A numerical method for solving the time fractional Schrödinger equation

Liu, Na; Jiang, Wei

doi:10.1007/s10444-017-9579-z

Cited by 26 publications

(13 citation statements)

References 15 publications

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“…In this test case, a linear time fractional Schrödinger model is considered as follows:

i^{C} {scriptD}_{t}^{false(α false)} u false(x, t false) + \frac{\partial^{2} u false(x, t false)}{\partial x^{2}} = f false(x, t false),

where

f false(x, t false) = - \frac{\sqrt{π} t^{\frac{1}{2} - α}}{2 normalΓ false(\frac{3}{2} - α false)} \sin false(2 π x false) - 4 \sqrt{t} π^{2} \cos false(2 π x false) + i ((), \frac{\sqrt{π} t^{\frac{1}{2} - α}}{2 normalΓ false(\frac{3}{2} - α false)} \cos false(2 π x false) - 4 \sqrt{t} π^{2} \sin false(2 π x false)) .

…”

Section: Numerical Simulations and Discussionmentioning

confidence: 99%

“…For validation purposes, the method proposed in this paper is tested against the reproducing kernel method, which is described in Liu and Jiang . At first, in order to approximate the solution of this model, the unknown function is considered in the following form:

u false(x, t false) = R false(x, t false) + i I false(x, t false),

where R ( x , t ) is the real part of the unknown function and I ( x , t ) is its imaginary part.…”

Section: Numerical Simulations and Discussionmentioning

confidence: 99%

“…A comparison between the reproducing kernel method proposed in Liu and Jiang and our method with 10 Levenberg‐Marquardt iterations in the sense of maximum error is reported in Table . As Table shows, the accuracy of the proposed method in this paper is acceptable.…”

Section: Numerical Simulations and Discussionmentioning

confidence: 99%

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Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Rasanan

Bajalan

Parand

et al. 2019

Math Methods in App Sciences

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By the rapid growth of available data, providing data-driven solutions for nonlinear (fractional) dynamical systems becomes more important than before. In this paper, a new fractional neural network model that uses fractional order of Jacobi functions as its activation functions for one of the hidden layers is proposed to approximate the solution of fractional differential equations and fractional partial differential equations arising from mathematical modeling of cognitive-decision-making processes and several other scientific subjects. This neural network uses roots of Jacobi polynomials as the training dataset, and the Levenberg-Marquardt algorithm is chosen as the optimizer. The linear and nonlinear fractional dynamics are considered as test examples showing the effectiveness and applicability of the proposed neural network. The numerical results are compared with the obtained results of some other networks and numerical approaches such as meshless methods. Numerical experiments are presented confirming that the proposed model is accurate, fast, and feasible.

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“…In this test case, a linear time fractional Schrödinger model is considered as follows:

i^{C} {scriptD}_{t}^{false(α false)} u false(x, t false) + \frac{\partial^{2} u false(x, t false)}{\partial x^{2}} = f false(x, t false),

where

f false(x, t false) = - \frac{\sqrt{π} t^{\frac{1}{2} - α}}{2 normalΓ false(\frac{3}{2} - α false)} \sin false(2 π x false) - 4 \sqrt{t} π^{2} \cos false(2 π x false) + i ((), \frac{\sqrt{π} t^{\frac{1}{2} - α}}{2 normalΓ false(\frac{3}{2} - α false)} \cos false(2 π x false) - 4 \sqrt{t} π^{2} \sin false(2 π x false)) .

…”

Section: Numerical Simulations and Discussionmentioning

confidence: 99%

u false(x, t false) = R false(x, t false) + i I false(x, t false),

where R ( x , t ) is the real part of the unknown function and I ( x , t ) is its imaginary part.…”

Section: Numerical Simulations and Discussionmentioning

confidence: 99%

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Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Rasanan

Bajalan

Parand

et al. 2019

Math Methods in App Sciences

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“…Li et al [14] solved the TFSE using a nonpolynomial spline. Liu and Jiang in [15] proposed a new scheme based on the reproducing kernel theory and collocation method for solving the TFSE. Esen and Orkun [16,17] proposed a cubic B-spline collocation method and a quadratic B-spline Galerkin method to obtain the numerical solutions of TFSEs, respectively.…”

Section: Introductionmentioning

confidence: 99%

Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

2022

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This paper proposes a numerical method to obtain an approximation solution for the time-fractional Schrödinger Equation (TFSE) based on a combination of the cubic trigonometric B-spline collocation method and the Crank-Nicolson scheme. The fractional derivative operator is described in the Caputo sense. The L1-approximation method is used for time-fractional derivative discretization. Using Von Neumann stability analysis, the proposed technique is shown to be conditionally stable. Numerical examples are solved to verify the accuracy and effectiveness of this method. The error norms L2 and L∞ are also calculated at different values of N and t to evaluate the performance of the suggested method.

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“…In recent years, the theory on them has been employed to get the accurate approximate solutions of various differential and integral operator equations. [4][5][6][7][8][9][10][11][12][13][14][15][16][17] Recently, Liu and Jiang, and Li [18][19][20][21] gave some techniques to solve fractional differential equations based on the Sobolev RKHS. Rosenfeld et al [22][23][24] developed Mittag-Leffer RKHS, and based on this space, presented kernelized ABM and meshless pseudospectral techniques for fractional problems.…”

Section: Introductionmentioning

confidence: 99%

An accurate numerical technique for fractional oscillation equations with oscillatory solutions

Wang

2021

Math Methods in App Sciences

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The solution of fractional oscillation equations exhibits highly oscillatory behavior. It is a challenging work to find efficient numerical methods for fractional oscillation equations. The objective of this paper is to develop an accurate numerical technique for fractional oscillation equations with oscillatory solutions. The present method combines the variation‐of‐constants formula with the reproducing kernel function approximation. The merit of our approach is that it can preserve the oscillatory structure of the solution to the considered fractional initial value problems. Illustrative examples clearly show the reliability and efficiency of the approach.

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A numerical method for solving the time fractional Schrödinger equation

Cited by 26 publications

References 15 publications

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Simulation of nonlinear fractional dynamics arising in the modeling of cognitive decision making using a new fractional neural network

Computational Solution of the Time-Fractional Schrödinger Equation by Using Trigonometric B-Spline Collocation Method

An accurate numerical technique for fractional oscillation equations with oscillatory solutions

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