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Year 2019, Volume: 7 Issue: 1, 7 - 15, 15.04.2019

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References

  • [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.

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

Year 2019, Volume: 7 Issue: 1, 7 - 15, 15.04.2019

Abstract

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.



References

  • [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.
There are 42 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Hami Gündoğdu

Ömer Faruk Gözükızıl

Publication Date April 15, 2019
Submission Date February 15, 2019
Acceptance Date April 3, 2019
Published in Issue Year 2019 Volume: 7 Issue: 1

Cite

APA Gündoğdu, H., & Gözükızıl, Ö. F. (2019). Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations. Konuralp Journal of Mathematics, 7(1), 7-15.
AMA Gündoğdu H, Gözükızıl ÖF. Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations. Konuralp J. Math. April 2019;7(1):7-15.
Chicago Gündoğdu, Hami, and Ömer Faruk Gözükızıl. “Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex MKdV Equations”. Konuralp Journal of Mathematics 7, no. 1 (April 2019): 7-15.
EndNote Gündoğdu H, Gözükızıl ÖF (April 1, 2019) Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations. Konuralp Journal of Mathematics 7 1 7–15.
IEEE H. Gündoğdu and Ö. F. Gözükızıl, “Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations”, Konuralp J. Math., vol. 7, no. 1, pp. 7–15, 2019.
ISNAD Gündoğdu, Hami - Gözükızıl, Ömer Faruk. “Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex MKdV Equations”. Konuralp Journal of Mathematics 7/1 (April 2019), 7-15.
JAMA Gündoğdu H, Gözükızıl ÖF. Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations. Konuralp J. Math. 2019;7:7–15.
MLA Gündoğdu, Hami and Ömer Faruk Gözükızıl. “Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex MKdV Equations”. Konuralp Journal of Mathematics, vol. 7, no. 1, 2019, pp. 7-15.
Vancouver Gündoğdu H, Gözükızıl ÖF. Double Laplace Decomposition Method and Exact Solutions of Hirota, Schrödinger and Complex mKdV Equations. Konuralp J. Math. 2019;7(1):7-15.
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