American Journal of Numerical Analysis
ISSN (Print): 2372-2118 ISSN (Online): 2372-2126 Website: http://www.sciepub.com/journal/ajna Editor-in-chief: Emanuele Galligani
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American Journal of Numerical Analysis. 2015, 3(3), 52-64
DOI: 10.12691/ajna-3-3-1
Open AccessArticle

Numerical Simulation of Time-fractional Fourth Order Differential Equations via Homotopy Analysis Fractional Sumudu Transform Method

Rishi Kumar Pandey1 and Hradyesh Kumar Mishra1,

1Department of Mathematics, Jaypee University of Engineering and Technology, Guna, INDIA

Pub. Date: November 13, 2015

Cite this paper:
Rishi Kumar Pandey and Hradyesh Kumar Mishra. Numerical Simulation of Time-fractional Fourth Order Differential Equations via Homotopy Analysis Fractional Sumudu Transform Method. American Journal of Numerical Analysis. 2015; 3(3):52-64. doi: 10.12691/ajna-3-3-1

Abstract

The work provides an incipient analytical technique called the Homotopy Analysis Fractional Sumudu Transform Method (HAFSTM) for solving time-fractional fourth order differential equations with variable coefficients. The HAFSTM is the cumulation of the homotopy analysis method (HAM) and sumudu transform method (STM). The numerical simulation of the proposed method has the sundry applications, it can solve linear and nonlinear boundary value quandaries without utilizing Adomian polynomial, and He’s polynomial, which can be considered a clear advantage of this incipient algorithm. The solutions obtained by proposing technique are very lucid and less computationally implementable.

Keywords:
Homotopy Analysis Method Fractional Sumudu Transform Method Fractional Partial Differential equation variable coefficients boundary value problem

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References:

[1]  G.K. Watugala, Sumudu transform—a new integral transform to solve differential equations and control engineering problems,” Mathematical Engineering in Industry, vol. 6(4) (1998) 319-329.
 
[2]  S. Weerakoon, Application of Sumudu transform to partial differential equations, International Journal of Mathematical Education in Science and Technology, vol. 25(2) (1994) 277-283.
 
[3]  S. Weerakoon, Complex inversion formula for Sumudu transform, International Journal of Mathematical Education in Science and Technology, vol. 29(4) (1998) 618-621.
 
[4]  M. A. Asiru, Further properties of the Sumudu transform and its applications, International Journal of Mathematical Education in Science and Technology, vol. 33(3) (2002) 441-449.
 
[5]  A. Kadem, Solving the one-dimensional neutron transport equation using Chebyshev polynomials and the Sumudu transform, Analele Universitatii din Oradea, vol. 12 (2005)153-171.
 
[6]  A. Kılıc¸man, H. Eltayeb, and K. A. M. Atan, A note on the comparison between Laplace and Sumudu transforms, Iranian Mathematical Society, vol. 37(1) (2011)131-141.
 
[7]  A. Kılıc¸man and H. E. Gadain, On the applications of Laplace and Sumudu transforms, Journal of the Franklin Institute, vol. 347(5) (2010) 848-862.
 
[8]  H. Eltayeb, A. Kılıc¸man, and B. Fisher, A new integral transform and associated distributions, Integral Transforms and Special Functions, vol. 21(5-6) (2010) 367-379.
 
[9]  A. Kılıc¸man and H. Eltayeb, “A note on integral transforms and partial differential equations,” Applied Mathematical Sciences, vol. 4(1-4) (2010) 109-118.
 
[10]  G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., vol. 135(1988) 501-544.
 
[11]  N.T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput., vol. 131 (2002)517-529.
 
[12]  S.S. Ray, R.K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. Math. Comput., vol. 167(2005) 561-571.
 
[13]  H. Jafari, V. Daftardar-Gejji, Revised Adomian decomposition method for solving systems of ordinary and fractional differential equations, Appl. Math. Comput., vol. 181(1) (2006)598-608.
 
[14]  H. Jafari, V. Daftardar-Gejji, Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method, Appl. Math. Comput., vol. 180 (2) (2006) 700-706.
 
[15]  A. Golbabai, M. Javidi, Application of homotopy perturbation method for solving eighth-order boundary value problems, Appl. Math. Comput., vol. 191(1) (2007) 334-346.
 
[16]  J. H. He, A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, Int. J. Non-Linear Mech., vol. 35(1)(2000) 37-43.
 
[17]  J. H. He, The homotopy perturbation method for non-linear oscillators with discontinuities, Appl. Math. Comput., vol. 151(1) (2004) 287-292.
 
[18]  J. H. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos Solitons Fractals, vol. 26(3) (2005) 695-700.
 
[19]  J. H. He, Asymptotology by homotopy perturbation method, Appl. Math. Comput., vol. 156 (3) (2004) 591-596.
 
[20]  J. H. He, Homotopy perturbation method for solving boundary problems, Phys. Lett. A, vol. 350(12) (2006) 87-88.
 
