Approximate solution of the two-dimensional Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives


Jalil Rashidinia, Ali Parsa, Raheleh Salehi

School of Mathematics‎, ‎Iran University of Science and Technology‎, ‎Narmak‎, ‎Tehran 168613114‎, ‎Iran

DOI: https://doi.org/10.17656/jzs.10578

Abstract

In this paper, a scheme based on Sinc and radial basis functions (RBF) is developed to approximate the solution of two-dimensional Rayleigh-Stokes problem for a heated generalized second-grade fluid with fractional derivatives‎. We use RBF and Sinc functions as basis functions to approximate spatial and time coordinates of the unknown function‎, respectively‎. ‎The error analysis is studied and discussed‎. An illustrative example verifies the effectiveness of our method and shows that one can obtain accurate results with only a small number of basis functions‎.


Key Words:
Rayleigh-Stokes problem;
Fractional calculus;
Sinc method;
Sinc quadrature;
RBFs.


References

[1] Baumann G‎. and ‎Stenger F‎. "‎Fractional calculus and Sinc methods‎". ‎Fractional Calculus and Applied Analysis, Vol. 14, pp ‎568-622‎. (2011).

[2] Carpinteri, A‎. and ‎Mainardi F‎."‎Fractals and Fractional Calculus in Continuum‎Mechanics"‎. ‎Springer-Verlag‎: ‎Wien‎, (1997)‎.‎‎‎

[3] Chen C‎., ‎Liu, F‎. and ‎Anh V‎. ‎"Numerical analysis of the Rayleigh-Stokes problem for a heated‎generalized second grade fluid with fractional derivatives‎". ‎Applied Mathematics and Computation, Vol. 204,pp 340-351. (2008).

[4] Chen C‎., ‎Liu, F. and‎Anh V‎. ‎"A Fourier method and an extrapolation technique for Stokes’ first‎problem for a heated generalized second grade fluid with‎fractional derivative‎". ‎Journal of Computational and Applied Mathematics, Vol. 223, pp ‎777-789. (2009).

[5] Das S‎. "Functional Fractional Calculus for System Identification and‎Controls‎". ‎Springer-Verlag Berlin Heidelberg‎: ‎Berlin, (2008)‎.

[6] Debnath L‎. ‎"A brief historical introduction to fractional calculus‎". International Journal of Mathematical Education in Science and Technology, Vol. 35, No. 4, pp 487-501. (2004).

[7] Fasshouer G.E‎.‎"Mesh free approximation methods with MATLAB‎". ‎USA‎: ‎World Scientific. (2007).

[8] Fedoseyer, A‎.,‎ Friedman M. J.‎and‎ Kansa E. J‎. "‎Improved multiquadrics method for elliptic partial differential equations via PDE collocation on the boundary"‎. ‎Comput‎. ‎Math‎. ‎Appl‎., Vol. 43, pp 439-455. (2002).

[9] Fornberg B‎, ‎Wright G. and ‎Larsson E‎. "‎Some observation regarding interpolants in the limit of flat radial basis functions‎".‎Comput‎. ‎Math‎. ‎Appl‎., Vol. 47, pp 37-55. (2004).

[10] Hilfer R‎. ‎"Applications of Fractional Calculus in Physics‎". ‎World Scientific‎, ‎Singapore. (2000)‎.‎

[11] Khan M‎. ‎"The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a‎fractional derivative model‎". ‎Nonlinear Analysis‎: ‎Real World Applications, Vol. 10, pp 3190-3195. (2009).

[12] Lund J. . and ‎Bowers K. L‎.‎"Sinc method for quadrature and differential equations‎". ‎SIAM‎, (‎1992)‎.‎

[13] Magin R.L‎.‎"Fractional Calculus in Bioengineering‎". Begell House‎: ‎Connecticut, (2007).

[14] Mohebbi A‎., ‎Abbaszadeh M‎. and ‎Dehghan M‎. ‎"Compact finite difference scheme and RBF‎meshless approach for solving 2D Rayleigh-Stokes problem for a heated generalized second grade fluid with‎fractional derivatives‎". ‎Comput‎. ‎Methods Appl‎. ‎Mech‎. ‎Engrg‎. Vol. 264, pp 163-177. (2013).

