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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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##### References:
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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. Chen, L. Liu, B. 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. [17] R. K. Dodd, J. C. Eilbeck, J. D. Gibbon and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1982. [18] E. H. Doha, A. H. Bhrawy, M. 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. Feit, J. 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. 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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 $\alpha(\mathbf{x}, t)$
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 $\alpha(\mathbf{x}, t)$
Modulus of the wavelet solutions in the spaces $(0.2, y)$ (up) and $(x, 0.4)$ (down) at $t = 1$ for some selections $\alpha(\mathbf{x}, t)$
The behavior of the real part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.0025$
The behavior of the imaginary part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.0025$
The behavior of the modulus part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.0025$
The behavior of the real part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.01$
The behavior of the imaginary part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.01$
The behavior of the modulus part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.01$
The behavior of the real part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.005$
The behavior of the imaginary part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.005$
The behavior of the modulus part of the wavelet solution and the corresponding AE function (up and down, respectively) at the final time where $\delta = 0.005$
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
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
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
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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