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doi: 10.3934/dcdss.2020295

A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation

1. 

Faculty of Mathematics, Yazd University, Yazd, 89195-741, Iran

2. 

Department of Mathematics, Shiraz University of Technology, Shiraz, 71555-313, Iran

3. 

Engineering School (DEIM), University of Tuscia, Viterbo, 01100, Italy

* Corresponding author: Carlo Cattani

Received  May 2019 Revised  June 2019 Published  June 2020

In this study, an efficient semi-discrete method based on the two-dimensional Legendre wavelets (2D LWs) is developed to provide approximate solutions for nonlinear variable-order time fractional two-dimensional (2D) Schrödinger equation. First, the variable-order time fractional derivative involved in the considered problem is approximated via the finite difference technique. Then, by help of the finite difference scheme and the theta-weighted method, a recursive algorithm is derived for the problem under examination. After that, the real functions available in the real and imaginary parts of the unknown solution of the problem are expanded via the 2D LWs. Finally, by applying the operational matrices of derivative, the solution of the problem is transformed to the solution of a linear system of algebraic equations in each time step which can simply be solved. In the proposed method, acceptable approximate solutions are achieved by employing only a small number of the basis functions. To illustrate the applicability, validity and accuracy of the wavelet method, some numerical test examples are solved using the suggested method. The achieved numerical results reveal that the method established based on the 2D LWs is very easy to implement, appropriate and accurate in solving the proposed model.

Citation: Masoumeh Hosseininia, Mohammad Hossein Heydari, Carlo Cattani. A wavelet method for nonlinear variable-order time fractional 2D Schrödinger equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020295
References:
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References:
[1]

M. A. AbdelkawyM. A. ZakyA. H. Bhrawy and D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Romanian Reports in Physics, 67 (2015), 773-791.   Google Scholar

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A. Atangana and J. F. Gómez–Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos, Solitons and Fractals, 114 (2018), 516-535.  doi: 10.1016/j.chaos.2018.07.033.  Google Scholar

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[13]

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[14]

Y. ChenL. LiuB. Li and Y. Sun, Numerical solution for the variable order linear cable equation with Bernstein polynomials, Appl. Math. Comput., 238 (2014), 329-341.  doi: 10.1016/j.amc.2014.03.066.  Google Scholar

[15]

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[16]

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[18]

E. H. DohaA. H. BhrawyM. A. Abdelkawy and RobertA. Van Gorder, Jacobi–Gauss–Lobatto collocation method for the numerical solution of $1+1$ nonlinear Schrödinger equations, J. Comput. Phys., 261 (2014), 244-255.  doi: 10.1016/j.jcp.2014.01.003.  Google Scholar

[19]

M. D. FeitJ. A. Fleck Jr. and A. Steiger, Solution of the Schrödinger equation by a spectral method, Computational Physics, 47 (1982), 412-433.  doi: 10.1016/0021-9991(82)90091-2.  Google Scholar

[20]

Z. Gao and S. Xie, Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional schrödinger equations, Appl. Numer. Math., 61 (2011), 593-614.  doi: 10.1016/j.apnum.2010.12.004.  Google Scholar

[21]

J. F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132 (2017), 1-13.  doi: 10.1140/epjp/i2017-11293-3.  Google Scholar

[22]

J. F. Gómez-Aguilar, H. Yépez–Martínez, J. Torres-Jiménez, T. Córdova-Fraga, R. F. Escobar-Jiménez and V. H. Olivares-Peregrino, Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel, Adv. Difference Equ., 2017 (2017), Paper No. 68, 18 pp. doi: 10.1186/s13662-017-1120-7.  Google Scholar

[23]

S. H. M. Hamed, E. A. Yousif and A. I. Arbab, Analytic and approximate solutions of the space-time fractional Schrödinger equations by homotopy perturbation sumudu transform method, Abstr. Appl. Anal., 2014 (2014), Art. ID 863015, 13pp. doi: 10.1155/2014/863015.  Google Scholar

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A. Hasegawa, Optical Solitons in Fibers, Berlin: Springer-Verlag, 1993. doi: 10.1117/12.2308783.  Google Scholar