[15] Oldham K.B. a ndSpanie J‎.‎"The Fractional Calculus". ‎Academic Press, New York. (1974)‎.

[16] Okayama T.‎,‎ Matsuo T. . and‎Sugihara M‎. ‎"Approximate Formulae for Fractional Derivatives by Means of Sinc Methods‎". ‎Journal of Concrete and Applicable Mathematics, Vol. 8, ‎pp 470 ‎- 488. (2010).

[17] Okayama T.‎,‎Matsuo T. and‎Sugihara M‎. ‎"Sinc-collocation methods for weakly singular Fredholm integral‎‎equations of the second kind". ‎Journal of Computational and Applied Mathematics, Vol. 234, ‎pp 1211-1227. (2010).

[18] Podlubny I‎. "‎Fractional Differential Equations‎".‎ Academic Press‎: ‎San Diego‎, ‎)1999)‎.‎

[19] Riley B. V‎. ‎"The numerical solution of Volterra integral equations with nonsmooth solutions‎based on sinc approximation‎". ‎Applied Numerical Mathematics, Vol. 9, pp 249-257. (1992).

[20] Sabatier J‎, ‎Agrawal OP‎, ‎Machado JAT‎. ‎"Advances in Fractional‎Calculus‎: ‎Theoretical Developments and Applications in Physics and Engineering‎". ‎Springer‎: Dordrecht. (2007)‎.

[21] Shan L.‎,‎Tong D‎. and‎ Xue L‎. ‎"Unsteady flow of non-Newtonian visco-elastic fluid in dual-porosity media with the fractional derivative‎". ‎Journal of Hydrodynamics, Vol. 21, pp 705-713‎. (2009).

[22] Shena F. ‎Tana W.‎,‎Zhaoc Y. . and‎Masuokad T‎. "‎The Rayleigh-Stokes problem for a heated generalized second‎grade fluid with fractional derivative model‎". ‎Nonlinear Analysis‎: ‎RealWorld Applications, Vol. 7, pp 1072‎- ‎1080‎. (2006).

[23] Salim T.O‎. and‎ El-Kahlout A‎. ‎"Solution of Fractional Order‎Rayleigh-Stokes Equations‎". ‎Adv‎. ‎Theor‎. ‎Appl‎. ‎Mech‎. Vol. 5,‎ pp 241‎- ‎254. (2008).

[24] Stenger F‎. ‎"Numerical Methods Based on Sinc and Analytic Functions". ‎Springer-Verlag‎:New York‎. (1993)‎.

[25] Stenger F‎. ‎"Handbook of Sinc Numerical Methods‎". CRC Press‎: ‎Boca‎Raton. (2011).

[26] Tian J. and Tong D‎. ‎"The flow analysis of fluids in fractal reservoir with the fractional derivative‎". ‎Journal of Hydrodynamics, Vol. 18. pp 287-293. (2006).

[27] Zakeri G.A‎. and‎ Navab M‎. ‎"Sinc collocation approximation of non-smooth solution of a nonlinear‎weakly singular Volterra integral equation‎". ‎Journal of Computational Physics, Vol. 229, pp ‎6548-6557. (2010).

[28] Zhuang P. and‎ Liu Q‎. "‎Numerical method of Rayleigh-Stokes problem for heated generalized‎second grade fluid with fractional derivative"‎. ‎Appl‎.‎Math‎. ‎Mech‎. ‎-Engl.‎‎Ed‎. Vol. 30, No. 12, pp ‎1533-1546. (2009).

[29] Chen C., Liu F., Burrage K. and Chen Y. ‎"Numerical methods of the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative".‎ IMA Journal of Applied Mathematics, pp1−21. (2012).

[30] Wenlland H‎. ‎"Scattered Data Approximation"‎. ‎Cambridg University Press, ‎New York. (2005).