Main Article Content
This paper introduces the Multistep Modified Reduced Differential Transform Method (MMRDTM). It is applied to approximate the solution for Nonlinear Schrodinger Equations (NLSEs) of power law nonlinearity. The proposed method has some advantages. An analytical approximation can be generated in a fast converging series by applying the proposed approach. On top of that, the number of computed terms is also significantly reduced. Compared to the RDTM, the nonlinear term in this method is replaced by related Adomian polynomials prior to the implementation of a multistep approach. As a consequence, only a smaller number of NLSE computed terms are required in the attained approximation. Moreover, the approximation also converges rapidly over a wide time frame. Two examples are provided for showing the ability and advantages of the proposed method to approximate the solution of the power law nonlinearity of NLSEs. For pictorial representation, graphical inputs are included to represent the solution and show the precision as well as the validity of the MMRDTM.
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Kivshar YS, Agrawal GP. Optical solitons: From Fibers to Photonic Crystals. San Diego: Academic Press; 2003.
Agrawal GP. Nonlinear Fiber Optics. San Diego: Academic Press; 2006.
Mollenauer LF, Gordon JP. Solitons in Optical Fibers: Fundamentals and Applications. New York: Academic Press; 2006.
Porsezian K, Kuriakose VC. Solitons in Nonlinear Optics: Advances and Applications. Eur J Phys-Special Topics. Berlin, Germany: Springer-Verlag; 2009.
Wang LH, Porsezian K, He JS. Breather and rogue wave solutions of a generalized nonlinear Schrödinger equation. Phys Rev E. 2013;87: 053202. DOI: 10.1103/PhysRevE.87.053202
Wazwaz AM. Partial Differential Equations: Methods and Applications. Leiden, The Netherlands.: Balkema Publishers; 2002.
Marasi HR, Sharifi N, Piri H. Modified differential transform method for singular Lane-Emden equations in integer and fractional order. J Appl Eng Math. 2015;5(1): 124–131.
Benhammouda B, Leal HV. A new multistep technique with differential transform method for analytical solution of some nonlinear variable delay differential equations. SpringerPlus. Springer International Publishing; 2016; DOI: 10.1186/s40064-016-3386-8
Jameel AF, Anakira NR, Rashidi MM, Alomari AK, Saaban A, Shakhatreh MA. Differential Transformation Method For Solving High Order Fuzzy Initial Value Problems. Ital J Pure Appl Math. 2018;39(March): 194–208.
Rao TRR. Numerical solution of Sine Gordon equations through reduced differential transform method. Global J Pure Appl Math. 2017;13(7): 3879–3888.
Wazwaz AM. A study on linear and nonlinear Schrödinger equations by the variational iteration method. Chaos Solitons Fractals 2008;37(4): 1136–1142. DOI: 10.1016/j.chaos.2006.10.009
Inc M, Korpinar ZS. On approximate solutions of bright optical soliton for Schrödinger equation of power law nonlinearity. J Adv Phys. 2017;6: 534–539. DOI: 10.1166/jap.2017.1359
Kilic B, Inc M. Optical solitons for the Schrödinger–Hirota equation with power law nonlinearity by the Bäcklund transformation. Optik. 2017;138: 64–67. DOI: 10.1016/j.ijleo.2017.03.017
Khater MMA, Attia RAM, Abdel-Aty AH, Abdou MA, Eleuch H, Lu D. Analytical and semi-analytical ample solutions of the higher-order nonlinear Schrödinger equation with the non-Kerr nonlinear term. Results Phys. 2020 Mar 1;16:103000. DOI: 10.1016/j.rinp.2020.103000
Ray SS. Numerical solutions and solitary wave solutions of fractional KdV equations using modified fractional reduced differential transform method. Comp Math Math Phys. 2013;53(12): 1870–1881. DOI: 10.1007/s10910-013-0210-3
El-Zahar ER. Applications of adaptive multistep differential transform method to singular perturbation problems arising in science and engineering. Appl Math Inf Sci. 2015;9(1): 223–232. DOI: 10.12785/amis/090128
Che Hussin CH, Kilicman A, Azmi A. Analytical Solutions Of Nonlinear Schrodinger Equations Using Multistep Modified Reduced Differential Transform Method. An Int J Adv Comput Tech. 2018;7(11): 2939–2944.
Che Hussin CH, Md Ismail AI, Kilicman A, Azmi A. Analytical Solutions Of Non-Linear Klein-Gordon Equations Using Multistep Modified Reduced Differential Transform Method. Therm Sci. 2019;23(1): S317–S326.
Hussin CH, Ismail AI, Kilicman A, Azmi A. Approximate analytical solutions of fractional nonlinear Schrodinger equations using multistep modified reduced differential transform method. Proc Int Conf Math Sci Tech 2018. 2019;2184: 0600003. DOI: 10.1063/1.5136435
Che Hussin CH, Ismail AIM, Kilicman A, Azmi A. Approximate analytical solutions of nonlinear korteweg-de vries equations using multistep modified reduced differential transform method. Math Stat. 2020;8(2): 9–16. DOI: 10.13189/ms.2020.081302
Kataria K K , Vellaisamy P. Simple parametrization methods for generating Adomian polynomials. Appl Anal Discret Math. 2016;10(1): 168–185.
Keskin Y, Oturanç G. Reduced differential transform method for partial differential equations. Int J Nonlinear Sci Numer Simul. 2009;10(6): 741–749. DOI: 10.1515/IJNSNS.2009.10.6.741.