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July  2021, 14(7): 2273-2295. 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  July 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (7) : 2273-2295. doi: 10.3934/dcdss.2020295
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show all references

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. 

[2]

E. A.-B. Abdel-Salama, E. A. Yousif and M. A. El-Aasser, On the solution of the space-time fractional cubic nonlinear schrödinger equation, Physics, 2017.

[3]

L. AcedoS. B. Yuste and K. Lindenberg, Reaction front in an $a+b\rightarrow c$ reaction-subdiffusion process, Phys. Rev. E, 69 (2004), 136-144. 

[4]

A. Arnold, Numerically absorbing boundary conditions for quantum evolution equations, VLSI Design, 6 (1998), Article ID 38298, 7 pages. doi: 10.1155/1998/38298.

[5]

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.

[6]

A. Atangana and J. F. Gómez-Aguilar, Numerical approximation of Riemann–Liouville definition of fractional derivative: From Riemann–Liouville to Atangana–Baleanu, Numer. Methods Partial Differential Equations, 34 (2018), 1502-1523.  doi: 10.1002/num.22195.

[7]

T. Bakkyaraj and R. Sahadevan, Approximate analytical solution of two coupled time fractional nonlinear schrödinger equations, Int. J. Appl. Comput. Math, 2 (2016), 113-135.  doi: 10.1007/s40819-015-0049-3.

[8]

E. BarkaiR. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132-138.  doi: 10.1103/PhysRevE.61.132.

[9]

D.A. BensonS. W. Wheatcraft and M. M. Meerschaert, The fractional-order governing equation of lévy motion, Water Resources Research, 36 (2000), 1413-1423.  doi: 10.1029/2000WR900032.

[10]

A. H. Bhrawya and M. A. Abdelkawy, A fully spectral collocation approximation for multi-dimensional fractional schrödinger equations, J. Comput. Phys., 294 (2015), 462-483.  doi: 10.1016/j.jcp.2015.03.063.

[11]

A. H. Bhrawy and M. A. Zaky, Numerical algorithm for the variable-order Caputo fractional functional differential equation, Nonlinear Dynam., 85 (2016), 1815-1823.  doi: 10.1007/s11071-016-2797-y.

[12]

A. H. Bhrawy and M. A. Zaky, Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dynam., 80 (2015), 101-116.  doi: 10.1007/s11071-014-1854-7.

[13]

C. Canuto, M. Hussaini, A. Quarteroni and T. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1998.

[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.

[15]

C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys, 12 (2003), 692-703.  doi: 10.1002/andp.200310032.

[16]

M. Dehghan and A. Shokri, A numerical method for two-dimensional Schrödinger equation using collocation and radial basis functions, Comput. Math. Appl., 54 (2007), 136-146.  doi: 10.1016/j.camwa.2007.01.038.

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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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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.

[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.

[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.

[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.

[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.

[30]

M. H. Heydari and Z. Avazzadeh, A new wavelet method for variable-order fractional optimal control problems, Asian J. Control, 20 (2018), 1804-1817.  doi: 10.1002/asjc.1687.

[31]

M. H. HeydariZ. Avazzadeh and M. Farzi Haromi, A wavelet approach for solving multi-term variable-order time fractional diffusion-wave equation, Appl. Math. Comput., 341 (2019), 215-228.  doi: 10.1016/j.amc.2018.08.034.

[32]

M. H. HeydariM. R. HooshmandaslC. Cattani and G. Hariharan, An optimization wavelet method for multi variable-order fractional differential equations, Fund. Inform., 151 (2017), 255-273.  doi: 10.3233/FI-2017-1491.

[33]

M. H. HeydariM. R. HooshmandaslF. M. Maalek Ghaini and C. Cattani, Wavelets method for the time fractional diffusion-wave equation, Phys. Lett. A, 379 (2015), 71-76.  doi: 10.1016/j.physleta.2014.11.012.

[34]

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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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