SOLUTION TO TIME FRACTIONAL COUETTE FLOW

Abstract

In this study, the Couette flow of a second grade uid is discussed in a porous layer when the bottom plate moves suddenly. The Laplace transform method is implemented to derive the analytical solution.The main object of this paper is to demonstrate how we can make signicant progress in treating a variety of problems in the theory of partial fractional dierential equations by combining theory of special functions and operational methods. In this article, it has been shown that the combined use of integral transforms and exponential operators methods provides a powerful tool to solve certain system of KdV. Constructive examples are also provided.

Keywords

• Fractional partial dierential equations,Riemann Liouville fractional deriva- tive,Time fractional Couette flow,Laplace transforms,Fourier transforms

References

1. Aghili, A., New results involving Airy polynomials, fractional calculus and solution to generalized heat equation. New trends in mathematical sciences.Vol. 3,2015.
2. Ansari, A., Fractional exponential operators and time fractional telegraph equation. Boundary value problems 2012,125. Springer.
3. Apelblat, A., Laplace transforms and their applications, Nova science publishers, Inc, New York, 2012. [4] Apelblat, A., Mass transfer with a chemical reaction of the first order analytical solutions, Chemical Engineering Journal 19 (1980) 19-37.
4. Dattoli, G.,Ottaviani,P.L., Torre,A., and Vazquez,L., Evolution operator equations: integration with algebraic and finitedifference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory,La Rivista del Nuovo Cimento 20 (1997) 2.pp. 1-132
5. Dattoli, G., Srivastava,H.M., and Zhukovsky,K.V., Operational methods and differential equations to initial value problems. Applied Mathematics and computations.184 (2007) pp 979-1001.
6. Elperin, T., Fominykh, A. and Orenbakh,Z., Lighthill- Levich problem for a non- Newtonian fluid, International Communications in Heat and Mass Transfer 32 (2005) 620-626.
7. Fang, T., A Note on the Incompressible Couette Flow with Porous Walls, International Communica- tions in Heat and Mass Transfer, Vol. 31, No. 1, 2004, pp. 31-41. http://dx.doi.org/10.1016/S0735- 1933(03)00199-4
8. Kilbas,A.A., Srivastava, H.M. and Trujillo, J.J.,Theory and applications of fractional differential equa- tions, North Holand Mathematics Studies,204,Elsevier Science Publishers,Amesterdam, Heidelberg and New York ,2006.

9. Khaled,A.R.A., and Vafai, K.,The Effect of the Slip Condition on Stokes and Couette Flows due to an Oscillating Wall: Exact Solutions, International Journal of Non-Linear Mechanics, Vol. 39, No. 5, 2004, pp. 795-809. http://dx.doi.org/10.1016/S0020-7462(03)00043-X
10. Podlubny,I., Fractional differential equations, Academic Press, San Diego, CA,1999.
11. Vallee, O., and Soares,M., Airy Functions and Applications to Physics. London: Imperial College Press; 2004.
12. Srivastav,H.M., On an extension of the Mittag - Leffler function, Yukohama math.j. Vol. 16, no. 2, 1968, pp. 77 - 88.