Abstract
In this article, we apply the newly introduced numerical method which is a combination of Sumudu transforms and Homotopy analysis method for the solution of time fractional third order dispersive type PDE equations. It is also discussed generalized algorithm, absolute convergence and analytic result of the finite number of independent variables including time variable.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Oldham, K.B., Spanier, J.: The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order. Academic Press, California (1974)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Part II. Geophy. J. R. Astron. Soc. 13, 529–39 (1967)
Ross, B.: The development of fractional calculus 1695–1900. Hist. Math. 4, 75–89 (1977)
Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amstrdam (2006)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Jan Van Mill, North–Holland (2006)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, New York (2010)
Caponetto, R., Dongola, G., Fortuna, L., Petras, I.: Fractional Order Systems: Modeling and Control Applications. WSPC, Singapore (2010)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integration and Derivatives: Theory and Application. G & B, Amsterdam (1993)
Uchaikin, V.V.: Self–similar anomalous diffusion and levy stable laws. Phys. Usp. 26(8), 821–849 (2003)
Tarasov, V.E.: Fractional generalization of gradient and Hamiltonian systems. J. Phys. A Math. Gen. 38(26), 5929–5943 (2005)
Laskin, N., Zaslavsky, G.M.: Nonlinear fractional dynamics of lattice with long–range interaction. Phys. A Stat. Mech. Appl. 368(1), 38–54 (2006)
Tarasov, V.E.: Gravitational field of fractal distribution of particles. Celest. Mech. Dyn. Astron. 94(1), 1–15 (2006)
Fujioka, J.: Lagrangian structure and Hamiltonian conservation in fractional optical solitons. Commun. Fract. Calc. 1(1), 1–14 (2010)
Tarasov, V.E.: Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg (2010)
El–Wakil, S.A., Abulwafa, E.M., El–Shewy, E.K., Mahmoud, A.A.: Time–fractional KdV equation for electron–acoustic waves in plasma of cold electron and two different temperature isothermal ions. Astrophys. Space Sci. 333(1), 269–267 (2011)
El–Wakil, S.A., Abulwafa, E.M., El–Shewy, E.K., Mahmoud, A.A.: Time–fractional study of electron acoustic solitary waves in plasma of cold electron and two isothermal ions. J. Plasma Phys. 78(6), 641–649 (2012)
Petr, I.: Fractional–Order Nonlinear Systems Modelin. Analysis and Simulation. Higher Education Press and Springer, Beijing (2011)
Klafter, J., Lim, S.C., Metzler, R. (eds.): Fractional Dynamics: Recent Advances. World Scientific, Singapore (2012)
Baleanu, D., Tenreiro Machado, J.A., Luo, A.C.J. (eds.): Fractional Dynamics and Control. Springer, New York (2012)
Zhou, J.K.: Differential Transformation Applications for Electrical Circuits. Wuhan Huazhong University Press (1986)
Chen, C.K., Ho, S.H.: Solving partial differential equations by two–dimensional differential transform method. Appl. Math. Comput. 106(2–3), 171–179 (1999)
Jang, M.J., Cheng, C.L., Liu, Y.C.: Two–dimensional differential transform for partial differential equations. Appl. Math. Comput. 121(2–3), 261–270 (2001)
Yang, S., Xiao, A., Su, H.: Convergence of the variational iteration method for solving multi–order fractional differential equations. Comput. Math. Appl. 60(10), 2871–2879 (2010)
Wazwaz, A.M.: The variational iteration method for analytic treatment for linear and nonlinear ODEs. Appl. Math. Comput. 212(1), 120–134 (2009)
Saravanan, A., Magesh, N.: An efficient computational technique for solving the Fokker–Planck equation with space and time fractional derivatives. J. King Saud Univ. Sci. 28, 160–166 (2016)
