Shifted Jacobi–Gauss-collocation with convergence analysis for fractional integro-differential equations

Doha, E. H.; Abdelkawy, Mohamed; Amin, A. Z. M.; Lopes, António M.

doi:10.1016/j.cnsns.2019.01.005

Cited by 47 publications

(13 citation statements)

References 38 publications

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“…Many researchers in different branches of physics, mathematics, and engineering are interested in studying fractional calculus due to its many real-life applications, such as ultrasonic wave propagation, 1 optimal control, 2,3 viscoelastic behavior modeling, 4 linear viscoelasticity, 5 variational problems, 6 fluid mechanics, 7 and other applications have been presented in previous studies. [8][9][10][11][12][13] Searching for numerical techniques to solve fractional differential equations has been strongly considered over the last few decades, some of these methods are the fractional finite volume method, 14 spectral collocation method, 15 random variable transformation technique, 16 operational matrix method, 17 Adomian decomposition method, 18 power series method, 19 nonpolynomial spline method, 20 Gauss-collocation method, 21 radial basis functions method, 22 modified Galerkin algorithm, 23 and other methods have been introduced; see previous studies. [24][25][26][27][28][29][30][31] The tau approach is considered as one of the highly accurate methods that have been used for different kinds of fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

Theoretical and spectral numerical study for fractional Van der Pol equation

Ezz-Eldien

2019

Math Methods in App Sciences

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This manuscript concerns with both theoretical and numerical study for a generalized form of fractional Van der Pol equations (FVDPEs). The Schauder fixed point and Banach contraction mapping principles are used for investigating the existence and uniqueness of the considered problem. The second novelty of this manuscript is using the tau method for solving a nonlinear problem (specially FVDPE). The convergence analysis of the suggested approach is also studied.Comparisons with other numerical approaches are introduced for testing the applicability of the current approach.

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Section: Introductionmentioning

confidence: 99%

Theoretical and spectral numerical study for fractional Van der Pol equation

Ezz-Eldien

2019

Math Methods in App Sciences

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“…The spectral method has become increasingly popular in numerical solutions of partial differential equations due to its high-order accuracy [29][30][31][32][33][34]. For time-dependent partial differential equations, if the spectral scheme is used in spatial, the difference scheme is usually adopted in time.…”

Section: Applications To First-order Hyperbolic Equationsmentioning

confidence: 99%

On Romanovski–Jacobi polynomials and their related approximation results

Abo‐Gabal

Zaky

Hafez

et al. 2020

Numerical Methods Partial

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The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of the F-distribution over the positive real line. We introduce some basic properties of the Romanovski-Jacobi polynomials, the Romanovski-Jacobi-Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski-Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes. KEYWORDS differentiation matrices, finite orthogonal polynomials, Gauss-type quadrature, Romanovski-Jacobi polynomials, spectral methods 1 INTRODUCTION The different families of classical orthogonal polynomials are nowadays part of the basic mathematical machinery of numerous physical, engineering, and mathematical algorithms and methodologies [1-3]. The classical hypergeometric polynomials of Hermite, Laguerre, and Jacobi are infinite types

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“…The local methods compute the solutions at particular points; in contrast, the global ones obtain the solutions overall the problem domain 6,7 . For example, the finite element and finite difference methods are local, 8‐13 while the spectral methods are global 14‐18 . The spectral methods gained importance due to their high convergence speed, accuracy, and applicability to either bounded or unbounded domains 16‐19 .…”

Section: Introductionmentioning

confidence: 99%

Accurate spectral algorithm for two‐dimensional variable‐order fractional percolation equations

Abdelkawy

Mahmoud

Abualnaja

et al. 2021

Math Methods in App Sciences

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A highly accurate spectral algorithm for (2 + 1) fractional percolation equations with variable order (VO‐FPEs) is considered. We propose a shifted Legendre–Gauss–Lobatto collocation (SL‐GL‐C) method in conjunction with shifted Chebyshev–Gauss–Radau collocation (SC‐GR‐C) method to solve the two‐dimensional VO‐FPEs. A complete theoretical formulation is presented, and numerical results are given to illustrate the performance and efficiency of the algorithm. The superiority of the scheme to tackle VO‐FPEs is revealed, even when dealing with non‐smooth time solutions.

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Shifted Jacobi–Gauss-collocation with convergence analysis for fractional integro-differential equations

Cited by 47 publications

References 38 publications

Theoretical and spectral numerical study for fractional Van der Pol equation

Theoretical and spectral numerical study for fractional Van der Pol equation

On Romanovski–Jacobi polynomials and their related approximation results

Accurate spectral algorithm for two‐dimensional variable‐order fractional percolation equations

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