Convergence analysis of space-time Jacobi spectral collocation method for solving time-fractional Schrödinger equations

https://doi.org/10.1016/j.amc.2019.06.003Get rights and content

Abstract

In this paper, the space-time Jacobi spectral collocation method (JSC Method) is used to solve the time-fractional nonlinear Schro¨dinger equations subject to the appropriate initial and boundary conditions. At first, the considered problem is transformed into the associated system of nonlinear Volterra integro partial differential equations (PDEs) with weakly singular kernels by the definition and related properties of fractional derivative and integral operators. Therefore, by collocating the associated system of integro-PDEs in both of the space and time variables together with approximating the existing integral in the equation using the Jacobi-Gauss-Type quadrature formula, then the problem is reduced to a set of nonlinear algebraic equations. We can consider solving the system by some robust iterative solvers. In order to support the convergence of the proposed method, we provided some numerical examples and calculated their L norm and weighted L2 norm at the end of the article.

Introduction

The Schro¨dinger equation is a significant development in the theory of quantum mechanics [1]. This equation is a powerful differential structure that is related to the quantum systems changes over the temporal when the effects of quantum are considerable.

It should be noted that, the classic variant of this equation is stated in terms of the integer first order temporal and second order spatial partial derivatives. Because of non-locality of fractional differential operators, the fractional variant of Schro¨dinger equation can better describe the physical and chemical events in real world applications [2], [3]. The notion of fractional Schro¨dinger equation was first introduced by Laskin [4], in which the Feyman path integral is extended. Because of nonlinearity, complexity and non-locality of the fractional Schro¨dinger equations, the classical methods for solving PDEs are not efficient. On the other hand, solution of such these equations has considerable importance for researchers to have a deterministic behavior for simulating the events accurately. Therefore, numerical and analytical schemes should be explored and extended to compute the solution of fractional Schro¨dinger equations successfully.

In recent years, in order to solve the fractional Schro¨dinger equation, Scientists proposed many analytical methods and numerical methods. Among the analytical methods, one can point out to the homotopy analysis method (HAM) [5]. Analytical methods are very straightforward for solving any nonlinear PDE, which do not need to discretization or linearization process. But, one of the disadvantages of these methods is being time consuming. Since, in these approaches the integration and differentiation processes are performed symbolically. Meanwhile, in numerical methods, we have two robust tools such as operational matrices of differentiation and Gaussian quadrature rules which accelerate (and reduce the computational time of) the process of differentiation and integration, respectively. Moreover, in some of the numerical approaches such as Krylov subspace methods [6], the no smooth solutions may be computed without any regularization tool, since the solution of these schemes depend on (and keep the behavior of) the right hand side functions of operator equations. Therefore, numerical methods are more attractive to be implemented by researchers. Among the numerical methods, one can point out to low order numerical methods such as finite elements [7], local discontinuous Galerkin (DG) [8], [9] and reproducing kernel [10] techniques which was proposed for solving time-fractional Schro¨dinger equations recently. It should be noted that, since the fractional differential operators are global, it is better to use some global numerical approaches to solve the considered equations. Taking into account that, if for solving a PDE, space variables are discretized by global methods such as radial basis function collocation technique and time variable is localized by local schemes such as Galerkin finite element methods (FEMs) to solve nonlinear time fractional parabolic problems [11] and finite difference methods (FDMs), this may results to an unbalanced numerical scheme [12], [13], [14] which has spectral accuracy in space variable and low order algebraic accuracy in time variable. Therefore, it is desirable to propose a balanced numerical scheme that has spectral accuracy in both time and space directions.

Fractional differential equations (FDEs) are easy to be implemented with respect to the spectral Galerkin methods specially for solving nonlinear FDEs [15]. These methods are successfully applied for solving nonlinear fractional boundary value problems (BVPs) [16], [18] and integral equations (IEs) [19], [20] with a rigorous convergence analysis and also regularized for solving FDEs with nonsmooth solutions [21]. Also, in [22], [23] the author solved fractional Schro¨dinger equations just in numerical implementation point of view via using spectral collocation method also used in [17]. In [24], [25] a linearized L1-Galerkin finite element method and linearized compact alternating direction implicit (ADI) schemes are proposed to solve the multidimensional nonlinear time-fractional Schro¨dinger equation, respectively. To the authors’ knowledge, space-time Jacobi spectral collocation methods (that supported by a rigorous convergence analysis) for solving nonlinear time-fractional Schro¨dinger equations have had few results. This motivate us to propose a Jacobi spectral collocation scheme together with a full convergence analysis for solving time-fractional Schro¨dinger equations which is a balanced approach and has spectral accuracy in both time and space directions.