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M. A. E. Herzallah and K. A. Gepreel, Approximate solution to the time–space fractional cubic nonlinear Schrödinger equation, Appl. Math. Model., 36 (2012), 5678-5685.  doi: 10.1016/j.apm.2012.01.012.  Google Scholar

[26]

M. H. Heydari, Wavelets Galerkin method for the fractional subdiffusion equation, Journal of Computational and Nonlinear Dynamics, 11 (2016), 061014, 7pp. doi: 10.1115/1.4034391.  Google Scholar

[27]

M. H. Heydari, A new direct method based on the Chebyshev cardinal functions for variable-order fractional optimal control problems, J. Franklin Inst., 355 (2018), 4970-4995.  doi: 10.1016/j.jfranklin.2018.05.025.  Google Scholar

[28]

M. H. Heydari and Z. Avazzadeh, Legendre wavelets optimization method for variable-order fractional Poisson equation, Chaos, Solitons and Fractals, 112 (2018), 180-190.  doi: 10.1016/j.chaos.2018.04.028.  Google Scholar

[29]

M. H. Heydari and Z. Avazzadeh, An operational matrix method for solving variable-order fractional biharmonic equation, Comput. Appl. Math., 37 (2018), 4397-4411.  doi: 10.1007/s40314-018-0580-z.  Google Scholar

[30]

