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doi: 10.3934/dcdss.2020146

Solving the time-fractional Schrodinger equation with the group preserving scheme

1. 

Department of Mathematics, University of Bonab, Bonab, Iran

2. 

Department of Mathematics, Firat University, Science Faculty, Elazig, 23119, Turkey

3. 

Department of Computer Engineering, Biruni University, Istanbul, Turkey

4. 

Department of Mathematics, Cankaya University, Ankara, Turkey

* Corresponding author: Mustafa Bayram

Received  May 2019 Revised  June 2019 Published  November 2019

In this paper a powerful numerical scheme is proposed to gain the numerical solutions of the time-fractional Schrodinger equation: $ i^{C}\mathcal{D}_{0^+, t}^{\alpha}w(x, t) + \vartheta\frac{\partial^2 w(x, t)}{\partial x^2}+ \delta \vert w(x, t)\vert^2 w(x, t)+P(x, t) w(x, t) = \mathcal{F}(x, t), 0<\alpha \leq 1 $, $ \vartheta $ and $ \delta $ are real constants, $ p(x, t) $ is trapping potential. The time fractional $ ^{C}\mathcal{D}_{0^+, t}^{\alpha} w $ is defined in Caputo definition. By using a fictitious $ \iota $ we can convert the variable $ w(x, t) $ into a new variable by: $ (1+\iota)^k w(x, t) = \Xi(x, t, \iota) $, where $ 0< k \leq 1 $. In new space with a semi-discretization of $ w(x, t, \iota) $ the group-preserving scheme is used to solve the problem.

Citation: Mohammad Parto-haghighi, Mustafa Inc, Mustafa Bayram, Dumitru Baleanu. Solving the time-fractional Schrodinger equation with the group preserving scheme. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020146
References:
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[2]

A. Ashyralyev and B. Hicdurmaz, On the numerical solution of fractional Schrodinger differential equations with the Dirichlet condition, Int. J. Comput. Math., 89 (2012), 1927-1936.  doi: 10.1080/00207160.2012.698841.  Google Scholar

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

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A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien and R. A. V. Gorder, A new Jacobi spectral collocation method for solving 1+1 fractional Schrodinger equations and fractional coupled Schrodinger systems, Eur. Phys. J. Plus, (2014), 129–260.  Google Scholar

[5]

A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien and R. A. Van Gorder, A new Jacobi spectral collocation method for solving fractional Schrodinger equations and fractional coupled Schrodinger systems, Eur. Phys. J. Plus, 129 (2014). Google Scholar

[6]

F. BulutA. Oruc and A. Esen, A Numerical Solutions of Fractional System of Partial Differential Equations By Haar Wavelets, Computer Modeling in Engineering and Sciences, 108 (2015), 263-284.   Google Scholar

[7]

S. ChenF. LiuP. Zhuang and V. Anh, Finite difference approximations for the fractional Fokker Planck equation, Appl. Math. Model., 33 (2009), 256-273.  doi: 10.1016/j.apm.2007.11.005.  Google Scholar

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M. DehghanM. Abbaszadeh and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear Sine Gordon and Klein-Gordon equations, Eng. Anal. Bound. Elem., 50 (2015), 412-434.  doi: 10.1016/j.enganabound.2014.09.008.  Google Scholar

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M. DehghanJ. Manafian and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Equations, 26 (2010), 448-479.  doi: 10.1002/num.20460.  Google Scholar

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R. EidS. I. MuslihD. Baleanu and E. Rabei, On fractional Schrodinger equation in dimensional fractional space, Nonlinear Anal., Real World Appl., 10 (2009), 1299-1304.  doi: 10.1016/j.nonrwa.2008.01.007.  Google Scholar

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M. Giona and H. E. Roman, Fractional diffusion equation for transport phenomena in random media, Physica, A 185 (1992), 87-97.   Google Scholar

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M. S. HashemiD. Baleanu and M. Parto-Haghighi, A lie group approach to solve the fractional poisson equation, Rom. J. Phys, 60 (2015), 1289-1297.   Google Scholar

[15]

M. S. Hashemi, D. Baleanu, M. Parto-Haghighi and E. Darvishi, Solving the time-fractional diffusion equation using a lie group integrator, Thermal Science, 19 (2015), S77–S83. Google Scholar

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R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

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M. M. Khader and K. M. Saad, A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method, Chaos Solitons & Fractals, 110 (2018), 169-177.  doi: 10.1016/j.chaos.2018.03.018.  Google Scholar

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M. M. Khader and K. M. Saad, A numerical study by using the Chebyshev collocation method for a problem of biological invasion: Fractional Fisher equation, Int. J. Biomath., 11 (2018), 1850099, 15 pp. doi: 10.1142/S1793524518500997.  Google Scholar

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J. W. KirchnerX. Feng and L. NeaC, Fractal stream chemistry and its implications for containant transport in catchments, Nature, 403 (2000), 524-526.   Google Scholar

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D. KumaraA. R. Seadawy and A. K. Joardare, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese J. Phys., 56 (2018), 75-85.   Google Scholar

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C. S. Liu, The Fictitious Time Integration Method to Solve the Space- and Time-Fractional Burgers Equations, CMC, 15 (2010), 221-240.   Google Scholar

[22]

C. S. Liu, Solving an inverse Sturm-Liouville problem by a Lie-Group method, Bound. Value Probl., 2008, Art. ID 749865, 18 pp. doi: 10.1155/2008/749865.  Google Scholar

[23]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. Google Scholar

[24]

A. MohebbM. Abbaszadeh and M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrodinger equation arising in quantum mechanics, Eng. Anal. Bound. Elem., 37 (2013), 475-485.  doi: 10.1016/j.enganabound.2012.12.002.  Google Scholar

[25]

A. M. Nagy, Numerical solution of time fractional nonlinear Klein Gordon equation using Sinc Chebyshev collocation method, Appl. Math. Comput., 310 (2017), 139-148.  doi: 10.1016/j.amc.2017.04.021.  Google Scholar

[26]

A. M. Nagy and N. H. Sweilam, An efficient method for solving fractional Hodgkin Huxley model, Phys. Lett. A, 378 (2014), 1980-1984.  doi: 10.1016/j.physleta.2014.06.012.  Google Scholar

[27]

Z. Odibat and S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. Appl., 58 (2009), 2199-2208.  doi: 10.1016/j.camwa.2009.03.009.  Google Scholar

[28]

I. Podlubny, Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[29]

S. Z. RidaH. M. El-Sherbiny and A. A. M. Arafa, On the solution of the fractional nonlinear Schrodinger equation, Phys. Lett. A, 372 (2008), 553-558.  doi: 10.1016/j.physleta.2007.06.071.  Google Scholar

[30]

K. M. Saad, A. Atangana and B. Dumitru, New Fractional derivatives with non-singular kernel applied to the Burger's Equation, Chaos, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.  Google Scholar

[31]

K. M. Saad, S. Deniz and D. Baleanu, On the new modified fractional analysis of Nagumo equation, Int. J. Biomath., 12 (2019), 1950034, 15 pp. doi: 10.1142/S1793524519500347.  Google Scholar

[32]

K. M. SaadB. Dumitru and A. Atangana, New fractional derivatives applied to the Korteweg-de Vries and Korteweg-de Vries-Burger's equations, Comput. Appl. Math., 37 (2018), 5203-5216.  doi: 10.1007/s40314-018-0627-1.  Google Scholar

[33]

S. SarwarS. AlkhalafS. Iqbal and M. A. Zahid, A note on optimal homotopy asymptotic method for the solutions of fractional order heat-and wave-like partial differential equations, Comput. Math. Appl., 70 (2015), 942-953.  doi: 10.1016/j.camwa.2015.06.017.  Google Scholar

[34]

S. Sarwar and S. Iqbal, Exact solution of non-linear fractional order klein-gordon partial differential equations using optimal homotopy asymptotic method, Nonlinear Science Letters A, 8 (2017), 340-348.   Google Scholar

[35]

S. Sarwar and S. mIqbal, Stability analysis, dynamical behavior and analytical solutions of nonlinear fractional differential system arising in chemical reaction, Chinese J. Phys., 56 (2018), 374-384.  doi: 10.1016/j.cjph.2017.11.009.  Google Scholar

[36]

S. Sarwar and M. M. Rashidi, Approximate solution of two-term fractional-order diffusion, wave-diffusion, and telegraph models arising in mathematical physics using optimal homotopy asymptotic method, Waves Random Complex Media, 26 (2016), 365-382.  doi: 10.1080/17455030.2016.1158436.  Google Scholar

[37]

S. Sarwar, M. A. Zahid and S. Iqbal, Mathematical study of Fractional Order Biological Model using Optimal Homotopy Asymptotic Method, Int. J. Biomath, 9 (2016), 1650081, 17 pp. doi: 10.1142/S1793524516500819.  Google Scholar

[38]

E. Shivanian, Spectral meshless radial point interpolation (SMRPI) method to twodimensional fractional telegraph equation, Math. Meth. Appl. Sci., 39 (2016), 1820-1835.  doi: 10.1002/mma.3604.  Google Scholar

[39]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.  Google Scholar

[40]

A. SohailK. Maqbool and R. Ellahi, Stability analysis for fractional-order partial differential equations by means of space spectral time Adams- Bashforth Moulton method, Numer. Methods Partial Differential Equations, 34 (2018), 19-29.  doi: 10.1002/num.22171.  Google Scholar

[41]

O. Tasbozan and A. Esen, Quadratic B-spline, galerkin method for numerical solutions of fractional telegraph equations, Bulletin of Mathematical Sciences and Applications, 18 (2017), 23-39.   Google Scholar

[42]

P. Wang and C. Huang, A conservative linearized difference scheme for the nonlinear fractional Schrodinger equation, Numer. Algorithms, 69 (2015), 625-641.  doi: 10.1007/s11075-014-9917-x.  Google Scholar

[43]

L. WeiH. DaiD. Zhang and Z. Si, Fully discrete local discontinuous galerkin method for solving the fractional telegraph equation, Calcolo, 51 (2014), 175-192.  doi: 10.1007/s10092-013-0084-6.  Google Scholar

[44]

L. WeiY. HeX. Zhang and S. Wang, Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrodinger equation, Finite Elem. Anal. Des., 59 (2012), 28-34.  doi: 10.1016/j.finel.2012.03.008.  Google Scholar

[45]

M. A. ZahidS. SarwarM. ArshadA. Asma and M. Arshad, New solitary wave solutions of generalized space-time fractional fifth order laxs and sawada kotera kdv type equations in mathematical physics, Journal of Advanced Physics, 7 (2018), 342-349.   Google Scholar

show all references

References:
[1]

F. B. Addaa and J. Cresson, Fractional differential equations and the Schrodinger equation, Appl. Math. Comput., 161 (2005), 323-345.  doi: 10.1016/j.amc.2003.12.031.  Google Scholar

[2]

A. Ashyralyev and B. Hicdurmaz, On the numerical solution of fractional Schrodinger differential equations with the Dirichlet condition, Int. J. Comput. Math., 89 (2012), 1927-1936.  doi: 10.1080/00207160.2012.698841.  Google Scholar

[3]

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

[4]

A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien and R. A. V. Gorder, A new Jacobi spectral collocation method for solving 1+1 fractional Schrodinger equations and fractional coupled Schrodinger systems, Eur. Phys. J. Plus, (2014), 129–260.  Google Scholar

[5]

A. H. Bhrawy, E. H. Doha, S. S. Ezz-Eldien and R. A. Van Gorder, A new Jacobi spectral collocation method for solving fractional Schrodinger equations and fractional coupled Schrodinger systems, Eur. Phys. J. Plus, 129 (2014). Google Scholar

[6]

F. BulutA. Oruc and A. Esen, A Numerical Solutions of Fractional System of Partial Differential Equations By Haar Wavelets, Computer Modeling in Engineering and Sciences, 108 (2015), 263-284.   Google Scholar

[7]

S. ChenF. LiuP. Zhuang and V. Anh, Finite difference approximations for the fractional Fokker Planck equation, Appl. Math. Model., 33 (2009), 256-273.  doi: 10.1016/j.apm.2007.11.005.  Google Scholar

[8]

M. DehghanM. Abbaszadeh and A. Mohebbi, An implicit RBF meshless approach for solving the time fractional nonlinear Sine Gordon and Klein-Gordon equations, Eng. Anal. Bound. Elem., 50 (2015), 412-434.  doi: 10.1016/j.enganabound.2014.09.008.  Google Scholar

[9]

M. DehghanJ. Manafian and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differential Equations, 26 (2010), 448-479.  doi: 10.1002/num.20460.  Google Scholar

[10]

M. DehghanS. A. Yousefi and A. Lotfi, The use of variational iteration method for solving the telegraph and fractional telegraph equations, Int. J. Numer. Methods Biomed. Eng., 27 (2011), 219-231.  doi: 10.1002/cnm.1293.  Google Scholar

[11]

J. Dong and M. Xu, Space-time fractional Schrodinger equation with time-independent potentials, J. Math. Anal. Appl., 344 (2008), 1005-1017.  doi: 10.1016/j.jmaa.2008.03.061.  Google Scholar

[12]

R. EidS. I. MuslihD. Baleanu and E. Rabei, On fractional Schrodinger equation in dimensional fractional space, Nonlinear Anal., Real World Appl., 10 (2009), 1299-1304.  doi: 10.1016/j.nonrwa.2008.01.007.  Google Scholar

[13]

M. Giona and H. E. Roman, Fractional diffusion equation for transport phenomena in random media, Physica, A 185 (1992), 87-97.   Google Scholar

[14]

M. S. HashemiD. Baleanu and M. Parto-Haghighi, A lie group approach to solve the fractional poisson equation, Rom. J. Phys, 60 (2015), 1289-1297.   Google Scholar

[15]

M. S. Hashemi, D. Baleanu, M. Parto-Haghighi and E. Darvishi, Solving the time-fractional diffusion equation using a lie group integrator, Thermal Science, 19 (2015), S77–S83. Google Scholar

[16]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. doi: 10.1142/9789812817747.  Google Scholar

[17]

M. M. Khader and K. M. Saad, A numerical approach for solving the fractional Fisher equation using Chebyshev spectral collocation method, Chaos Solitons & Fractals, 110 (2018), 169-177.  doi: 10.1016/j.chaos.2018.03.018.  Google Scholar

[18]

M. M. Khader and K. M. Saad, A numerical study by using the Chebyshev collocation method for a problem of biological invasion: Fractional Fisher equation, Int. J. Biomath., 11 (2018), 1850099, 15 pp. doi: 10.1142/S1793524518500997.  Google Scholar

[19]

J. W. KirchnerX. Feng and L. NeaC, Fractal stream chemistry and its implications for containant transport in catchments, Nature, 403 (2000), 524-526.   Google Scholar

[20]

D. KumaraA. R. Seadawy and A. K. Joardare, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese J. Phys., 56 (2018), 75-85.   Google Scholar

[21]

C. S. Liu, The Fictitious Time Integration Method to Solve the Space- and Time-Fractional Burgers Equations, CMC, 15 (2010), 221-240.   Google Scholar

[22]

C. S. Liu, Solving an inverse Sturm-Liouville problem by a Lie-Group method, Bound. Value Probl., 2008, Art. ID 749865, 18 pp. doi: 10.1155/2008/749865.  Google Scholar

[23]

R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006. Google Scholar

[24]

A. MohebbM. Abbaszadeh and M. Dehghan, The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrodinger equation arising in quantum mechanics, Eng. Anal. Bound. Elem., 37 (2013), 475-485.  doi: 10.1016/j.enganabound.2012.12.002.  Google Scholar

[25]

A. M. Nagy, Numerical solution of time fractional nonlinear Klein Gordon equation using Sinc Chebyshev collocation method, Appl. Math. Comput., 310 (2017), 139-148.  doi: 10.1016/j.amc.2017.04.021.  Google Scholar

[26]

A. M. Nagy and N. H. Sweilam, An efficient method for solving fractional Hodgkin Huxley model, Phys. Lett. A, 378 (2014), 1980-1984.  doi: 10.1016/j.physleta.2014.06.012.  Google Scholar

[27]

Z. Odibat and S. Momani, The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. Appl., 58 (2009), 2199-2208.  doi: 10.1016/j.camwa.2009.03.009.  Google Scholar

[28]

I. Podlubny, Fractional Differential Equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, 198. Academic Press, Inc., San Diego, CA, 1999.  Google Scholar

[29]

S. Z. RidaH. M. El-Sherbiny and A. A. M. Arafa, On the solution of the fractional nonlinear Schrodinger equation, Phys. Lett. A, 372 (2008), 553-558.  doi: 10.1016/j.physleta.2007.06.071.  Google Scholar

[30]

K. M. Saad, A. Atangana and B. Dumitru, New Fractional derivatives with non-singular kernel applied to the Burger's Equation, Chaos, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.  Google Scholar

[31]

K. M. Saad, S. Deniz and D. Baleanu, On the new modified fractional analysis of Nagumo equation, Int. J. Biomath., 12 (2019), 1950034, 15 pp. doi: 10.1142/S1793524519500347.  Google Scholar

[32]

K. M. SaadB. Dumitru and A. Atangana, New fractional derivatives applied to the Korteweg-de Vries and Korteweg-de Vries-Burger's equations, Comput. Appl. Math., 37 (2018), 5203-5216.  doi: 10.1007/s40314-018-0627-1.  Google Scholar

[33]

S. SarwarS. AlkhalafS. Iqbal and M. A. Zahid, A note on optimal homotopy asymptotic method for the solutions of fractional order heat-and wave-like partial differential equations, Comput. Math. Appl., 70 (2015), 942-953.  doi: 10.1016/j.camwa.2015.06.017.  Google Scholar

[34]

S. Sarwar and S. Iqbal, Exact solution of non-linear fractional order klein-gordon partial differential equations using optimal homotopy asymptotic method, Nonlinear Science Letters A, 8 (2017), 340-348.   Google Scholar

[35]

S. Sarwar and S. mIqbal, Stability analysis, dynamical behavior and analytical solutions of nonlinear fractional differential system arising in chemical reaction, Chinese J. Phys., 56 (2018), 374-384.  doi: 10.1016/j.cjph.2017.11.009.  Google Scholar

[36]

S. Sarwar and M. M. Rashidi, Approximate solution of two-term fractional-order diffusion, wave-diffusion, and telegraph models arising in mathematical physics using optimal homotopy asymptotic method, Waves Random Complex Media, 26 (2016), 365-382.  doi: 10.1080/17455030.2016.1158436.  Google Scholar

[37]

S. Sarwar, M. A. Zahid and S. Iqbal, Mathematical study of Fractional Order Biological Model using Optimal Homotopy Asymptotic Method, Int. J. Biomath, 9 (2016), 1650081, 17 pp. doi: 10.1142/S1793524516500819.  Google Scholar

[38]

E. Shivanian, Spectral meshless radial point interpolation (SMRPI) method to twodimensional fractional telegraph equation, Math. Meth. Appl. Sci., 39 (2016), 1820-1835.  doi: 10.1002/mma.3604.  Google Scholar

[39]

J. SinghD. KumarZ. Hammouch and A. Atangana, A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Appl. Math. Comput., 316 (2018), 504-515.  doi: 10.1016/j.amc.2017.08.048.  Google Scholar

[40]

A. SohailK. Maqbool and R. Ellahi, Stability analysis for fractional-order partial differential equations by means of space spectral time Adams- Bashforth Moulton method, Numer. Methods Partial Differential Equations, 34 (2018), 19-29.  doi: 10.1002/num.22171.  Google Scholar

[41]

O. Tasbozan and A. Esen, Quadratic B-spline, galerkin method for numerical solutions of fractional telegraph equations, Bulletin of Mathematical Sciences and Applications, 18 (2017), 23-39.   Google Scholar

[42]

P. Wang and C. Huang, A conservative linearized difference scheme for the nonlinear fractional Schrodinger equation, Numer. Algorithms, 69 (2015), 625-641.  doi: 10.1007/s11075-014-9917-x.  Google Scholar

[43]

L. WeiH. DaiD. Zhang and Z. Si, Fully discrete local discontinuous galerkin method for solving the fractional telegraph equation, Calcolo, 51 (2014), 175-192.  doi: 10.1007/s10092-013-0084-6.  Google Scholar

[44]

L. WeiY. HeX. Zhang and S. Wang, Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrodinger equation, Finite Elem. Anal. Des., 59 (2012), 28-34.  doi: 10.1016/j.finel.2012.03.008.  Google Scholar

[45]

M. A. ZahidS. SarwarM. ArshadA. Asma and M. Arshad, New solitary wave solutions of generalized space-time fractional fifth order laxs and sawada kotera kdv type equations in mathematical physics, Journal of Advanced Physics, 7 (2018), 342-349.   Google Scholar

Figure 1.  Plot of exact and approximate solutions of real part for example 1
Figure 2.  Plot of exact and approximate solutions of imaginary part for example 1
Figure 3.  Contour plot of error of real part for example 1
Figure 4.  Contour plot of error of imaginary part for example 1
Figure 5.  Plot of exact and approximate solutions of real part for example 2
Figure 6.  Plot of exact and approximate solutions of imaginary part for example 2
Figure 7.  Contour plot of error of real part for example 2
Figure 8.  Contour plot of error of imaginary part for example 2
Figure 9.  Plot of exact and approximate solutions of real part for example 3
Figure 10.  Plot of exact and approximate solutions of imaginary part for example 3
Figure 11.  Contour plot of error of real part for example 3
Figure 12.  Contour plot of error of imaginary part for example 3
Figure 13.  Plot of exact and approximate solutions of real part for example 4
Figure 14.  Plot of exact and approximate solutions of imaginary part for example 4
Figure 15.  Contour plot of error of real part for example 4
Figure 16.  Contour plot of error of imaginary part for example 4
Figure 17.  Plot of exact and approximate solutions of real part for example 5
Figure 18.  Plot of exact and approximate solutions of imaginary part for example 5
Figure 19.  Contour plot of error of real part for example 5
Figure 20.  Contour plot of error of imaginary part for example 5
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