WITHDRAWN: A novel Legendre wavelet Petrov–Galerkin method for fractional Volterra integro-differential equations

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

“…The comparisons show that the best case of α for this equation is α = 1 2 . Also, we can see the absolute error is the same or slightly smaller than that in [38] when α = 1 2 , m = 4.…”

A method for fractional Volterra integro-differential equations by Laguerre polynomials

“…They found the approximate solution for N = 5, but they did not state the numerical results of the errors of their method. Besides, Sahu et al [32] found the approximate solution with the maximum absolute error 1.1 × 10 -16 by the Legendre wavelet Petrov-Galerkin method for N = 6. If the results are compared, it can be said that the proposed method is better than the other methods since the exact solution is found for N = 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]. 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…”

Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis

“…In literature, approximations of a special type Abel integral equations appear in Khan and Gondal 11 by two‐step Laplace decomposition algorithm, in Eshaghi et al 12 by using fractional order Legendre functions and pseudospectral method, and in Kilbas and Saigo 13 based on asymptotic behavior of the solution. On the other hand, few specific class of fractional order Volterra integro‐differential equations are also considered for numerical analysis, based on Wavelet methods in Sahu and Saha Ray, 14 using fast iterative refinement method in Grace and Deif, 15 and using spline collocation method in Pedas and Tamme 16 . In recent days, semi analytical methods became popular over numerical methods in order to avoid of finding suitable discretization 17 of a general model.…”