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Figure 1.  Behavior of the real part of the wavelet solutions in the spaces $ (0.2, y) $ (up) and $ (x, 0.4) $ (down) at $ t = 1 $ for some selections $ \alpha(\mathbf{x}, t) $
Figure 2.  The behavior of the real part of the wavelet solutions in the spaces $ (0.2, y) $ (up) and $ (x, 0.4) $ (down) at $ t = 1 $ for some selections $ \alpha(\mathbf{x}, t) $
Figure 3.  Modulus of the wavelet solutions in the spaces $ (0.2, y) $ (up) and $ (x, 0.4) $ (down) at $ t = 1 $ for some selections $ \alpha(\mathbf{x}, t) $
Figure 4.  The behavior of the real part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.0025 $
Figure 5.  The behavior of the imaginary part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.0025 $
Figure 6.  The behavior of the modulus part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.0025 $
Figure 7.  The behavior of the real part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.01 $
Figure 8.  The behavior of the imaginary part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.01 $
Figure 9.  The behavior of the modulus part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.01 $
Figure 10.  The behavior of the real part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.005 $
Figure 11.  The behavior of the imaginary part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.005 $
Figure 12.  The behavior of the modulus part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $ \delta = 0.005 $
Table 1.  The obtained error values by the presented wavelet method in case of $ \alpha(\mathbf{x}, t) = 1 $ with three values of $ \delta t $
$ \delta t=0.1 $ $ \delta t=0.01 $ $ \delta t=0.005 $
$ t $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $
0.1 1.0972E-4 2.7299E-4 2.9405E-4 5.4640E-5 1.3650E-4 5.6319E-4 2.7320E-5 6.8250E-5 7.3515E-5
0.3 1.4037E-4 2.9682E-4 3.2834E-4 7.0180E-5 1.4841E-4 7.1732E-4 3.5090E-5 7.4200E-5 8.2779E-5
0.5 7.2400E-4 1.2708E-3 1.5000E-3 3.6200E-4 6.3540E-4 7.3128E-4 1.8100E-4 3.1770E-4 3.6564E-4
0.7 2.1281E-4 3.2733E-3 3.3000E-3 1.0640E-4 1.6367E-3 1.600E-4 5.3200E-5 8.1835E-4 9.7607E-4
0.9 1.5228E-3 1.0266E-3 1.8000E-3 7.6140E-4 5.1130E-4 9.1715E-4 3.8070E-4 2.5565E-4 4.5857E-4
1.0 8.3072E-4 1.0078E-3 1.3000E-3 4.1536E-4 5.0390E-4 6.5302E-4 2.0768E-4 2.5159E-4 3.2651E-4
$ \delta t=0.1 $ $ \delta t=0.01 $ $ \delta t=0.005 $
$ t $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $
0.1 1.0972E-4 2.7299E-4 2.9405E-4 5.4640E-5 1.3650E-4 5.6319E-4 2.7320E-5 6.8250E-5 7.3515E-5
0.3 1.4037E-4 2.9682E-4 3.2834E-4 7.0180E-5 1.4841E-4 7.1732E-4 3.5090E-5 7.4200E-5 8.2779E-5
0.5 7.2400E-4 1.2708E-3 1.5000E-3 3.6200E-4 6.3540E-4 7.3128E-4 1.8100E-4 3.1770E-4 3.6564E-4
0.7 2.1281E-4 3.2733E-3 3.3000E-3 1.0640E-4 1.6367E-3 1.600E-4 5.3200E-5 8.1835E-4 9.7607E-4
0.9 1.5228E-3 1.0266E-3 1.8000E-3 7.6140E-4 5.1130E-4 9.1715E-4 3.8070E-4 2.5565E-4 4.5857E-4
1.0 8.3072E-4 1.0078E-3 1.3000E-3 4.1536E-4 5.0390E-4 6.5302E-4 2.0768E-4 2.5159E-4 3.2651E-4
Table 2.  The obtained error values by the presented wavelet method with three different values of $ \delta t $
$ \delta t=0.01 $ $ \delta t=0.005 $ $ \delta t=0.0025 $
$ t $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $
0.1 2.9620E-5 1.0970E-4 1.1363E-4 1.4703E-5 5.4867E-5 5.6803E-5 7.3241E-6 2.7443E-5 2.8404E-5
0.3 8.8820E-5 3.2366E-4 3.3563E-4 4.4328E-5 1.6168E-4 1.6765E-4 2.2127E-5 8.0924E-5 8.3895E-5
0.5 1.3329E-4 4.7940E-4 4.9758E-4 6.6425E-5 2.3966E-4 2.4869E-4 3.3150E-5 1.1985E-4 1.2435E-4
