations are given as Adomian decomposition [11], Bessel collocation [12, 13], CAS wavelets [14], Chebyshev pseudo-spectral [15], cubic B-spline wavelets [16], Euler wavelet [17], fractional differential transform [18], homotopy analysis [19], homotopy perturbation [20,21,22,23], Jacobi spectral-collocation [24, 25], Legendre collocation [26], Legendre wavelet [27], linear and quadratic interpolating polynomials [28], modification of hat functions [29], multi-domain pseudospectral [30], normalized systems functions [31], novel Legendre wavelet Petrov–Galerkin method [32], operational Tau [33], piecewise polynomial collocation [34], quadrature rules [35], reproducing kernel [36], second Chebyshev wavelet [37], second kind Chebyshev polynomials [38], sinccollocation [39, 40], spline collocation [41], Taylor expansion [27], and variational iteration [20, 23]
In this paper, a method based on the Laguerre polynomials is presented to find the solutions of linear fractional Volterra integro-differential equation in the form x
Summary
The fractional calculus represents a powerful tool in applied mathematics to study numerous problems from different fields of science and engineering such as mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology, and bioengineering [1]. Since the fractional calculus has attracted much more interest among mathematicians and other scientists, the solutions of the fractional differential and integro-differential equations have been studied frequently in recent years [2,3,4,5,6,7,8,9,10]. Laguerre polynomials are used to solve some integer order integro-differential equations. Laguerre polynomials are used to solve the fractional Fredholm integro-differential equation [51]. There has not been a method in the literature for fractional Volterra integro-differential equations in terms of Laguerre polynomials. In this paper, a method based on the Laguerre polynomials is presented to find the solutions of linear fractional Volterra integro-differential equation in the form x
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