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

Local Discontinuous Galerkin Method for Nonlinear Ginzburg- Landau Equation

The Ginzburg-Landau equation has been applied widely in many fields. It describes the amplitude evolution of instability waves in a large variety of dissipative systems in fluid mechanics, which are close to criticality. In this chapter, we develop a local discontinuous Galerkin method to solve the nonlinear Ginzburg-Landau equation. The nonlinear Ginzburg-Landau problem has been expressed as a system of low-order differential equations. Moreover, we prove stability and optimal order of convergence OhN+1 for Ginzburg-Landau equation where h and N are the space step size and polynomial degree, respectively. The numerical experiments confirm the theoretical results of the method

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

Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations

Dynamical low-rank integrators for matrix differential equations recently attracted a lot of attention and have proven to be very efficient in various applications. In this paper, we propose a novel strategy for choosing the rank of the projector-splitting integrator of Lubich and Oseledets adaptively. It is based on a combination of error estimators for the local time-discretization error and for the low-rank error with the aim to balance both. This ensures that the convergence of the underlying time integrator is preserved. The adaptive algorithm works for projector-splitting integrator methods for first-order matrix differential equations and also for dynamical low-rank integrators for second-order equations, which use the projector-splitting integrator method in its substeps. Numerical experiments illustrate the performance of the new integrators

Theoretical analysis (Convergence and stability) of a difference approximation for multiterm time fractional convection diffusion-wave equations with delay

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. © 2020 by the authors. Licensee MDPI, Basel, Switzerland.The first author wishes to acknowledge the support of RFBR Grant 19-01-00019

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

Convergence and stability estimates in difference setting for time-fractional parabolic equations with functional delay

A class of one-dimensional time-fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation–extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work. © 2019 Wiley Periodicals, Inc.Russian Foundation for Basic Research, RFBR: 19‐01‐00019A1‐S‐45928The authors want to thank the associate editor in charge of handling this manuscript and anonymous reviewers for all their comments and criticisms. Their suggestions contributed decisively to improve the overall quality of this work. The first two authors wish to acknowledge the support of RFBR Grant 19‐01‐00019. Meanwhile, the last author wishes to acknowledge the financial support of the National Council for Science and Technology of Mexico through grant A1‐S‐45928

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”)

Exponential integrators: tensor structured problems and applications

The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as phi-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed