American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdss.2020295

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

Masoumeh Hosseininia 1, , Mohammad Hossein Heydari 2,3, and  Carlo Cattani 2,3,,

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.

Keywords: Variable-order time fractional derivative, two-dimensional (2D) Schrödinger equation, two-dimensional Legendre wavelets (2D LWs).
Mathematics Subject Classification: 26A33.
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:
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[23]

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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

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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) $
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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) $
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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) $
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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 $
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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 $
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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 $
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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 $
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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 $
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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 $
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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 $
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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 $
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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 $
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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
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$ \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
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$ \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
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$ 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
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$ 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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