[21]  Sh. S. Behzadi, Iterative methods for solving nonlinear Fokker-Plank equation, Int. J. Industrial Mathematics, vol. 3 (2011) 143-156.
 
[22]  Sh. S. Behzadi, Numerical solution of Sawada-Kotera equation by using iterative methods, Int. J. Industrial Mathematics, vol. 4 (2012)269-288.
 
[23]  M. Ghanbari, Approximate analytical solutions of fuzzy linear Fredholm integral equations by HAM, Int. J. Industrial Mathematics, vol. 4 (2012)53-67.
 
[24]  S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
 
[25]  M. Zurigat, S. Momani, A. Alawneh, Analytical approximate solutions of systems of fractional algebraic–differential equations by homotopy analysis method, Comput. Math. Appl., vol. 59 (3) (2010)1227-1235.
 
[26]  G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Acad. Publ., Boston, 1994.
 
[27]  Sh. S. Behzadi, Iterative methods for solving nonlinear Fokker-Plank equation, Int. J. Industrial Mathematics, vol. 3 (2011) 143-156.
 
[28]  J.H. He, Variational iteration method- a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics, vol. 34 (1999) 699-708.
 
[29]  J.H. He, X.H. Wu, Variational iteration method: new development and applications, Computers & Mathematics with Applications, vol. 54 (2007) 881-894.
 
[30]  J.H. He, G.C.Wu, F. Austin, The variational iteration method which should be followed, Nonlinear Science Letters A, vol. 1 (2009) 1-30.
 
[31]  E. Hesameddini, H. Latifizadeh, An optimal choice of initial solutions in the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10 (2009) 1389-1398.
 
[32]  E. Hesameddini, H. Latifizadeh, Reconstruction of variational iteration algorithms using the Laplace transform, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10 (2009) 1377-1382.
 
[33]  Yasir Khan, An effective modification of the Laplace decomposition method for nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10 (2009) 1373-1376.
 
[34]  S.A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics, vol. 1 (2001) 141-155.
 
[35]  AR. Vahidi, Gh. A. Cordshooli, On the Laplace transform decomposition algorithm for solving nonlinear differential equations, Int. J. Industrial Mathematics, vol. 3(2011) 17-23.
 
[36]  Y. Khan, Qingbiao. Wu, Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers & Mathematics with Applications, vol.61(8) (2011) 1963-1967.
 
[37]  J. Singh, D. Kumar, Sushila, Homotopy perturbation sumudu transform method for nonlinear equations, Adv. Theor. Appl. Mech., vol. 4(2011) 165-175.
 
[38]  S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Applied Mathematical Modelling vol. 38(13) (2014) 3154-3163.
 
[39]  M. S. Mohamed, F. Al-malki, M. Al-humyani, Homotopy Analysis Transform Method for Time-Space Fractional Gas Dynamics Equation, Gen. Math. Notes, vol. 24(1) (2014) 1-16.
 
[40]  S. Kumar, M. M. Rashidi, New analytical method for gas dynamics equation arising in shock fronts, Computer Physics Communications, vol. 185 (7) (2014)1947-1954.
 
[41]  A. Kılıc¸man, H. Eltayeb, and R. P. Agarwal, “On Sumudu transform and system of differential equations,” Abstract and Applied Analysis, Article ID598702, 11 pages, 2010.
 
[42]  J. Zhang, “A Sumudu based algorithm for solving differential equations,” Academy of Sciences of Moldova, vol. 15(3) (2007) 303-313.
 
[43]  V. B. L. Chaurasia and J. Singh, “Application of Sumudu transform in Sch¨odinger equation occurring in quantum mechanics,” Applied Mathematical Sciences, vol. 4(57-60), (2010)2843-2850.
 
[44]  N. A. Khan, N.U. Khan, M. Ayaz,A. Mahmood, N. Fatima, Numerical study of time- fractional fourth-order differential equations with variable coefficients, Journal of King Saud University – Science, vol. 23(1)(2011) 91-98.
 
[45]  Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnamica, 24 (1999) 207-233.
 
[46]  O. L. Moustafa, On the Cauchy problem for some fractional order partial differential equations, Chaos Solitons& Fractals, 18 (2003) 135–140.
 
[47]  I. Podlubny, Fractional Differential Equations, Academic, New York, 1999.
 
[48]  A.M. Wazwaz, Analytical treatment of variable coefficients fourth-order parabolic partial differential equations. Applied Mathematics Computation, vol. 123(2001) 219–227.
 
[49]  A.M. Wazwa, Exact solutions for variable coefficient fourth order parabolic partial differential equations in higher dimensional spaces, Applied Mathematics Computation vol. 130 (2002) 415-424.
 
[50]  J. Biazar, H. Ghazvini, He’s variational iteration method for fourth-order parabolic equation, Computers Mathematics with Applications vol.54 (2007) 1047-1054.