Nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations within Sumudu transform. (English) Zbl 1275.65066

Atangana, Abdon; Baleanu, Dumitru

Abstr. Appl. Anal. 2013, Article ID 160681, 8 p. (2013).

Summary: We solve the system of nonlinear fractional Jaulent-Miodek and Whitham-Broer-Kaup equations via the Sumudu transform homotopy method (STHPM). The method is easy to apply, accurate, and reliable.

MSC:

65M99	Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
35R11	Fractional partial differential equations
44A10	Laplace transform

Keywords:

Laplace transform; nonlinear fractional Jaulent-Miodek equation; Whitham-Broer-Kaup equations; Sumudu transform homotopy method

Software:

BVPh

References:

[1]	Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics. Fractals and Fractional Calculus in Continuum Mechanics, CISM Courses and Lectures, 378, 291-348, (1997), Vienna, Austria: Springer, Vienna, Austria · Zbl 0917.73004
[2]	Atangana, A.; Alabaraoye, E., Solving system of fractional partial differential equations arisen in the model of HIV infection of CD4 cells and attractor one-dimensional Keller-Segel equation, Advances in Difference Equations, 2013, (2013) · Zbl 1380.92026
[3]	Atangana, A.; Kılıçman, A., Analytical solutions of the space-time fractional derivative of advection dispersion equation, Mathematical Problems in Engineering, 2013, (2013) · Zbl 1299.35058
[4]	Atangana, A.; Botha, J. F., Analytical solution of the ground water flow equation obtained via homotopy decomposition method, Journal of Earth Science & Climatic Change, 3, 115, (2012)
[5]	Mahmudov, N. I., Approximate controllability of fractional sobolev-type evolution equations in banach spaces, Abstract and Applied Analysis, 2013, (2013) · Zbl 1271.93021
[6]	Mohebbi, A.; Asgari, Z., Efficient numerical algorithms for the solution of “good” Boussinesq equation in water wave propagation, Computer Physics Communications, 182, 12, 2464-2470, (2011) · Zbl 1457.65147
[7]	Xie, F.; Yan, Z.; Zhang, H., Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations, Physics Letters A, 285, 1-2, 76-80, (2001) · Zbl 0969.76517
[8]	Kupershmidt, B. A., Mathematics of dispersive water waves, Communications in Mathematical Physics, 99, 1, 51-73, (1985) · Zbl 1093.37511
[9]	Ma, W.-X.; Li, C.-X.; He, J., A second Wronskian formulation of the Boussinesq equation, Nonlinear Analysis. Theory, Methods & Applications, 70, 12, 4245-4258, (2009) · Zbl 1159.37425
[10]	Hong, T.; Wang, Y. Z.; Huo, Y. S., Bogoliubov quasiparticles carried by dark solitonic excitations in non-uniform Bose-Einstein condensates, Chinese Physics Letters, 15, 550-552, (1998)
[11]	Zhang, J.-f., Multiple soliton solutions of the dispersive long-wave equations, Chinese Physics Letters, 16, 1, 4-5, (1999)
[12]	Das, G. C.; Sarma, J.; Uberoi, C., Explosion of a soliton in a multicomponent plasma, Physics of Plasmas, 4, 6, 2095-2100, (1997)
[13]	Lou, S. Y., A direct perturbation method: nonlinear Schrodinger equation with loss, Chinese Physics Letters, 16, 659-661, (1999)
[14]	Atangana, A.; Cloot, A. H., Stability and convergence of the space fractional variable-order Schrödinger equation, Advances in Difference Equations, 2013, (2013) · Zbl 1380.35157
[15]	Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus Models and Numerical Methods. Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3, xxiv+400, (2012), Hackensack, NJ, USA: World Scientific, Hackensack, NJ, USA · Zbl 1248.26011
[16]	Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, xvi+523, (2006), Amsterdam, The Netherlands: Elsevier Science B. V., Amsterdam, The Netherlands · Zbl 1092.45003
[17]	Oldham, K. B.; Spanier, J., The Fractional Calculus, Mathematics in Science and Engineering, 111, xiii+234, (1974), New York, NY, USA: Academic Press, New York, NY, USA