0.7 1.1238E-4 3.9372E-4 4.0944E-4 5.5570E-5 1.9680E-4 2.0450E-4 2.7615E-5 9.8391E-5 1.0219E-4
0.9 9.3184E-5 3.9345E-4 4.0433E-4 4.8381E-5 1.9775E-4 2.0358E-4 2.4638E-5 9.9143E-5 1.0216E-4
1.0 3.3547E-4 1.3286E-3 1.4000E-3 1.7082E-4 6.6725E-4 6.8877E-4 8.6191E-5 3.3438E-4 3.4531E-4
$ \delta t=0.01 $ $ \delta t=0.005 $ $ \delta t=0.0025 $
$ t $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $
0.1 2.9620E-5 1.0970E-4 1.1363E-4 1.4703E-5 5.4867E-5 5.6803E-5 7.3241E-6 2.7443E-5 2.8404E-5
0.3 8.8820E-5 3.2366E-4 3.3563E-4 4.4328E-5 1.6168E-4 1.6765E-4 2.2127E-5 8.0924E-5 8.3895E-5
0.5 1.3329E-4 4.7940E-4 4.9758E-4 6.6425E-5 2.3966E-4 2.4869E-4 3.3150E-5 1.1985E-4 1.2435E-4
0.7 1.1238E-4 3.9372E-4 4.0944E-4 5.5570E-5 1.9680E-4 2.0450E-4 2.7615E-5 9.8391E-5 1.0219E-4
0.9 9.3184E-5 3.9345E-4 4.0433E-4 4.8381E-5 1.9775E-4 2.0358E-4 2.4638E-5 9.9143E-5 1.0216E-4
1.0 3.3547E-4 1.3286E-3 1.4000E-3 1.7082E-4 6.6725E-4 6.8877E-4 8.6191E-5 3.3438E-4 3.4531E-4
Table 3.  The obtained error values by the presented wavelet method with $ k = 0 $ and three different values of $ M $ with $ \delta t = 0.01 $
$ M=8 $ $ M=10 $ $ M=12 $
$ t $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $
0.1 1.5350E-5 2.9309E-5 3.3085E-5 1.5040E-5 3.1348E-5 3.4769E-5 1.5840E-5 3.0785E-5 3.4621E-5
0.3 6.3197E-6 5.3790E-5 5.4160E-5 6.2715E-6 5.3587E-5 5.3953E-5 5.7390E-6 5.4213E-5 5.4516E-5
0.5 3.1960E-5 4.8457E-5 5.8084E-5 3.4046E-5 4.8869E-5 5.9559E-5 3.4143E-5 4.7114E-5 5.8185E-5
0.7 2.4709E-5 3.2107E-5 4.0514E-5 2.4347E-5 3.3151E-5 4.1131E-5 2.5224E-5 3.6278E-5 4.4185E-5
0.9 3.2024E-5 4.0227E-5 5.1417E-5 3.1606E-5 4.0445E-5 5.1330E-5 3.1359E-5 3.9262E-5 5.0248E-5
1.0 4.7832E-5 3.7982E-5 6.1078E-5 4.6654E-5 3.8333E-4 3.8616E-4 4.4804E-4 3.6276E-5 4.4951E-4
$ M=8 $ $ M=10 $ $ M=12 $
$ t $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $
0.1 1.5350E-5 2.9309E-5 3.3085E-5 1.5040E-5 3.1348E-5 3.4769E-5 1.5840E-5 3.0785E-5 3.4621E-5
0.3 6.3197E-6 5.3790E-5 5.4160E-5 6.2715E-6 5.3587E-5 5.3953E-5 5.7390E-6 5.4213E-5 5.4516E-5
0.5 3.1960E-5 4.8457E-5 5.8084E-5 3.4046E-5 4.8869E-5 5.9559E-5 3.4143E-5 4.7114E-5 5.8185E-5
0.7 2.4709E-5 3.2107E-5 4.0514E-5 2.4347E-5 3.3151E-5 4.1131E-5 2.5224E-5 3.6278E-5 4.4185E-5
0.9 3.2024E-5 4.0227E-5 5.1417E-5 3.1606E-5 4.0445E-5 5.1330E-5 3.1359E-5 3.9262E-5 5.0248E-5
1.0 4.7832E-5 3.7982E-5 6.1078E-5 4.6654E-5 3.8333E-4 3.8616E-4 4.4804E-4 3.6276E-5 4.4951E-4
Table 4.  The obtained error values by the presented wavelet method with $ k = 1 $ and two different values of $ M $ with $ \delta t = 0.005 $
$ M=4 $ $ M=5 $
$ t $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $
0.2 7.0513E-5 4.6682E-4 4.7212E-4 7.0460E-5 4.6155E-4 4.6690E-4
0.4 2.5190E-5 4.0126E- 4 4.0205E-4 2.4703E-5 3.9546E-4 3.9623E-4
0.6 7.0915E-5 2.8288E-4 2.9163E-4 6.5807E-5 2.8002E-4 2.8765E-4
0.8 1.5620E-4 1.6841E-4 2.2970E-4 1.5549E-4 1.6815E-4 2.2902E-4
1.0 2.4920E-4 8.7739E-5 2.6419E-4 2.5251E-4 8.8278E-5 2.6750E-4
$ M=4 $ $ M=5 $
$ t $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $ $ \varepsilon_{real} $ $ \varepsilon_{image} $ $ \left| \varepsilon\right| $
0.2 7.0513E-5 4.6682E-4 4.7212E-4 7.0460E-5 4.6155E-4 4.6690E-4
0.4 2.5190E-5 4.0126E- 4 4.0205E-4 2.4703E-5 3.9546E-4 3.9623E-4
0.6 7.0915E-5 2.8288E-4 2.9163E-4 6.5807E-5 2.8002E-4 2.8765E-4
0.8 1.5620E-4 1.6841E-4 2.2970E-4 1.5549E-4 1.6815E-4 2.2902E-4
1.0 2.4920E-4 8.7739E-5 2.6419E-4 2.5251E-4 8.8278E-5 2.6750E-4
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