The time-fractional PDE we will considered as follows:iμψ(x,y,t)tμ=a12ψ(x,y,t)x2+a22ψ(x,y,t)y2+γ|ψ(x,y,t)|2ψ(x,y,t)+δR(x,y,t),0<μ<1,(x,y,t)Ω1×Ω2×Ω3,where Ω1=[0,L1],Ω2=[0,L2] and Ω3=[0,T], with the initial time conditions,ψ(x,y,0)=ζ1(x,y),(x,y)Ω1×Ω2,and two-dimensional boundary space conditions,ψ(0,y,t)=ζ2(y,t),ψ(L1,y,t)=ζ3(y,t),(y,t)Ω2×Ω3,ψ(x,0,t)=ζ4(x,t),ψ(x,L2,t)=ζ5(x,t),(x,t)Ω1×Ω3,while ζ1, ζ2, ζ3, ζ4, ζ5 and R(x, y, t) are given functions.

It should be noted that the definition of Caputo fractional derivative as follows,μτμ(g(τ))={mg(τ)τm,μ=mN,1Γ(mμ)0τg(m)(s)(τs)μm+1ds,m1<μ<m,where μtμ(·) denotes the of order μ derivative.

The definition of Riemann-Liouville (R-L) fractional integral as follows, indicated by lτμ,lτμ(g(τ))=1Γ(μ)0τ(τs)μ1g(s)ds,s>0.

Moreoverlτμ(μτμg(τ))=g(τ)i=0m1g(i)(0)τii!,m1<μ<m.

In the next section, we proposed space-time Jacobi spectral collocation method to solve the Eq. (1.1). Some useful lemmas and preliminaries are provided in Section 3, such as the error estimation of interpolation by Jacobi-Gauss points. Rigorous convergence analysis associated to the used numerical method in weighted L2 norms and L norms is provided in Section 4. In Section 5, algorithm implementation and numerical results are given. Finally, in the last section, some concluding remarks and considered problems for the future research works are stated.

Section snippets

Jacobi spectral collocation methods

We first split all of the known and unknown functions into real and imaginary parts and transform the basic equation into a system of coupled time-fractional PDEs in Caputo sense. In the next step, we impose the Riemann-Liouville fractional integral operator on both sides of the equations to change this system into a nonlinear system of Volterra integro-PDEs with weakly singular kernels that contains initial conditions. Finally, both of the space and time variables are collocated and the

Some preliminaries and useful Lemmas

In the next section, we will introduction the convergence analysis associated to the proposed space-time JSC method to solve (1.1) and (1.2). Therefore, we need to some definitions and lemmas for stablishing the proof of the main theorem. These lemmas include error of the Gauss quadrature rules, estimation of the interpolation errors, Lebesgue constant corresponding to the Legendre series, and finally the Gronwall inequality.

Definition 3.1

[26]

Let I is a bounded interval in R and Lp(I) is a measurable function

Convergence analysis of Jacobi spectral collocation method

The main purpose of this section is to make a specific convergence analysis for the numerical scheme presented in the previous article. According to the convergence result, we show that the Jacobi collocation method we use is the approximate solution obtained in (2.15) exponentially approximates the exact solution. First, we will derive the error estimate for the function in L-norm.

Here, we assume that the kernel function K(x, s, U(s)) has the two following properties which are required for

Example 1. 1D linear equation.

The domain is (x,t)(1,1)×(0,2),i3/4t3/4ψ(x,t)+2x2ψ(x,t)=f1(x,t),(x,t)(1,1)×(0,2),ψ(x,0)=0,x(1,1),ψ(1,t)=ψ(1,t)=t2,t(0,2),wheref1=1625πΓ(34)t5/4(icosπxsinπx)+t2(π2cosπxiπ2sinπx).

The exact solution isψ(x,t)=t2(cosπx+isinπx).

In the space direction x, we use PM+2 Lagrange-Gauss-Lobatto orthogonal polynomials, where the node x0=1 and xM+1=1. In the time direction t, we use PN Jacobi orthogonal polynomials with the index (μ1,0)=(1/4,0), i.e., (1+s)1/4(1s)0, where s=t1. The

Conclusions

In this paper, space-time JSC method has been implemented to solve 1D and 2D time-fractional Schro¨dinger equations with the appropriate initial and boundary conditions. In this regard, we first transform the basic equation into the associated system of nonlinear weakly singular integro-PDEs and then apply the collocation scheme together with approximating the existing integrals with high accurate Gaussian quadrature rules, which reduce the basic equation into the corresponding system of

Acknowledgments

The author would like to thank the referees for the helpful suggestions. The work was supported by NSFC Project (11671342, 91430213, 11671157, 11771369), Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2018JJ2374, 2018WK4006) and Key Project of Hunan Provincial Department of Education (17A210).

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