International Journal of Nonlinear Analysis and Applications
Numerical investigation for solving non-linear partial differential equation using Sumudu-Elzaki transform decomposition method
Document Type : Research Paper
Authors
1 Department of Accounting, Al-Esraa University College, Baghdad, Iraq
2 Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, Baghdad, Iraq
Abstract
In this study, we used a powerful method, named as Sumudu-Elzaki transform method (SETM) together with Adomian polynomials (APs), which can be used to solve non-linear partial differential equations. We will give the essential clarification of this method by expanding some numerical examples to exhibit the viability and the effortlessness of this technique which can be used to solve other non-linear problems.
Keywords
[1] S.A. Ahmed, T.M. Elzaki and A.A. Hassan, Solution of integral differential equations by new double integral transform (laplace–Sumudu transform), Abstr. Appl. Anal. 2020 (2020) 4725150.
[2] Z. Ahmed, M.I. Idrees, F.B. Belgacem and Z. Perveen, On the convergence of double Sumudu transform, J. Nonlinear Sci. Appl. 13 (2020) 154–162.
[3] A.A. Alderremy, T.M. Elzaki and M. Chamekh, New transform iterative method for solving some Klein-Gordon equations, Results Phys. 10 (2018) 655–659.
[4] M.G. AL-Safi, W.R. Abd AL-Hussein and A.G.N. Al-Shammari, A new approximate solution for the Telegraph equation of space-fractional order derivative by using Sumudu method, Iraqi J. Sci., 59(3A) (2018) 1301–1311.
[5] A. Atangana and A. Kilicman, The use of Sumudu transform for solving certain nonlinear fractional heat-like equations, Abstr. Appl. Anal. 2013 (2013) 737481.
[6] S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton, N.J., USA, Princeton University Press, 1949.
[7] R.N. Bracewell and R. Bracewell, The Fourier Transform and Its Applications, 3rd edition, Boston, USA, McGraw-Hill, 2000.
[8] H. Bulut, H.M. Baskonus and F.B.M. Belgacem, The analytical solution of some fractional ordinary differential equations by the Sumudu transform method, Abstr. Appl. Anal. 2013 (2013) 1–6.
[9] A.M. Elsheikh and T. M. Elzaki, Modified variation iteration method and laplace–Elzaki transform for solve Hirota, Schrodinger and Complex Mkdv Equation, World J. Engin. Res. Tech. 6(6) (2020) 128–136.
[10] H. Eltayeb, A. Kilicman and S. Mesloub, Modified laplace decomposition method for solving system of equations Emden-Fowler type, J. Comput. Theor. Nanosci. 12 (2015) 5297–5301.
[11] H. Eltayeb, A. Kilicman and S. Mesloub, Application of double laplace Adomian decomposition method for solving linear singular one dimensional thermo-elasticity coupled system, J. Nonlinear Sci. Appl. 10 (2016) 278–289.
[12] T.M. Elzaki, Double laplace variational iteration method for solution of nonlinear convolution partial differential equations, Arch. Des. Sci. 65 (2012) 588.
[13] T.M. Elzaki and H. Kim, The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method, Int. J. Math. Anal., 9 (2015) 1065–1071.
[14] T.M. Elzaki and A. Mousa, On the convergence of triple Elzaki transform, SN Appl. Sci. 1 (2019) 275.
[15] H. Gadain and I. Bachar, On a nonlinear singular one-dimensional parabolic equation and double laplace decom position method, Adv. Mech. Engin. 9 (2017) 1–7.
[16] H. Gadain, S. Mesloub and A. Kilicman, Application of double laplace decomposition method to solve a singular one-dimensional pseudo hyperbolic equation, Adv. Mech. Engin. 9 (2017) 1–9.
[17] M.A. Hassan and T.M. Elzaki, Double Elzaki transform decomposition method for solving non-linear partialdifferential equations, J. Appl. Math. Phys. 8 (2020) 1463–1471.
[18] M.A. Hassan and T.M. Elzaki, Double Elzaki transform decomposition method for solving third order KortewegDe-Vries equations, J. Appl. Math. and Phys. 9 (2021) 21–30.
[19] J.-H. He and X.-H. Wu, Variational iteration method: new development and applications, Comput. and Math.
With Appl. 54(7-8) (2007) 881–894.
[20] M.I. Idrees, Z. Ahmed, M. Awais and Z. Perveen, On the convergence of double Elzaki transform, Int. J. Advanced
and Appl. Sci. 5 (2018) 19–24.
[21] A. Kilicman and H. Eltayeb, On new integral transform and differential equations, J. Math. Probl. Eng. 2010
(2010) 1–13.
[22] O.H. Mohammed and H.A. Salim, Computational methods based laplace decomposition for solving nonlinear
system of fractional order differential equations, Alexandria Eng. J. 57 (2018) 3549–3557.
[23] R.I. Nuruddeen, Elzaki decomposition method and its applications in solving linear and nonlinear Schrodinger
equations, Sohag J. Math. 4 (2017) 31–35.[24] T. Patel and R. Meher, Adomian decomposition Sumudu transform method for convective fin with temperaturedependent internal heat generation and thermal conductivity of fractional order energy balance equation, Int. J.
Appl. Comput. Math. 3 (2017) 1879–1895.
[25] H.M. Srivastava, A.K. Golmankhaneh, D. Baleanu and X.Y. Yang, Local fractional Sumudu transform with
application to IVPs on Cantor sets, Abstr. Appl. Anal. 2014 (2014) 1–7.
[26] A.M. Wazwaz, Partial Differential Equations Solitary Waves’ Theory, Berlin, Springer, 2009.