Balaji, S.: Legendre wavelet operational matrix method for solution of fractional order Riccati differential equation. J. Egyptian Math. Soc. 23(2), 263–270 (2015)
Erturk, V.S., Momani, S., Odibat, Z.: Application of generalized differential transform method to multi–order fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 13(8), 1642–1654 (2008)
Zhao, J., Tang, B., Kumar, S., Hou, Y.: The extended fractional sub–equation method for nonlinear fractional differential equations. Math. Probl. Eng 2012(11), 924956 (2012)
He, J.H.: Asymptotology by homotopy perturbation method. Appl. Math. Comput. 156(3), 591–596 (2004)
He, J.H.: The homotopy perturbation method for non–linear oscillators with discontinuities. Appl. Math. Comput. 151(1), 287–292 (2004)
He, J.H.: A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int. J. Non–Linear Mech. 35(1), 37–43 (2000)
He, J.H.: Application of homotopy perturbation method to nonlinear wave equations. Chaos Solit Fractals. 26(3), 695–700 (2000)
Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai, Jiao Tong University (1992)
Liao, S.J.: Beyond Perturbation Introduction to the Homotopy Analysis Method. Chapman & Hall CRC Press, Washington DC (2004)
Liao, S.J.: Homotopy Analysis Method in Nonlinear Differential Equations. Springer, Heidelberg (2012)
Liao, S.J.: Advances in the Homotopy Analysis Method. World Scientific Press, Singapore (2014)
Kumar, S.: An analytical algorithm for nonlinear fractional Fornberg–Whitham equation arising in wave breaking based on a new iterative method. Alexand. Eng. J. 53(1), 225–231 (2014)
Kumar, S., Singh, J., Kumar, D., Kapoor, S.: New homotopy analysis transform algorithm to solve volterra integral equation. Ain Shams Eng. J. 5(1), 243–246 (2014)
Yin, X.B., Kumar, S., Kumar, D.: A modified homotopy analysis method for solution of fractional wave equations. Adv. Mech. Eng. 7(12), 1–8 (2015)
Kumar, S., Kumar, A., Baleanu, D.: Two analytical methods for time–fractional nonlinear coupled Boussinesq–Burger‘s equations arise in propagation of shallow water waves. Nonlinear Dyn. 85(2), 699–715 (2016)
Kumar, S., Kumar, D., Singh, J.: Fractional modelling arising in unidirectional propagation of long waves in dispersive media. Adv. Nonlinear Anal (2016). doi:10.1515/anona-2013-0033. ISSN (Online) 2191–950X, ISSN (Print) 2191–9496
Arikoglu, A., Ozkol, I.: Solution of a fractional differential equations by using differential transform method. Chaos Solit. Fractals. 34(4), 1473–1481 (2007)
Erturk, V.S., Momani, S.: Solving systems of fractional differential equations using differential transform method. J. Comput. Appl. Math. 215(1), 142–151 (2008)
Pandey, R.K., Mishra, H.K.: Homotopy analysis fractional Sumudu transform method for time–fractional fourth order differential equations with variable coefficients. Am. J. Numer. Anal. 3(3), 52–64 (2015)
Waleed, H.: Solving nth –order integro–differential equations using the combined laplace transform–Adomian decomposition method. Am. J. Appl. Math. 4(6), 882–886 (2013)
Haghighi, A.R., Dadvand, A., Ghejlo, H.H.: Solution of the fractional diffusion equation with the Riesz fractional derivative using McCormack method. Commun. Adv. Comput. Sci. App. 2014(11), 00024 (2012)
Watugala, G.K.: Sumudu transform–a new integral transform to solve differential equations and control engineering problems. Math. Eng. Ind. 6(4), 319–329 (1998)
Weerakoon, S.: Application of Sumudu transform to partial differential equations. Int. J. Math. Educ. Sci. Tech. 25(2), 277–283 (1994)
Weerakoon, S.: Complex inversion formula for Sumudu transform. Int. J. Math. Educ. Sci. Tech. 29(4), 618–621 (1998)
Belgacem, F.B.M., Karaballi, A.A., Kalla, S.L.: Analytical investigations of the Sumudu transform and applications to integral production equations. Math. Probl. Eng. 2003(3), 103–118 (2003)
