A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels

Abstract

Please cite this article as: S. Nemati, S. Sedaghat, I. Mohammadi, A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels, Journal of Computational and Applied Mathematics (2016), http://dx. AbstractA spectral method based on operational matrices of the second kind Chebyshev polynomials (SKCPs) is employed for solving fractional integro-differential equations with weakly singular kernels. Firstly, properties of shifted SKCPs, operational matrix of fractional integration and product operational matrix are introduced and then utilized to reduce the given equation to the solution of a system of linear algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent linear Volterra integral equation of the second kind with the same initial conditions. Approximate solution is achieved by expanding the functions in terms of SKCPs and employing operational matrices. Unknown coefficients are determined by solving final system of linear equations. An estimation of the error is given. Finally, illustrative examples are included to demonstrate the high precision, fast computation and good performance of the new scheme.

Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel

“…Example 2 Consider the following fractional-order integro-differential equation with weakly singular kernel [39,40]: with the condition y(0) = 0. The exact solution of this equation is y(x) = x 4 3 + x 3 .…”

“…In this problem we apply the FEFs method to solve Eq. (30) with various values of m. Table 2 shows the root-mean-square errors for the second kind Chebyshev polynomials method (SKCP) [40] and the FEFs method with α = 1, 2 3 . From Table 2, we can see that both of them obtain good approximations with the exact solution and the FEFs method has smaller root-mean-square errors.…”

A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro‐differential equations with weakly singular kernels

“…This kind of equations appears in the field of elasticity and fracture mechanics, 20 radiative equilibrium, 21 heat conduction problem, 22 and so on. During the recent years, a few methods have been used to solve the fractional integro-differential equations with weakly singular kernels numerically, such as the second kind Chebyshev polynomials, 23 spline collocation, 24 second kind Chebyshev wavelets (SCW) method, 25 Legendre wavelets (LW), 26 an operational Jacobi Tau method, 27 hat functions, 28 and the shifted Jacobi polynomials. 29 The various applications of fractional integro-differential equations with weakly singular kernels is our main motivation from the present work.…”

New Analytical Solutions for Time-Fractional Kolmogorov-Petrovsky-Piskunov Equation with Variety of Initial Boundary Conditions

“…Analytical and approximate series solutions for the nonlinear fractional differential equations are fundamental importance for seeking solutions of the most complex phenomena that are modeled. There are many methods that have also been proposing for solving analytical and approximate series solutions: the transform methods, including Laplace, Fourier, and Mellin transforms [5]; the Tau method [6]; the Adomian decomposition method [7]; the variational iteration method [7,8]; the Sumudu decomposition method [9]; the blockpulse functions [10]; shifted Chebyshev polynomials [11]; shifted Legendre polynomials [12]; Chebyshev wavelets [13,14]; and Legendre wavelets [15].…”