Year 2019, Volume 7, Issue 1, Pages 7 - 15 2019-04-15

Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations

Hami Gündoğdu [1] , Ömer Faruk Gözükızıl [2]

37 105

In this paper, we put forward a powerful method, named as the double Laplace decomposition method, for obtaining exact solutions of nonlinear partial differential equations subject to initial conditions. We especially interested in Hirota, Schrödinger and complex modified KdV equations with their initial conditions. The double Laplace deceomposition method is applied to these equations. We then gain complex-valued solutions, yield the given initial conditions. Moreover, we give some nonlinear partial equations to demonstrate that this method effective, useful, and powerful tool for getting real-valued functions.



Double Laplace transform, decomposition method, exact solution, Hirota equation, Schrödinger equation, complex mKdV equation
  • [1] N. Damil, M. Potier-Ferry, A. Najah, R. Chari, and H. Lahmam, An iterative method based upon Pade approximamants, Comm. In Num. Meth.s In Eng., Vol:15, (1999), 701-708.
  • [2] G. L. Liu, New research directions in singular perturbation theory: artificial parameter approach and inverse-perturbation technique, Proceeding of the 7th Conference of the Modern Mathematics and Mechanics, Shanghai, Vol:61, (1997), 47-53.
  • [3] J. M. Cadou, N. Moustaghfir, E. H. Mallil, N. Damil, and M. Potier-Ferry, Linear iterative solvers based on pertubration techniques, Comput. Ren. Math., Vol:332, (2001), 457-462.
  • [4] E. Mallil, H. Lahmam, N. Damil, and M. Potier-Ferry, An iterative process based on homotopy and perturbation techniques, Comput. Meth. In Appl. Mech. Eng., Vol:190, (2000), 1845-1858.
  • [5] J.H. He, An approximate solution technique depending upon artificial parameter, Comm. In Non. Sci. And Num. Simul., Vol:3, (1998), 92-97.
  • [6] C. M. Bender, K. S. Pinsky, and L. M. Simmons, A new perturbative approach to nonlinear problems, J. Of Math. Phys., Vol:30, (1989), 1447-1455.
  • [7] H. Gündoğdu, and Ö . F. Gözükızıl, Obtaining the solution of Benney-Luke Equation by Laplace and adomian decomposition methods, S.A.U J. Of Sci., Vol:21, (2017), 1524-1528.
  • [8] G. Adomian, Nonlinear stochastic systems theory and applications to physics, Kluwer Academic Publishers, 1989.
  • [9] G. Adomian, Solving frontier problems of physics: the decomposition method, Kluwer Academic Publishers-Plenum, Springer Netherlands, 1994.
  • [10] G. Adomian, Solution of physical problems by decomposition, Comput. And Math. with Appl., Vol:27, (1994), 145-154.
  • [11] G. Adomian, Solution of nonlinear P.D.E, Appl.Math. Lett., Vol:11, (1998), 121-123.
  • [12] G. Adomian, and R. Rach, Inhomogeneous nonlinear partial differential equations with variable coefficients, Appl.Math. Lett., Vol:5, (1992), 11-12.
  • [13] G. Adomian, and R. Rach, Modified decomposition solutions of nonlinear partial differential equations, Appl.Math. Lett., Vol:5, (1992), 29-30.
  • [14] G. Adomian, and R. Rach, A modified decomposition series, Comput. And Math. with Appl., Vol:23, (1992), 17-23.
  • [15] H. Gündoğdu, and Ö . F. Gözükızıl, Solving Nonlinear Partial Differential Equations by Using Adomian Decomposition Method,Modified Decomposition Method and Laplace Decomposition Method, MANAS J. Of Eng, Vol:5, (2017), 1-13.
  • [16] S. A. Khuri, A laplace decomposition algorithm applied to class of nonlinear differential equations, J. Of Math. Anal.And Appl., Vol:1, (2001), 141-155.
  • [17] K. Majid, M. Hussain, J. Hossein, and K. Yasir, Application of Laplace decomposition method to solve nonlinear coupled partial differential equations, W. Appl. Sci. J., Vol:9, (2010), 13-19.
  • [18] H. Hosseinzadeh, H. Jafari, and M. Roohani, Application of Laplace decomposition method for solving Klein-Gordon equation, W. Appl. Sci. J., Vol:8, (2010), 809-813.
  • [19] A. Aghili, and B. P. Moghaddam, Certain theorems on two dimensional Laplace transform and nonhomogeneous parabolic partial differential equations, Surveys in Math. and Appl., Vol:6, 2011, 165-174.
  • [20] H. Eltayeb, and A. Kilicman, A note on solutions of wave, Laplace’s and heat equations with convolution terms by using a double Laplace transform, Appl. Math. Lett., Vol:21, (2008), 1324-1329.
  • [21] A. Kilicman, H. Eltayeb, A note on defining singular integral as distribution and partial differential equations with convolution term, Math. and Comput. Mod., Vol:49, (2009), 327-336.
  • [22] T. Elzaki, Dobule Laplace variational iteration method for solution of nonlinear concolution partial differential equations, Arch. Des Sci. Vol:65, No.12 (2012), 588-593.
  • [23] H. Eltayeb, A. Kilicman, A note on double Laplace transform and telegraphic equations, Abst. and Appl.Anal. Vol: 2013.
  • [24] L. Debnath, The double Laplace transforms and their properties with applications to Functional, Integral and Partial Differential Equations, Int. J. Appl. Comput. Math, (2016).
  • [25] R. Dhunde, and G. L. Waghmare, Solving partial integro-differential equations using double Laplace transform method, American J. of Comput. and Appl. Math., Vol:5, No.1 (2015), 7-10.
  • [26] R. Hirota, Exact envelope-soliton solutions of a nonlinear wave equation, J. Math. Phys. Vol.14, (1973), 805-809.
  • [27] P. Wang, B. Tian, W.J. Liu, M. Li, and K. Sun, Soliton solutions for a generalized inhomogeneous variable-coefficient Hirota equation with symbolic computation, Stud. Appl. Math. vol. 125, (2010), 213–222.
  • [28] E. Fan, and J. Zhang, Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A. Vol:305, (2002), 383–392.
  • [29] P. Wang, B. Tian, W.J. Liu, M. Li, and K. Sun, Soliton solutions for a generalized inhomogeneous variable-coefficient Hirota equation with symbolic computation, Stud. Appl. Math. Vol:125, (2010), 213–222.
  • [30] L. Li, Z. Wu, L. Wang, and J. He, High-order rogue waves for the Hirota equation, Ann. Phys. Vol:334, (2013), 198–211.
  • [31] J.J. Shu, Exact n-envelope-soliton solutions of the Hirota equation, Opt. Appl. Vol:33 (2003), 539–546.
  • [32] M. Eslami, M.A. Mirzazadeh, A. Neirameh, New exact wave solutions for Hirota equation, Pramana – J. Phys. Vol:84, (2015), 3–8.
  • [33] V. E. Zakharov and A. B. Shabat, Exact theory on two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Sov. Phys. Vol:34, (1972), 62-69.
  • [34] W.X. Ma, and M. Chen, Direct search for exact solutions to the nonlinear Schr¨odinger equation, Appl. Math. Comput. Vol:215, (2009), 2835–2842.
  • [35] Y. Zhou, M. Wang, and T. Miao, The periodic wave solutions and solitary for a class of nonlinear partial differential equations, Phys. Lett. A. Vol:323, (2004), 77–88.
  • [36] H. Eleuch, Y. V. Rostovtsev, and M. O. Scully, New analytic solution of Schr¨odinger’s equation, EPL, Vol: 89, No.5 (2010), 50004.
  • [37] M. Sindelka, H. Eleuch, and Y. V. Rostovtsev, Analytical approach to 1D bound state problems, Eur. Phys. J. D Vol: 66, (2012), 224.
  • [38] Ablowitz, M. J., Clarkson, P. A., Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, New York, 1991.
  • [39] R. F. Rpdriguez,J.A Reyes, A. Espinosa-Ceron, J. Fujioka, and B. A. Malomed, Standard and embedded solitons in nematic optical fibers, Phys. Rev. E. Vol.68, (2003), 036606.
  • [40] J.S. He, L.H. Wang, L.J. Li, K. Porsezian, and R. Erd´elyi, Few-cycle optical rogue waves: complex modified Korteweg-de Vries equation, Phys. Rev. E. Vol:89, (2014), 062917.
  • [41] A. Estrin, and T. J. Higgins, The solution of boundary value problems by multiple Laplace transformation, J. Franklin Ins., Vol:252, No.2 (1951),153-167.
  • [42] L. Debnath, The double Laplace transforms and their properties with applications to functional, integral and partial differential equations, Int. J. Appl. Comput. Math., Vol:2, (2016), 223-241.
Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Author: Hami Gündoğdu (Primary Author)
Institution: SAKARYA UNIVERSITY
Country: Turkey


