WITHDRAWN: Legendre–Gauss–Lobatto collocation method for solving multi-dimensional Fredholm integral equations

Bhrawy, A. H.; Abdelkawy, M. A.; Machado, J. A. Tenreiro; Amin, A. Z. M.

doi:10.1016/j.camwa.2016.04.011

Cited by 24 publications

(12 citation statements)

References 45 publications

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“…Then we choose the coefficients such that the error is minimized. For spectral collocation method [6,8,9], the approximate solution is compelled to satisfy the discussed problem as possible. In other words, the residuals is letting to be zero at confirmed collocation points.…”

Section: Introductionmentioning

confidence: 99%

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

Doha

Abdelkawy

Amin

et al. 2019

NAMC

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In this manuscript, we introduce a spectral technique for approximating the variable-order fractional Riccati equation (VO-FRDEs). Firstly, the solution and its space fractional derivatives is expanded as shifted Chebyshev polynomials series. Then we determine the expansion coefficients by reducing the VO-FRDEs and its conditions to a system of algebraic equations. We show the accuracy and applicability of our numerical approach through four numerical examples.

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Section: Introductionmentioning

confidence: 99%

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

Doha

Abdelkawy

Amin

et al. 2019

NAMC

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“…Fredholm integral equations for 1-D backward heat conduction problems were studied in [1]. There are many methods to obtain the solution of an integral equation, such as the meshless method [2], the Legendre-Gauss-Lobatto collocation method [3], the piecewise linear maximum entropy method [4], etc. In this paper, it is our intention to numerically solve the Fredholm integral equation of the second kind by combining the classical Monte Carlo method and the Monte Carlo method with a control variate [5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo method with control variate for integral equations

Tian

2018

THERM SCI

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“…2016b; Tatari and Haghighi 2014), integral (Bhrawy et al. 2016c; Rahmoune 2013), integro-differential (Jiang and Ma 2013; Ma and Huang 2014) and delay differential (Bhrawy et al. 2015a; Reutskiy 2015) equations.…”

Section: Introductionmentioning

confidence: 99%

A space–time spectral collocation algorithm for the variable order fractional wave equation

et al. 2016

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The variable order wave equation plays a major role in acoustics, electromagnetics, and fluid dynamics. In this paper, we consider the space–time variable order fractional wave equation with variable coefficients. We propose an effective numerical method for solving the aforementioned problem in a bounded domain. The shifted Jacobi polynomials are used as basis functions, and the variable-order fractional derivative is described in the Caputo sense. The proposed method is a combination of shifted Jacobi–Gauss–Lobatto collocation scheme for the spatial discretization and the shifted Jacobi–Gauss–Radau collocation scheme for temporal discretization. The aforementioned problem is then reduced to a problem consists of a system of easily solvable algebraic equations. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.

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WITHDRAWN: Legendre–Gauss–Lobatto collocation method for solving multi-dimensional Fredholm integral equations

Cited by 24 publications

References 45 publications

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

Approximate solutions for solving nonlinear variable-order fractional Riccati differential equations

Monte Carlo method with control variate for integral equations

A space–time spectral collocation algorithm for the variable order fractional wave equation

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