Belgacem, F.B.M., Karaballi, A.A.: Sumudu transform fundamental properties investigations and applications. J. Appl. Math. Stoch. Anal. 2006(23), 91083 (2006)
Belgacem, F.B.M.: Introducing and analysing deeper Sumudu properties. Nonlinear Studies 13(1), 23–41 (2006)
Hussain, M.G.M., Belgacem, F.B.M.: Transient solutions of Maxwell‘s equations based on Sumudu transform. Prog. Electromagn. Res. 74, 273–289 (2007)
Belgacem, F.B.M.: Sumudu applications to Maxwell‘s equations. Prog. Electromagn. Res. 5(4), 355–360 (2009)
Belgacem, F.B.M.: Sumudu transform applications to bessel functions and equations. Appl. Math. Sci. 4(74), 3665–3686 (2010)
Gupta, V.G., Sharma, B., Belgacem, F.B.M.: On the solutions of generalized fractional kinetic equations. Appl. Math. Sci. 5(19), 899–910 (2011)
Katatbeh, Q.D., Belgacem, F.B.M.: Applications of the Sumudu transform to fractional differential equations. Nonlinear Studies 18(1), 99–112 (2011)
Guerrero, F., Santonja, F.J., Villanueva, R.J.: Solving a model for the evolution of smoking habit in Spain with homotopy analysis method. Nonlinear Anal. Real World Appl. 14(1), 549–558 (2013)
Jafari, H., Firoozjaee, M.: A multistage homotopy analysis method for solving non–linear Riccati differential equations. IJRRAS 4(2), 128–132 (2010)
Alomari, A.K., Noorani, M.S.M., Nazar, R., Li, C.P.: Homotopy analysis method for solving fractional Lorenz system. Commun. Nonlinear Sci. Numer. Simulat. 15(7), 1864–1872 (2010)
Elsaid, A.: Homotopy analysis method for solving a class of fractional partial differential equations. Commun. Nonlinear Sci. Numer. Simulat. 16(9), 3655–3664 (2011)
Mohyud–Din, S.T., Yldrm, A., Ylkl, E.: Homotopy analysis method for space– and time–fractional kdv equation. Int. J. Numer. Method H. 22(7), 928–941 (2012)
Fadravi, H.H., Nik, H.S., Buzhabadi, R.: Homotopy analysis method for solving foam drainage equation with space– and time–fractional derivatives. Int. J. Differ. Equ. 2011(12), 237045 (2011)
Djidjeli, K., Twizell, E.H.: Global extrapolations of numerical methods for solving a third–order dispersive partial differential equations. Int. J. Comput. Math. 41(1–2), 81–89 (1999)
Twizell, E.H.: Computational Methods for Partial Differential Equations. Ellis Horwood New York: Chichester and John Wiley and Sons (1984)
Lewson, J.D., Morris, J.L.I.: The extrapolation of first–order methods for parabolic partial differential equations. I. SIAM J. Numer. Anal. 15(6), 1212–12124 (1978)
Mengzhao, Q.: Difference scheme for the dispersive equation. Computing 31, 261–261 (1983)
Wazwaz, A.M.: An analytic study on the third–order dispersive partial differential equation. Appl. Math. Comput. 142(2–3), 511–520 (2003)
Kanth, A.S.V.R., Aruna, K.: Solution of fractional third–order dispersive partial differential equations. J. Egyptian Math. Soc. 2(3), 190–199 (2015)
Luchko, Y., Gorenflo, R.: An operational method for solving fractional differential equations with the Caputo derivatives. Acta. Math. Vietnamica. 24(2), 207–233 (1999)
Moustafa, O.L.: On the Cauchy problem for some fractional order partial differential equations. Chaos Solit Fractals. 18(1), 135–140 (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Helge Holden
Rights and permissions
About this article
Cite this article
Pandey, R.K., Mishra, H.K. Homotopy analysis Sumudu transform method for time—fractional third order dispersive partial differential equation. Adv Comput Math 43, 365–383 (2017). https://doi.org/10.1007/s10444-016-9489-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-016-9489-5
Keywords
- Dispersive partial differential equation
- Homotopy analysis method
- Homotopy analysis Sumudu transform method
- Linear and nonlinear partial differential equation