Abstract
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.
References
[1] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, (John Wiley & Sons, New York, 1993) Search in Google Scholar
[2] A.Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, (Elsevier, San Diego, 2006) Search in Google Scholar
[3] I. Podlubny, Fractional Differential Equations, (Academic Press,San Diego, 1999) Search in Google Scholar
[4] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, J. Comput. Appl. Math. 264, 65 (2014) 10.1016/j.cam.2014.01.002Search in Google Scholar
[5] T. Abdeljawad, J. Comput. Appl. Math. 279, 57 (2015) 10.1016/j.cam.2014.10.016Search in Google Scholar
[6] N. Benkhettoua, S. Hassania, D.F.M. Torres, J. King Saud Univ.- Sci. (In Press)(2015) Search in Google Scholar
[7] M.A.Hammad, R. Khalil, Int. J. of Pure Appl.Math. 94, 215 (2014) Search in Google Scholar
[8] W.S. Chung, J. Comput. Appl. Math. 290, 150 (2015) 10.1016/j.cam.2015.04.049Search in Google Scholar
[9] H. Bateman, Monthly Weather Rev. 43, 163 (1915) 10.1175/1520-0493(1915)43<163:SRROTM>2.0.CO;2Search in Google Scholar
[10] J.M. Burger, Trans. Roy. Neth. Acad. Sci. Amsterdam 17, 1 (1939) Search in Google Scholar
[11] J.M. Burger, , Adv. Appl. Mech. 1, 171 (1948) 10.2307/1438425Search in Google Scholar
[12] L. Debnath, Partial Differential Equations for Scientists and Engineers, (Birkh¨auser, Boston, 1997) 10.1007/978-1-4899-2846-7Search in Google Scholar
[13] D.T. Blackstock, J. Acoust. Soc. Am. 36, (1964) Search in Google Scholar
[14] Z.A. Goldberg, Sov. Phys. Jetp 15, 179 (1962) Search in Google Scholar
[15] L.A. Pospelov, Sov. Phys. Acoust. 11, 302 (1966) Search in Google Scholar
[16] T. Oziş, A. Özdeş, J. Comput. Appl. Math. 71, 163 (1996) Search in Google Scholar
[17] E.N. Aksan, A. Ozdeş, Appl. Math. Comput. 156, 395 (2004) 10.1016/j.amc.2003.07.027Search in Google Scholar
[18] S. Kutluay, A.R. Bahadir, A. Ozdeş, J J. Comput. Appl.Math. 103, 251 (1999) 10.1016/S0377-0427(98)00261-1Search in Google Scholar
[19] E. Varoğlu, W.D.L. Finn, Internat. J. Numer. Methods Engrg. 16, 171 (1980) 10.1002/nme.1620160112Search in Google Scholar
[20] J. Caldwell, P. Wanless, A.E. Cook, App. Math. Model. 5, 189 (1981) 10.1016/0307-904X(81)90043-3Search in Google Scholar
[21] D.J. Evans, A.R. Abdullah, Computing 32, 239(1984) 10.1007/BF02243575Search in Google Scholar
[22] R.C. Mittal, P. Singhal, Commun. Numer. Meth. Engng. 9, 397 (1993) 10.1002/cnm.1640090505Search in Google Scholar
[23] A. Esen, O. Tasbozan, Mediterr. J. Math., DOI 10.1007/s00009- 015-0555-x (2015) Search in Google Scholar
[24] A. Esen, N.M. Yagmurlu,O. Tasbozan, Appl.Math. Inf. Sci. 7, 1951 (2013) 10.12785/amis/070533Search in Google Scholar
[25] E.A.-B. Abdel-Salam, E.A. Yousif, Y.A.S. Arko, E.A.E. Gumma, J.App. Math. 2014, (2014) 10.1155/2014/218092Search in Google Scholar
[26] M. Inc, J. Math. Anal. Appl. 345, 476 (2008) Search in Google Scholar
[27] S.J. Liao, Ph.D thesis, Shanghai Jiao Tong University(Shanghai, China, 1992) Search in Google Scholar
[28] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, (Chapman and Hall/CRC Press, Boca Raton, 2003) Search in Google Scholar
[29] S.J. Liao, Commun. Nonlinear Sci. Numer. Simulat. 2, 95 (1997) Search in Google Scholar
[30] S.J. Liao, Appl. Math. Comput. 147, 499 (2004) 10.1016/S0096-3003(02)00790-7Search in Google Scholar
[31] S.J. Liao, Commun. Nonlinear Sci. Numer. Simulat. 14, 983 (2009) Search in Google Scholar
[32] O. Tasbozan, A. Esen and N.M. Yagmurlu, Open J. of Appl. Sciences 2, 193 (2012) 10.4236/ojapps.2012.23029Search in Google Scholar
[33] O. Tasbozan, A. Kurt, Sohag J. Math. 2, 57 (2015) Search in Google Scholar
[34] A. Kurt, O. Tasbozan, Int. J. Pure Appl. Math. 98, 503 (2015) 10.12732/ijpam.v98i4.9Search in Google Scholar
[35] E. Hopf, Comm. Pure Appl. Math. 3, 201 (1950) 10.1002/cpa.3160030302Search in Google Scholar
[36] T. Oziş, E.N. Aksan, A. Ozdeş, Appl. Math. Comput. 139, 417 (2003) 10.1016/S0096-3003(02)00204-7Search in Google Scholar
[37] J.D. Cole, Quart. Apply. Math. 9, 225 (1951) 10.1090/qam/42889Search in Google Scholar
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