Computational aspects of fractional Romanovski–Bessel functions

Abo‐Gabal, Howayda; Zaky, Mahmoud A.; Hendy, Ahmed S.; Doha, E. H.

doi:10.1007/s40314-021-01515-5

Cited by 5 publications

(2 citation statements)

References 14 publications

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“…Spectral techniques (see, e.g., previous works [42][43][44][45][46][47][48][49] ) are approaches used in applied mathematics and scientific computing to numerically approximate the solutions of both linear and nonlinear differential and integral equations. The spectral tau, Galerkin, and collocation schemes are three well-known types of spectral techniques.…”

Section: The Numerical Schemementioning

confidence: 99%

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

Mostafa

Zaky

Hafez

et al. 2022

Math Methods in App Sciences

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We present a class of orthogonal functions on infinite domain based on Jacobi polynomials. These functions are generated by applying a tanh transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo‐derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced tanh Jacobi functions, we develop an efficient spectral tanh Jacobi collocation method for the numerical simulation of nonlinear Schrödinger equations on the infinite domain without using artificial boundary conditions. The applicability and accuracy of the solution method are demonstrated by two numerical examples for solving the nonlinear Schrödinger equation and the nonlinear Ginzburg–Landau equation.

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Section: The Numerical Schemementioning

confidence: 99%

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

Mostafa

Zaky

Hafez

et al. 2022

Math Methods in App Sciences

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“…Liu et al [11] presented and discussed a finite difference/finite element algorithm for casting about for numerical solutions to a time-fractional fourth-order reaction-diffusion problem with a nonlinear reaction term, which is based on a finite difference approximation in time and a finite element method in spatial direction. Moreover, many numerical methods have been developed for solving different classes of fractional differential equations such as spectral methods [12][13][14][15][16][17][18][19], finite difference methods [20][21][22][23][24][25], finite element methods [26,27], finite volume method [28,29], and matrix transfer technique [30,31]. The study of delay differential equations with fractional derivatives is rapidly expanding these days since they are frequently employed in modeling of elastic media and stress-strain behavior for the torsional model, control difficulties, high-speed machining communications, and so on [1].…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

et al. 2021

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The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2−1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time.

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An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

Omran

Zaky

Hendy

et al. 2021

Journal of Function Spaces

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In this paper, we construct and analyze a linearized finite difference/Galerkin–Legendre spectral scheme for the nonlinear multiterm Caputo time fractional-order reaction-diffusion equation with time delay and Riesz space fractional derivatives. The temporal fractional orders in the considered model are taken as 0 < β 0 < β 1 < β 2 < ⋯ < β m < 1 . The problem is first approximated by the L 1 difference method on the temporal direction, and then, the Galerkin–Legendre spectral method is applied on the spatial discretization. Armed by an appropriate form of discrete fractional Grönwall inequalities, the stability and convergence of the fully discrete scheme are investigated by discrete energy estimates. We show that the proposed method is stable and has a convergent order of 2 − β m in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results.

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Computational aspects of fractional Romanovski–Bessel functions

Cited by 5 publications

References 14 publications

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

Tanh Jacobi spectral collocation method for the numerical simulation of nonlinear Schrödinger equations on unbounded domain

Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

An Efficient Hybrid Numerical Scheme for Nonlinear Multiterm Caputo Time and Riesz Space Fractional-Order Diffusion Equations with Delay

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