[27] X.Y. Yang, Y. yang, C. Cattani and M. Zhu, A new technique for solving 1-D Burgers equation, Thermal Sci. 21
(2017) 129–136.
[28] D. Ziane and M.H. Cherif, Resolution of nonlinear partial differential equations by Elzaki transform decomposition
method, J. Approx. Theory Appl. Math. 5 (2015) 18–30.
[2] Z. Ahmed, M.I. Idrees, F.B. Belgacem and Z. Perveen, On the convergence of double Sumudu transform, J. Nonlinear Sci. Appl. 13 (2020) 154–162.
[3] A.A. Alderremy, T.M. Elzaki and M. Chamekh, New transform iterative method for solving some Klein-Gordon equations, Results Phys. 10 (2018) 655–659.
[4] M.G. AL-Safi, W.R. Abd AL-Hussein and A.G.N. Al-Shammari, A new approximate solution for the Telegraph equation of space-fractional order derivative by using Sumudu method, Iraqi J. Sci., 59(3A) (2018) 1301–1311.
[5] A. Atangana and A. Kilicman, The use of Sumudu transform for solving certain nonlinear fractional heat-like equations, Abstr. Appl. Anal. 2013 (2013) 737481.
[6] S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton, N.J., USA, Princeton University Press, 1949.
[7] R.N. Bracewell and R. Bracewell, The Fourier Transform and Its Applications, 3rd edition, Boston, USA, McGraw-Hill, 2000.
[8] H. Bulut, H.M. Baskonus and F.B.M. Belgacem, The analytical solution of some fractional ordinary differential equations by the Sumudu transform method, Abstr. Appl. Anal. 2013 (2013) 1–6.
[9] A.M. Elsheikh and T. M. Elzaki, Modified variation iteration method and laplace–Elzaki transform for solve Hirota, Schrodinger and Complex Mkdv Equation, World J. Engin. Res. Tech. 6(6) (2020) 128–136.
[10] H. Eltayeb, A. Kilicman and S. Mesloub, Modified laplace decomposition method for solving system of equations Emden-Fowler type, J. Comput. Theor. Nanosci. 12 (2015) 5297–5301.
[11] H. Eltayeb, A. Kilicman and S. Mesloub, Application of double laplace Adomian decomposition method for solving linear singular one dimensional thermo-elasticity coupled system, J. Nonlinear Sci. Appl. 10 (2016) 278–289.
[12] T.M. Elzaki, Double laplace variational iteration method for solution of nonlinear convolution partial differential equations, Arch. Des. Sci. 65 (2012) 588.
[13] T.M. Elzaki and H. Kim, The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method, Int. J. Math. Anal., 9 (2015) 1065–1071.
[14] T.M. Elzaki and A. Mousa, On the convergence of triple Elzaki transform, SN Appl. Sci. 1 (2019) 275.
[15] H. Gadain and I. Bachar, On a nonlinear singular one-dimensional parabolic equation and double laplace decom position method, Adv. Mech. Engin. 9 (2017) 1–7.
[16] H. Gadain, S. Mesloub and A. Kilicman, Application of double laplace decomposition method to solve a singular one-dimensional pseudo hyperbolic equation, Adv. Mech. Engin. 9 (2017) 1–9.
[17] M.A. Hassan and T.M. Elzaki, Double Elzaki transform decomposition method for solving non-linear partialdifferential equations, J. Appl. Math. Phys. 8 (2020) 1463–1471.
[18] M.A. Hassan and T.M. Elzaki, Double Elzaki transform decomposition method for solving third order KortewegDe-Vries equations, J. Appl. Math. and Phys. 9 (2021) 21–30.
[19] J.-H. He and X.-H. Wu, Variational iteration method: new development and applications, Comput. and Math.
With Appl. 54(7-8) (2007) 881–894.
[20] M.I. Idrees, Z. Ahmed, M. Awais and Z. Perveen, On the convergence of double Elzaki transform, Int. J. Advanced
and Appl. Sci. 5 (2018) 19–24.
[21] A. Kilicman and H. Eltayeb, On new integral transform and differential equations, J. Math. Probl. Eng. 2010
(2010) 1–13.
[22] O.H. Mohammed and H.A. Salim, Computational methods based laplace decomposition for solving nonlinear
system of fractional order differential equations, Alexandria Eng. J. 57 (2018) 3549–3557.
[23] R.I. Nuruddeen, Elzaki decomposition method and its applications in solving linear and nonlinear Schrodinger
equations, Sohag J. Math. 4 (2017) 31–35.[24] T. Patel and R. Meher, Adomian decomposition Sumudu transform method for convective fin with temperaturedependent internal heat generation and thermal conductivity of fractional order energy balance equation, Int. J.
Appl. Comput. Math. 3 (2017) 1879–1895.
[25] H.M. Srivastava, A.K. Golmankhaneh, D. Baleanu and X.Y. Yang, Local fractional Sumudu transform with
application to IVPs on Cantor sets, Abstr. Appl. Anal. 2014 (2014) 1–7.
[26] A.M. Wazwaz, Partial Differential Equations Solitary Waves’ Theory, Berlin, Springer, 2009.
[27] X.Y. Yang, Y. yang, C. Cattani and M. Zhu, A new technique for solving 1-D Burgers equation, Thermal Sci. 21
(2017) 129–136.
[28] D. Ziane and M.H. Cherif, Resolution of nonlinear partial differential equations by Elzaki transform decomposition
method, J. Approx. Theory Appl. Math. 5 (2015) 18–30.
)
Volume 13, Issue 1
March 2022Pages 963-973
- Receive Date: 11 June 2021
- Revise Date: 25 July 2021
- Accept Date: 13 August 2021