On Solving Some Classes of Nonlinear Fractional Differentional Equations Using Fractal Index Method
Science Journal of Applied Mathematics and Statistics
Volume 2, Issue 6, December 2014, Pages: 112-115
Received: Nov. 28, 2014;
Accepted: Dec. 6, 2014;
Published: Dec. 17, 2014
Views 2612 Downloads 160
Sayed K. Elagan, Mathematics Department, Faculty of Science, Taif University, Taif, Saudi Arabia; Mathematics Department, Faculty of Science, Menofya University, Shebin Elkom, Egypt
Mohamed S. Mohamed, Mathematics Department, Faculty of Science, Taif University, Taif, Saudi Arabia; Mathematics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt
Khaled A. Gepreel, Mathematics Department, Faculty of Science, Taif University, Taif, Saudi Arabia; Mathematics Department, Faculty of Science, Zagazig University, Kuala Lumpur, Egypt
Rabha W. Ibrahim, Institute of Mathematical Sciences, University Malaya, 50603, Kuala Lumpur, Malaysia
Afaf Elesimy, Mathematics Department, Faculty of Science, Taif University, Taif, Saudi Arabia
We provide a new solution of diffusion fractional differential equation using fractal index and fractional sub-equation method. Also we shall impose a new solution for fraction Birnolli equation of arbitrary order using the fractal index method. As a result many exact solutions are obtained. It is shown that our considered method provides a very effective tool for solving fractional differentional equations.
Sayed K. Elagan,
Mohamed S. Mohamed,
Khaled A. Gepreel,
Rabha W. Ibrahim,
On Solving Some Classes of Nonlinear Fractional Differentional Equations Using Fractal Index Method, Science Journal of Applied Mathematics and Statistics.
Vol. 2, No. 6,
2014, pp. 112-115.
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
J. Sabatier, O. P. Agrawal, and J. A. Machado, Advance in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
V. Lakshmikantham, S. Leela, J. Vasundhara, Theory of Fractional Dynamic Systems. Cambridge Academic Publishers, Cambridge 2009.
D. Baleanu, B. Guvenc and J. A. Tenreiro, New Trends in Nanotechnology and Fractional Calculus Applications, Springer, New York, NY, USA, 2010.
P. R. Gordoa, A. Pickering, Z. N. Zhu, Bucklund transformations for a matrix second Painlev equation, Physics Letters A, 374 (34) (2010) 3422-3424.
R. Molliq, B. Batiha, Approximate analytic solutions of fractional Zakharov-Kuznetsov equations by fractional complex transform, International Journal of Engineering and Technology, 1 (1) (2012) 1-13.
R.W. Ibrahim, Complex transforms for systems of fractional differential equations, Abstract and Applied Analysis Volume 2012, Article ID 814759, 15 pages.
S. Sivasubramanian, M. Darus, R. W. Ibrahim, On the starlikeness of certain class of analytic functions, Mathematical and Computer Modelling, vol. 54, no. -2(2011) pp. 112118.
R. W. Ibrahim, An application of Lauricella hypergeometric functions to the generalized heat equations, Malaya Journal of Matematik, 1(2014) 43-48.
Zheng-Biao Li and Ji-Huan He. "Application of the ractional Complex Transform to Fractional Differential Equations"Nonlinear Science Letters A- Mathematics, Physics and Mechanics 2.3 (2011): 121-126.
Ji-Huan He a, S.K. Elagan and Z.B. Li c, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus Physics Letters A 376 (2012) 257–259.
RabhaW. Ibrahima, and S. K. Elagan, On solutions for classes of fractional differential equations, Malaya J. Mat. 2(4) (2014) 411–418.
J. R. Macdonald, L. R. Evangelista, E. K. Lenzi, and G. Barbero, J. Phys. Chem. C, 115(2011) 7648-7655.
P. A. Santoro, J. L. de Paula, E. K. Lenzi, L. R. Evangelista, J. Chem. Phys. 135(114704)(2011) 1-5.
J.T. Machado, V. Kiryakova, F. Mainardi, Commun. Nonlinear Sci. 16(2011) 1140- 1153.
R. W. Ibrahim, On holomorphic solution for space- and time-fractional telegraph equations in complex domain, Journal of Function Spaces and Applications 2012, Article ID 703681, 10 pages.
R.W. Ibrahim, Numerical solution for complex systems of fractional order, Journal of Applied Mathematics 2012, Article ID 678174, 11 pages.
K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag Berlin Heidelberg, 2010.
S. Zhang, H.Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 375 (2011) 1069-1073.
A. N. Kochubei, The Cauchy problem for evolution equations of fractional order, Differential Equations 25 (1989) 967-974.
A. N. Kochubei, Diffusion of fractional order, Differential Equations 26 (1990) 485-492.
R. Metzler, J. Klafter, The random walks guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 (2000) 1-77.
G. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos. Chaotic advection, tracer dynamics and turbulent dispersion. Phys. D 76 (1994) 110-122.
F. Mainardi, G. Pagnini and R. Gorenflo; Some aspects of fractional diffusion equations of single and distributed order, App. Math. Compu., 187( 1) (2007) 295-305.
Khaled A. Gepreel and Mohamed S. Mohamed, Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation, Chinese physics B, Vol. 22, Issue. 1, 2013, 010201-6.
Hossam A. Ghany and Mohamed S. Mohamed, White noise functional solutions for the wick-type stochastic fractional Kdv-Burgers-Kuramoto Equations, Journal of the Chinese Journal of Physics. Vol. 50, Issue. 4, 2012, pp. 619-627.