On the Solution of Burgers’ Equation with the New Fractional Derivative

• Autorzy: Ali Kurt , Yücel Çenesiz , Orkun Tasbozan
• Języki publikacji: EN

Abstrakty

EN

Firstly in this article, the exact solution of a time fractional Burgers’ equation, where the derivative is conformable fractional derivative, with dirichlet and initial conditions is found byHopf-Cole transform. Thereafter the approximate analytical solution of the time conformable fractional Burger’s equation is determined by using a Homotopy Analysis Method(HAM). This solution involves an auxiliary parameter ~ which we also determine. The numerical solution of Burgers’ equation with the analytical solution obtained by using the Hopf-Cole transform is compared.

Słowa kluczowe

EN

Hopf-Cole Transform; Time Fractional Burgers’Equation; Conformable Fractional Derivative; Time FractionalHeat Equation; Homotopy Analysis Method

• Czasopismo: Open Physics
• Rocznik: 2015
• Tom: 13
• Numer: 1

Daty

otrzymano

2015-08-25

zaakceptowano

2015-11-10

online

2015-12-04

Twórcy

autor

Ali Kurt

• Department of Mathematics,
  Mustafa Kemal University, Hatay, Türkiye

autor

Yücel Çenesiz

• Department of Mathematics, Selçuk University,
  Konya, Türkiye

autor

Orkun Tasbozan

• Department of Mathematics,
  Mustafa Kemal University, Hatay, Türkiye

Bibliografia

• [1] K.S. Miller, B. Ross, An Introduction to the Fractional Calculusand Fractional Differential Equations, (John Wiley & Sons, NewYork, 1993)
• [2] A.Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applicationsof Fractional Differential Equations, (Elsevier, San Diego, 2006)
• [3] I. Podlubny, Fractional Differential Equations, (AcademicPress,San Diego, 1999)
• [4] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, J. Comput. Appl.Math. 264, 65 (2014)
• [5] T. Abdeljawad, J. Comput. Appl. Math. 279, 57 (2015)
• [6] N. Benkhettoua, S. Hassania, D.F.M. Torres, J. King Saud Univ.-Sci. (In Press)(2015)
• [7] M.A.Hammad, R. Khalil, Int. J. of Pure Appl.Math. 94, 215 (2014)
• [8] W.S. Chung, J. Comput. Appl. Math. 290, 150 (2015)
• [9] H. Bateman, Monthly Weather Rev. 43, 163 (1915)[Crossref]
• [10] J.M. Burger, Trans. Roy. Neth. Acad. Sci. Amsterdam 17, 1 (1939)
• [11] J.M. Burger, , Adv. Appl. Mech. 1, 171 (1948)
• [12] L. Debnath, Partial Differential Equations for Scientists and Engineers,(Birkh¨auser, Boston, 1997)
• [13] D.T. Blackstock, J. Acoust. Soc. Am. 36, (1964)
• [14] Z.A. Goldberg, Sov. Phys. Jetp 15, 179 (1962)
• [15] L.A. Pospelov, Sov. Phys. Acoust. 11, 302 (1966)
• [16] T. Oziş, A. Özdeş, J. Comput. Appl. Math. 71, 163 (1996)
• [17] E.N. Aksan, A. Ozdeş, Appl. Math. Comput. 156, 395 (2004)
• [18] S. Kutluay, A.R. Bahadir, A. Ozdeş, J J. Comput. Appl.Math. 103,251 (1999)
• [19] E. Varoğlu, W.D.L. Finn, Internat. J. Numer. Methods Engrg. 16,171 (1980)[Crossref]
• [20] J. Caldwell, P. Wanless, A.E. Cook, App. Math. Model. 5, 189(1981)[Crossref]
• [21] D.J. Evans, A.R. Abdullah, Computing 32, 239(1984)[Crossref]
• [22] R.C. Mittal, P. Singhal, Commun. Numer. Meth. Engng. 9, 397(1993)[Crossref]
• [23] A. Esen, O. Tasbozan, Mediterr. J. Math., DOI 10.1007/s00009-015-0555-x (2015)[Crossref]
• [24] A. Esen, N.M. Yagmurlu,O. Tasbozan, Appl.Math. Inf. Sci. 7, 1951(2013)[Crossref]
• [25] E.A.-B. Abdel-Salam, E.A. Yousif, Y.A.S. Arko, E.A.E. Gumma,J.App. Math. 2014, (2014)
• [26] M. Inc, J. Math. Anal. Appl. 345, 476 (2008)
• [27] S.J. Liao, Ph.D thesis, Shanghai Jiao Tong University(Shanghai,China, 1992)
• [28] S.J. Liao, Beyond Perturbation: Introduction to the HomotopyAnalysis Method, (Chapman and Hall/CRC Press, Boca Raton,2003)
• [29] S.J. Liao, Commun. Nonlinear Sci. Numer. Simulat. 2, 95 (1997)[Crossref]
• [30] S.J. Liao, Appl. Math. Comput. 147, 499 (2004)
• [31] S.J. Liao, Commun. Nonlinear Sci. Numer. Simulat. 14, 983(2009)[Crossref]
• [32] O. Tasbozan, A. Esen and N.M. Yagmurlu, Open J. of Appl. Sciences2, 193 (2012)
• [33] O. Tasbozan, A. Kurt, Sohag J. Math. 2, 57 (2015)
• [34] A. Kurt, O. Tasbozan, Int. J. Pure Appl. Math. 98, 503 (2015)
• [35] E. Hopf, Comm. Pure Appl. Math. 3, 201 (1950)[Crossref]
• [36] T. Oziş, E.N. Aksan, A. Ozdeş, Appl. Math. Comput. 139, 417(2003)
• [37] J.D. Cole, Quart. Apply. Math. 9, 225 (1951)[Crossref]

Identyfikatory

DOI

10.1515/phys-2015-0045