Global consistency analysis of L1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations

Hendy, Ahmed S.; Zaky, Mahmoud A.

doi:10.1016/j.apnum.2020.05.002

Cited by 50 publications

(22 citation statements)

References 37 publications

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“…Summing (38) for all 0 ≤ q ≤ m, we find that the Romanovski-Jacobi polynomials are orthogonal in the Sobolev space B m , (Λ), namely,…”

Section: Orthogonal Projectionsmentioning

confidence: 99%

“…For time-dependent partial differential equations, if the spectral scheme is used in spatial, the difference scheme is usually adopted in time. However, when the exact solution is very smooth, the accuracy of the approximate solution would be limited by the finite difference scheme in time [35][36][37][38][39]. The aim of this section is to introduce spectral Galerkin methods based on Romanovski-Jacobi polynomials for the approximation of solutions of the Cauchy problem for the linear hyperbolic equations in one and two space dimensions.…”

Section: Applications To First-order Hyperbolic Equationsmentioning

confidence: 99%

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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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“…Summing (38) for all 0 ≤ q ≤ m, we find that the Romanovski-Jacobi polynomials are orthogonal in the Sobolev space B m , (Λ), namely,…”

Section: Orthogonal Projectionsmentioning

confidence: 99%

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

Self Cite

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“…Fractional differential equations represent more complex models, but mostly it is difficult to solve them analytically. Therefore different researchers are looking for numerical methods, e.g., finite element method, spectral method, and finite difference method, to find the solution to these fractional differential equations [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The finite difference method is relatively simple and easy; that is why it has been seen more in the literature for the solution of fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

2020

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In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

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“…An efficient spectral method that is based on Jacobi-Gauss-Radau collocation is applied in order to solve a system of multi-dimensional distributed-order generalized Schrödinger equations in [14]. A combined difference and Galerkin-Legendre spectral method in [15] is used to solve time fractional diffusion equations with nonlinear source term. A Legendre spectral-collocation method for the numerical solution of distributed-order fractional initial value problems is designed in [16].…”

Section: Introductionmentioning

confidence: 99%

Theoretical Analysis (Convergence and Stability) of a Difference Approximation for Multiterm Time Fractional Convection Diffusion-Wave Equations with Delay

Hendy

Staelen

2020

Mathematics

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In this paper, we introduce a high order numerical approximation method for convection diffusion wave equations armed with a multiterm time fractional Caputo operator and a nonlinear fixed time delay. A temporal second-order scheme which is behaving linearly is derived and analyzed for the problem under consideration based on a combination of the formula of L2−1σ and the order reduction technique. By means of the discrete energy method, convergence and stability of the proposed compact difference scheme are estimated unconditionally. A numerical example is provided to illustrate the theoretical results.

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Global consistency analysis of L1-Galerkin spectral schemes for coupled nonlinear space-time fractional Schrödinger equations

Cited by 50 publications

References 37 publications

On Romanovski–Jacobi polynomials and their related approximation results

On Romanovski–Jacobi polynomials and their related approximation results

A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

Theoretical Analysis (Convergence and Stability) of a Difference Approximation for Multiterm Time Fractional Convection Diffusion-Wave Equations with Delay

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