Abstract
In this paper, we present a numerical scheme for finding numerical solution of a class of weakly singular nonlinear fractional integro-differential equations. This method exploits the alternative Legendre polynomials. An operational matrix, based on the alternative Legendre polynomials, is derived to be approximated the singular kernels of this class of the equations. The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. Moreover, results of essential numerical applications have also been documented in a graphical as well as tabular form to elaborate the effectiveness and accuracy of the proposed method.
Highlights
IntroductionThe operational alternative Legendre method is introduced and employed to solve a class of nonlinear fractional integro-differential equations with weakly singular kernel: t
In this study, the operational alternative Legendre method is introduced and employed to solve a class of nonlinear fractional integro-differential equations with weakly singular kernel: tDαt y(t) F(y(t)) + (t − s)− βG(y(s))ds + f(t), α > 0, 0 ≤ β < 1, t ∈ [0, 1], (1)y(i)(0) y(0i), i 0, 1, 2, . . . , ⌈α⌉ − 1, (2)where F: C([0, 1]) ⟶ R and G: C([0, 1]) ⟶ R are considered to be nonlinear
A spectral method based on the Chebyshev polynomials of the second kind has been applied in [11]
Summary
The operational alternative Legendre method is introduced and employed to solve a class of nonlinear fractional integro-differential equations with weakly singular kernel: t. Heydari et al have utilized the Chebyshev wavelet method to solve systems of the linear and nonlinear singular fractional Volterra integro-differential equations (see [6]). Zhao et al have proposed the piecewise polynomial collocation method for solving the fractional integro-differential equations with weakly singular kernels (see [8]). Application of the alternative Legendre polynomials is extended to solve the nonlinear weakly singular fractional order integro-differential equations. For this purpose, the fractional operational matrices of integration and product are derived. Error analysis and convergence analysis of the proposed method are presented
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