Alikhanov Legendre–Galerkin spectral method for the coupled nonlinear time-space fractional Ginzburg–Landau complex system

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg-Landau system is proposed and analyzed. The Alikhanov L2-1 sigma difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Gronwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims

The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation

The nontrivial behavior of wave packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difficulty comes from the fact that the Riesz fractional derivative is inherently a prehistorical operator. In contrast, nonlinear Schrödinger equation with both time and space nonlocal operators, which is the cornerstone in the modeling of a new type of fractional quantum couplers, is still in high demand of attention. This paper is devoted to numerically study the propagation of solitons through a new type of quantum couplers which can be called time-space fractional quantum couplers. The numerical methodology is based on the finite-difference/Galerkin Legendre spectral method with an easy to implement numerical algorithm. The time-fractional derivative is considered to describe the decay behavior and the nonlocal memory of the model. We conduct numerical simulations to observe the performance of the tunable decay and the sharpness behavior of the time-space fractional strongly coupled nonlinear Schrödinger model as well as the performance of the numerical algorithm. Numerical simulations show that the time and space fractional-order operators control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional-order parameters

The impact of memory effect on space fractional strong quantum couplers with tunable decay behavior and its numerical simulation

The nontrivial behavior of wave packets in the space fractional coupled nonlinear Schrödinger equation has received considerable theoretical attention. The difficulty comes from the fact that the Riesz fractional derivative is inherently a prehistorical operator. In contrast, nonlinear Schrödinger equation with both time and space nonlocal operators, which is the cornerstone in the modeling of a new type of fractional quantum couplers, is still in high demand of attention. This paper is devoted to numerically study the propagation of solitons through a new type of quantum couplers which can be called time-space fractional quantum couplers. The numerical methodology is based on the finite-difference/Galerkin Legendre spectral method with an easy to implement numerical algorithm. The time-fractional derivative is considered to describe the decay behavior and the nonlocal memory of the model. We conduct numerical simulations to observe the performance of the tunable decay and the sharpness behavior of the time-space fractional strongly coupled nonlinear Schrödinger model as well as the performance of the numerical algorithm. Numerical simulations show that the time and space fractional-order operators control the decay behavior or the memory and the sharpness of the interface and undergo a seamless transition of the fractional-order parameters. © 2021, The Author(s).This study was supported financially by RFBR Grant (19-01-00019), the National Research Centre of Egypt (NRC) and Ghent university

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

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<β0<β1<β2<⋯<βm<1. The problem is first approximated by the L1 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-βm in time and an exponential rate of convergence in space. We finally provide some numerical experiments to show the efficacy of the theoretical results. © 2021 A. K. Omran et al.A. K. Omran is funded by a scholarship under the joint executive program between the Arab Republic of Egypt and Russian Federation. M. A. Zaky wishes to acknowledge the support of the Nazarbayev University Program (091019CRP2120). M. A. Zaky wishes also to acknowledge the partial support of the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”)

A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients

In this paper, a weak Galerkin finite element method for solving the time fractional reaction-convection diffusion problem is proposed. We use the well known L1 discretization in time and a weak Galerkin finite element method on uniform mesh in space. Both continuous and discrete time weak Galerkin finite element method are considered and analyzed. The stability of the discrete time scheme is proved. The error estimates for both schemes are given. Finally, we give some numerical experiments to show the efficiency of the proposed method

A pseudo-spectral scheme for systems of two-point boundary value problems with left and right sided fractional derivatives and related integral equations

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left- and right-sided fractional derivatives. The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations. Then, a Legendre-based spectral collocation method is developed for solving the transformed system. Therefore, we can make good use of the advantages of the Gauss quadrature rule. We present the construction and analysis of the collocation method. These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler-Lagrange equations. Two numerical examples are given to confirm the convergence analysis and robustness of the scheme. © 2021 Tech Science Press. All rights reserved.Russian Foundation for Basic Research, РФФИ: 19-01-00019The Russian Foundation for Basic Research (RFBR) Grant No. 19-01-00019

Numerical approximation and simulation of the stochastic wave equation on the sphere

Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schr\uf6dinger equation on the unit sphere

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions