Shehu transform on time-fractional Schrödinger equations – an analytical approach

References

[1] K. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier, 1974.Search in Google Scholar

[2] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, 1993.Search in Google Scholar

[3] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives and Some of Their Applications, vol. 1, Science and Technica, 1987.Search in Google Scholar

[4] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.Search in Google Scholar

[5] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, 2006.Search in Google Scholar

[6] M. D. Ortigueira, Fractional Calculus for Scientists and Engineers, vol. 84, Springer Science & Business Media, 2011.10.1007/978-94-007-0747-4Search in Google Scholar

[7] S. Das, Functional Fractional Calculus, Springer Science & Business Media, 2011.10.1007/978-3-642-20545-3Search in Google Scholar

[8] R. Hilfer, Ed. Applications of Fractional Calculus in Physics, World Scientific, 2000.10.1142/3779Search in Google Scholar

[9] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, vol. 35, New York, Springer, 2003.10.1007/978-0-387-21746-8Search in Google Scholar

[10] L. Debnath, “Recent applications of fractional calculus to science and engineering,” Int. J. Math. Math. Sci., vol. 2003, no. 54, pp. 3413–3442, 2003. https://doi.org/10.1155/s0161171203301486.Search in Google Scholar

[11] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, World Scientific, 2010.10.1142/p614Search in Google Scholar

[12] D. Baleanu, Z. B. Güvenç, and J. T. Machado, Eds. New Trends in Nanotechnology and Fractional Calculus Applications, New York, Springer, 2010, p. C397.10.1007/978-90-481-3293-5Search in Google Scholar

[13] R. Herrmann, Fractional Calculus: An Introduction for Physicists, 2011.10.1142/8072Search in Google Scholar

[14] A. Papoulis, “A new method of inversion of the Laplace transform,” Q. Appl. Math., vol. 14, no. 4, pp. 405–414, 1957. https://doi.org/10.1090/qam/82734.Search in Google Scholar

[15] A. Kılıçman and H. E. Gadain, “On the applications of Laplace and Sumudu transforms,” J. Franklin Inst., vol. 347, no. 5, pp. 848–862, 2010. https://doi.org/10.1016/j.jfranklin.2010.03.008.Search in Google Scholar

[16] T. M. Elzaki, “On the connections between Laplace and Elzaki transforms,” Adv. Theor. Appl. Math., vol. 6, no. 1, pp. 1–11, 2011.Search in Google Scholar

[17] M. S. Rawashdeh and S. Maitama, “Solving coupled system of nonlinear PDE’s using the natural decomposition method,” Int. J. Pure Appl. Math., vol. 92, no. 5, pp. 757–776, 2014. https://doi.org/10.12732/ijpam.v92i5.10.Search in Google Scholar

[18] S. Maitama and W. Zhao, “New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations,” Int. J. Nonlinear Anal. Appl., vol. 17, no. 2, pp. 167–219, 2019.Search in Google Scholar

[19] D. Ziane, R. Belgacem, and A. Bokhari, “A new modified Adomian decomposition method for nonlinear partial differential equations,” Open Journal of Mathematical Analysis (OMA), vol. 3, no. 2, pp. 81–90, 2019. https://doi.org/10.30538/psrp-oma2019.0041.Search in Google Scholar

[20] L. Akinyemi and O. S. Iyiola, “Exact and approximate solutions of time‐fractional models arising from physics via Shehu transform,” Math. Methods Appl. Sci., vol. 43, no. 12, pp. 7442–7464, 2020. https://doi.org/10.1002/mma.6484.Search in Google Scholar

[21a] R. Belgacem, D. Baleanu, and A. Bokhari, Shehu Transform and Applications to Caputo-Fractional Differential Equations, 2019.Search in Google Scholar

[b] G. H. Hardy, “Gösta Mittag-Leffler, 1846–1927,” Proc. R. Soc. Lond. (A), vol. 119, 1928.Search in Google Scholar

[22] A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler function and its properties,” J. Math. Anal. Appl., vol. 336, no. 2, pp. 797–811, 2007. https://doi.org/10.1016/j.jmaa.2007.03.018.Search in Google Scholar

[23] O. S. Iyiola, E. O. Asante-Asamani, and B. A. Wade, “A real distinct poles rational approximation of generalized Mittag-Leffler functions and their inverses: applications to fractional calculus,” J. Comput. Appl. Math., vol. 330, pp. 307–317, 2018. https://doi.org/10.1016/j.cam.2017.08.020.Search in Google Scholar

[24] Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, 2003.10.1016/B978-012410590-4/50012-7Search in Google Scholar

[25] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys., vol. 71, no. 3, p. 463, 1999. https://doi.org/10.1103/revmodphys.71.463.Search in Google Scholar

[26] J. Belmonte-Beitia and G. F. Calvo, “Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials,” Phys. Lett., vol. 373, no. 4, pp. 448–453, 2009. https://doi.org/10.1016/j.physleta.2008.11.056.Search in Google Scholar

[27] T. Xu, B. Tian, L. L. Li, X. Lü, and C. Zhang, “Dynamics of Alfvén solitons in inhomogeneous plasmas,” Phys. Plasmas, vol. 15, no. 10, p. 102307, 2008. https://doi.org/10.1063/1.2997340.Search in Google Scholar

[28] M. Naber, “Time fractional Schrödinger equation,” J. Math. Phys., vol. 45, no. 8, pp. 3339–3352, 2004. https://doi.org/10.1063/1.1769611.Search in Google Scholar

[29] S. Wang and M. Xu, “Generalized fractional Schrödinger equation with space-time fractional derivatives,” J. Math. Phys., vol. 48, no. 4, p. 043502, 2007. https://doi.org/10.1063/1.2716203.Search in Google Scholar

[30] S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear Schrödinger equation,” Phys. Lett., vol. 372, no. 5, pp. 553–558, 2008. https://doi.org/10.1016/j.physleta.2007.06.071.Search in Google Scholar

[31] R. K. Saxena, R. Saxena, and S. L. Kalla, “Computational solution of a fractional generalization of the Schrödinger equation occurring in quantum mechanics,” Appl. Math. Comput., vol. 216, no. 5, pp. 1412–1417, 2010. https://doi.org/10.1016/j.amc.2010.02.041.Search in Google Scholar

[32] J. Wang, Y. Zhou, and W. Wei, “Fractional Schrödinger equations with potential and optimal controls,” Nonlinear Anal. R. World Appl., vol. 13, no. 6, pp. 2755–2766, 2012. https://doi.org/10.1016/j.nonrwa.2012.04.004.Search in Google Scholar

[33] N. A. Khan, M. Jamil, and A. Ara, Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method, International Scholarly Research Notices, 2012, 2012.10.5402/2012/197068Search in Google Scholar

[34] K. M. Hemida, K. A. Gepreel, and M. S. Mohamed, “Analytical approximate solution to the time-space nonlinear partial fractional differential equations,” Int. J. Pure Appl. Math., vol. 78, no. 2, pp. 233–243, 2012.Search in Google Scholar

[35] S. H. 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,” in Abstract and Applied Analysis, vol. 2014, Hindawi, 2014.10.1155/2014/863015Search in Google Scholar

[36] S. O. Edeki, G. O. Akinlabi, and S. A. Adeosun, “Analytic and numerical solutions of time-fractional linear Schrödinger equation,” Commun. Math. Appl., vol. 7, no. 1, pp. 1–10, 2016.Search in Google Scholar

[37] A. Mohebbi, M. Abbaszadeh, and M. Dehghan, “The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics,” Eng. Anal. Bound. Elem., vol. 37, no. 2, pp. 475–485, 2013. https://doi.org/10.1016/j.enganabound.2012.12.002.Search in Google Scholar

[38] K. Shah, M. Junaid, and N. Ali, “Extraction of Laplace, Sumudu, Fourier and Mellin transform from the natural transform,” J. Appl. Environ. Biol. Sci, vol. 5, no. 9, pp. 108–115, 2015.Search in Google Scholar

[39] A. M. Malik and O. H. Mohammed, “Two efficient methods for solving fractional Lane–Emden equations with conformable fractional derivative,” J. Egypt. Math. Soc., vol. 28, no. 1, pp. 1–11, 2020. https://doi.org/10.1186/s42787-020-00099-z.Search in Google Scholar

[40] S. Ali, S. Bushnaq, K. Shah, and M. Arif, “Numerical treatment of fractional order Cauchy reaction diffusion equations,” Chaos, Solit. Fractals, vol. 103, pp. 578–587, 2017. https://doi.org/10.1016/j.chaos.2017.07.016.Search in Google Scholar

[41] K. Shah, H. Naz, M. Sarwar, and T. Abdeljawad, “On spectral numerical method for variable-order partial differential equations,” AIMS Math., vol. 7, no. 6, pp. 10422–10438, 2022. https://doi.org/10.3934/math.2022581.Search in Google Scholar

[42] A. Bashan, N. M. Yagmurlu, Y. Ucar, and A. Esen, “An effective approach to numerical soliton solutions for the Schrödinger equation via modified cubic B-spline differential quadrature method,” Chaos, Solit. Fractals, vol. 100, pp. 45–56, 2017. https://doi.org/10.1016/j.chaos.2017.04.038.Search in Google Scholar