[18]	Ross, B., Fractional Calculus and Its Applications. Fractional Calculus and Its Applications, Lecture Notes in Mathematics, 457, vi+381, (1975), Berlin, Germany: Springer, Berlin, Germany
[19]	Luchko, Y.; Gorenflo, R., The initial value problem for some fractional differential equations with the Caputo derivative, Preprint Series, A08-98, (1998), Fachbreich Mathematik and Informatik, Freic Universitat Berlin
[20]	Atangana, A.; Secer, A., A note on fractional order derivatives and table of fractional derivatives of some special functions, Abstract and Applied Analysis, 2013, (2013) · Zbl 1276.26010
[21]	Podlubny, I., Fractional Differential Equations. Fractional Differential Equations, Mathematics in Science and Engineering, 198, xxiv+340, (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0924.34008
[22]	Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications, xxxvi+976, (1993), Yverdon, Switzerland: Gordon and Breach Science, Yverdon, Switzerland · Zbl 0818.26003
[23]	Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience, xvi+366, (1993), New York, NY, USA: John Wiley & Sons, New York, NY, USA · Zbl 0789.26002
[24]	Jumarie, G., On the representation of fractional Brownian motion as an integral with respect to \((\text{d} t)^a,\) Applied Mathematics Letters, 18, 7, 739-748, (2005) · Zbl 1082.60029
[25]	Watugala, G. K., Sumudu transform: a new integral transform to solve differential equations and control engineering problems, International Journal of Mathematical Education in Science and Technology, 24, 1, 35-43, (1993) · Zbl 0768.44003
[26]	Weerakoon, S., Application of Sumudu transform to partial differential equations, International Journal of Mathematical Education in Science and Technology, 25, 2, 277-283, (1994) · Zbl 0812.35004
[27]	Singh, J.; Kumar, D.; Kılıçman, A., Homotopy perturbation method for fractional gas dynamics equation using Sumudu transform, Abstract and Applied Analysis, 2013, (2013) · Zbl 1262.76082
[28]	Eltayeb, H.; Kılıçman, A., A note on the Sumudu transforms and differential equations, Applied Mathematical Sciences, 4, 21-24, 1089-1098, (2010) · Zbl 1207.34014
[29]	Kılıçman, A.; Eltayeb, H.; Agarwal, P. R., On Sumudu transform and system of differ ential equations, Abstract and Applied Analysis, 2010, (2010) · Zbl 1197.34001
[30]	Gupta, V. G.; Sharma, B., Application of Sumudu transform in reaction-diffusion systems and nonlinear waves, Applied Mathematical Sciences, 4, 9–12, 435-446, (2010) · Zbl 1195.26012
[31]	Eltayeb, H.; Kılıçman, A., A note on the Sumudu transforms and differential equations, Applied Mathematical Sciences, 4, 21–24, 1089-1098, (2010) · Zbl 1207.34014
[32]	Belgacem, F. B. M.; Karaballi, A. A.; Kalla, S. L., Analytical investigations of the Sumudu transform and applications to integral production equations, Mathematical Problems in Engineering, 3-4, 103-118, (2003) · Zbl 1068.44001
[33]	Abdon, A., New cass of boundary value problems, Informations Sciences Letters, 1, 2, 67-76, (2012)
[34]	Wu, G.-C., Adomian decomposition method for non-smooth initial value problems, Mathematical and Computer Modelling, 54, 9-10, 2104-2108, (2011) · Zbl 1235.65105
[35]	Wu, G. C.; Baleanu, D., Variational iteration method for the Burgers’ flow with fractional derivatives-New Lagrange multipliers, Applied Mathematical Modelling, 5, 1012-1018, (2012)
[36]	Wu, G.-C.; Baleanu, D., Variational iteration method for fractional calculus—a universal approach by Laplace transform, Advances in Difference Equations, 2013, (2013) · Zbl 1365.34022
[37]	Xu, H.; Liao, S.-J.; You, X.-C., Analysis of nonlinear fractional partial differential equations with the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14, 4, 1152-1156, (2009) · Zbl 1221.65286
[38]	He, J.-H., Homotopy perturbation method with an auxiliary term, Abstract and Applied Analysis, 2012, (2012) · Zbl 1235.65096