Author: Ömer Faruk Gözükızıl
Institution: SAKARYA ÜNİVERSİTESİ
Country: Turkey


Dates

Publication Date: April 15, 2019

Bibtex @research article { konuralpjournalmath527635, journal = {Konuralp Journal of Mathematics (KJM)}, issn = {}, eissn = {2147-625X}, address = {Mehmet Zeki SARIKAYA}, year = {2019}, volume = {7}, pages = {7 - 15}, doi = {}, title = {Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations}, key = {cite}, author = {Gündoğdu, Hami and Gözükızıl, Ömer Faruk} }
APA Gündoğdu, H , Gözükızıl, Ö . (2019). Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations. Konuralp Journal of Mathematics (KJM), 7 (1), 7-15. Retrieved from http://dergipark.org.tr/konuralpjournalmath/issue/31492/527635
MLA Gündoğdu, H , Gözükızıl, Ö . "Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations". Konuralp Journal of Mathematics (KJM) 7 (2019): 7-15 <http://dergipark.org.tr/konuralpjournalmath/issue/31492/527635>
Chicago Gündoğdu, H , Gözükızıl, Ö . "Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations". Konuralp Journal of Mathematics (KJM) 7 (2019): 7-15
RIS TY - JOUR T1 - Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations AU - Hami Gündoğdu , Ömer Faruk Gözükızıl Y1 - 2019 PY - 2019 N1 - DO - T2 - Konuralp Journal of Mathematics (KJM) JF - Journal JO - JOR SP - 7 EP - 15 VL - 7 IS - 1 SN - -2147-625X M3 - UR - Y2 - 2019 ER -
EndNote %0 Konuralp Journal of Mathematics (KJM) Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations %A Hami Gündoğdu , Ömer Faruk Gözükızıl %T Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations %D 2019 %J Konuralp Journal of Mathematics (KJM) %P -2147-625X %V 7 %N 1 %R %U
ISNAD Gündoğdu, Hami , Gözükızıl, Ömer Faruk . "Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations". Konuralp Journal of Mathematics (KJM) 7 / 1 (April 2019): 7-15.
AMA Gündoğdu H , Gözükızıl Ö . Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations. Konuralp J. Math.. 2019; 7(1): 7-15.
Vancouver Gündoğdu H , Gözükızıl Ö . Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations. Konuralp Journal of Mathematics (KJM). 2019; 7(1): 15-7.