Chebyshev wavelet method to nonlinear fractional Volterra–Fredholm integro-differential equations with mixed boundary conditions

Abstract

This research work addresses the numerical solutions of nonlinear fractional integro-differential equations with mixed boundary conditions, using Chebyshev wavelet method. The basic idea of this work started from the Caputo definition of fractional differential operator. The fractional derivatives are replaced by Caputo operator, and the solution is approximated by wavelet family of functions. The numerical scheme by Chebyshev wavelet method is constructed through a very simple and straightforward way. The numerical results of the current method are compared with the exact solutions of the problems, which show that the proposed method has a strong agreement with the exact solutions of the problems. The numerical solutions of the present method are also compared with steepest decent method and Adomian decomposition method. The comparison with other methods reveals that this method has the highest degree of accuracy than those methods.

The Polynomial Least Squares Method for Nonlinear Fractional Volterra and Fredholm Integro-Differential Equations

“…and after minimizing the functional J(c 2 ), we find the minimum as c 2 = 1. Thus,x = t 2 , which is the exact solution of the problem while, again, the previous methods in [19] (Chebyshev Wavelets Method), Ref. [20] (also a Chebyshev Wavelets Method) and [24] (Newton-Kantorovitch Method) were only able to find approximate solutions.…”

“…This means that PLSM is able to find, in a very simple manner, the exact solution of the problem,x(t) = x e (t) = t. We remark that the previous methods in [13] (Shifted Chebyshev Polynomials Method and Adomian Decomposition Method) and [19] (Chebyshev Wavelets Method) were only able to find approximate solutions.…”

“…By minimizing the functional J(c 1 , c 2 , c 3 ) (with an expression too long to be included here), we obtain c 1 = 0, c 2 = 0 and c 3 = 1, thus finding the exact solutionx(t) = x e (t) = t 3 . We remark that the previous methods in [13,19] were only able to find approximate solutions.…”

Local radial basis function scheme for solving a class of fractional integro‐differential equations based on the use of mixed integral equations

“…Sinc‐collocation method and sinc‐Galerkin method have been introduced for solving Volterra‐Fredholm integro‐differential equations of fractional order. Chebyshev wavelets, Haar wavelets, CAS wavelets and Legendre wavelets as very well localized functions have been used to obtain the numerical solutions of fractional integro‐differential equations.…”

Fractional conformable attractors with low fractality

“…Many dynamic systems can be described by means of fractional derivatives. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] Fractional-order differential equations have been developed by using different computational approaches and asymptotics expansions or perturbation methods such as the Adomian decomposition method, [15][16][17] Aboodh decomposition method, 18 homotopy perturbation method, 19,20 fractional natural decomposition method, 21 Chebyshev wavelet method, 22 Laplace Adomian decomposition method, 23,24 Bernstein operational matrix method, 25,26 and q-homotopy analysis transform method. 27,28 Chaos theory is a branch of mathematics with several applications in physics, engineering, economics, and biology.…”