A numerical method for the fractional Schrödinger type equation of spatial dimension two

Ford, Neville J.; Rodrigues, M. M.; Vieira, N.

doi:10.2478/s13540-013-0028-5

Cited by 28 publications

(10 citation statements)

References 13 publications

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“…Let R k 1 be the time discretization error of (3.4). Then we can obtain 7) in whichq = min{σ − β, 1} when the first-order extrapolation is applied. Letting ε k = u k − U k and subtracting (3.5) from (3.6) gives…”

Section: An Error Estimate Of the Time Discrete Systemmentioning

confidence: 99%

“…The stability and convergence of L1 finite difference methods were obtained for a timefractional nonlinear predator-prey model under the restriction LT β < 1/Γ(1 − β) in [27], where L is the Lipschitz constant of the nonlinear function, depending upon an upper bound of numerical solutions [17]. Such a condition implied that the numerical results just held locally in time and certain time step restriction condition (see, e.g., [2,7]) were also required. Similar restrictions also appear in the numerical analysis for the other fractional nonlinear equations (see, e.g.…”

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Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations

Yang

Zeng

2019

Commun. Appl. Math. Comput.

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In this paper, a new type of the discrete fractional Grönwall inequality is developed, which is applied to analyze the stability and convergence of a Galerkin spectral method for a linear timefractional subdiffusion equation. Based on the temporal-spatial error splitting argument technique, the discrete fractional Grönwall inequality is also applied to prove the unconditional convergence of a semi-implicit Galerkin spectral method for a nonlinear time-fractional subdiffusion equation.1 method for the initial value problem. For example, piecewise linear interpolation yields the widely applied L1 method [22,19]. High-order interpolation can also be applied, see [3,12,30]. The FLMM [23,24,25] provides another general framework for constructing high-order methods to discretize the fractional integral and derivative operators. The FLMM inherits the stability properties of linear multistep methods for initial value problem, which greatly facilitates the analysis of the resulting numerical scheme, in a way often strikingly opposed to standard quadrature formulas [25]. Up to now, the FLMM has been widely applied to discretize the model (1.1) and its variants.It is well known that the classical discrete Grönwall inequality plays an important role in the analysis of the numerical methods for time-dependent partial differential equations (PDEs). Due to the lack of a generalized discrete Grönwall type inequality for the time-stepping methods of the time-dependent fractional differential equations (FDEs), the analysis of the numerical methods for time-dependent FDEs is more complicated. Recently, a discrete fractional Grönwall inequality has been established by Liao et al. [13,19,20,21] for interpolation methods to solve linear and nonlinear time-dependent FDEs. Jin et al. [8] proposed a criterion for showing the fractional discrete Grönwall inequality and verified it for the L1 scheme and convolution quadrature generated by backward difference formulas.Till now, there have been some works on the numerical analysis of nonlinear time-dependent FDEs. The stability and convergence of L1 finite difference methods were obtained for a timefractional nonlinear predator-prey model under the restriction LT β < 1/Γ(1 − β) in [27], where L is the Lipschitz constant of the nonlinear function, depending upon an upper bound of numerical solutions [17]. Such a condition implied that the numerical results just held locally in time and certain time step restriction condition (see, e.g., [2,7]) were also required. Similar restrictions also appear in the numerical analysis for the other fractional nonlinear equations (see, e.g. [16,15]). In order to avoid such a restriction, the temporal-spatial error splitting argument (see, e.g., [10]) is extended to the numerical analysis of the nonlinear time-dependent FDEs (see, e.g., [14,17]). Li et al. proposed unconditionally convergent L1-Galerkin finite element methods (FEMs) for nonlinear time-fractional Schödinger equations [14] and nonlinear time-fractional subdiffusion equations[17], respectively....

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Section: An Error Estimate Of the Time Discrete Systemmentioning

confidence: 99%

mentioning

confidence: 99%

Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations

Yang

Zeng

2019

Commun. Appl. Math. Comput.

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“…. A function u is called a strong solution of (5) if u ∈ C(R + ; D(A)) and g 1−α * u belongs to C 1 ((0, ∞); H), and (5) holds for all t > 0.…”

Section: Preliminariesmentioning

confidence: 99%

The Time Fractional Schrödinger Equation on Hilbert Space

Górka

Prado

Trujillo

2017

Integr. Equ. Oper. Theory

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Abstract. We study the linear fractional Schrödinger equation on a Hilbert space, with a fractional time derivative of order 0 < α < 1, and a self-adjoint generator A. Using the spectral theorem we prove existence and uniqueness of strong solutions, and we show that the solutions are governed by an operator solution family {Uα(t)} t≥0 . Moreover, we prove that the solution family Uα(t) converges strongly to the family of unitary operators e −itA , as α approaches to 1.Mathematics Subject Classification. 35Q41, 35R11, 81S99; Secondary 47D08, 47D03, 81C10.

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“…The fractional Riemann-Liouville derivative is composed by using the Hadamard finite-part integral in this research and the piecewise linear interpolation polynomial is utilized to approximate the integral. This method is applied to design numerical methods for solving some fractional partial differential equations [28,29]. Yan et al modified the method in [27] by using the piecewise quadratic interpolation polynomial [30].…”

Section: Introductionmentioning

confidence: 99%

A novel high-order algorithm for the numerical estimation of fractional differential equations

Asl

Javidi

Yan

2018

Journal of Computational and Applied Mathematics

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This paper uses polynomial interpolation to design a novel high-order algorithm for the numerical estimation of fractional differential equations. The Riemann-Liouville fractional derivative is expressed by using the Hadamard finite-part integral and the piecewise cubic interpolation polynomial is utilized to approximate the integral. The detailed error analysis is presented an it is established that the convergence order of the algorithm is O(h 4−α). Asymptotic expansion for the error of presented algorithm is also investigated. Some numerical examples are provided and compared with the exact solution to show that the numerical results are in well agreement with the theoretical ones and also to illustrate the accuracy and efficiency of the proposed algorithm.

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A numerical method for the fractional Schrödinger type equation of spatial dimension two

Cited by 28 publications

References 13 publications

Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations

Numerical Analysis of Linear and Nonlinear Time-Fractional Subdiffusion Equations

The Time Fractional Schrödinger Equation on Hilbert Space

A novel high-order algorithm for the numerical estimation of fractional differential equations

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