A NUMERICAL SOLUTION FOR THE FRACTIONAL RAYLEIGH-STOKES ‎PROBLEM BY SPACE-TIME RADIAL BASIS FUNCTIONS

Subject Areas : Statistics

Nafiseh Noghrei ¹ , Asghar Kerayechian ² , Alireza Soheili ³

1 - Department of Applied Mathematics‎, ‎School of Mathematical Sciences‎, ‎Ferdowsi University of Mashhad‎, ‎Mashhad‎, ‎Iran.
2 - Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
3 - Department of Mathematics, Faculty of Mathematics, Ferdowsi University of Mashhad, Iran.

Received: 2020-01-24 Accepted : 2020-10-14 Published : 2021-09-23

Keywords: فرمول‌بندی مکان- زمان, حسابان کسری, تابع پایه شعاعی گاوسین, روش سینک,

Abstract :

In this paper, we approximate the solution of two-dimensional Rayleigh-Stokes problem ‎for a heated generalized second grade fluid with fractional derivatives. This approximation is ‎based on the space-time radial basis functions (RBFs) and the Sinc quadrature rule. In this ‎method, we use Gaussian radial basis function and don't distinguish between time and place ‎variables and the collocation points have both the coordinates of time and space. We use the ‎Sinc quadrature rule with single exponential transformation to approximate the integral part of ‎fractional derivatives. The chosen fractional derivatives is Riemann – Liouville.‎This method is implemented on two examples with different values of the fractional ‎derivative order. Obtained results illustrate the effectiveness of our method and sh ow that ‎one can obtain accurate results with a small number of the collocation points for the radial ‎basis function. It should be noted that all calculations in this paper have been done using ‎Mathematica software.‎

References:

[1] K. B. Oldham, J. Spanie. The Fractional Calculus. Academic Press. New York (1974)

 

[2] K. S. Miller, B. Ross. An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley. New York (1993)

 

[3] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo. Theory and Applications of Fractional Differential Equations. Elsevier. Amsterdam (2006)

 

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

 

[5] A. Carpinteri, F. Mainardi. Fractals and Fractional Calculus in Continuum Mechanics. Springer-Verlag. Wien (1997)

 

[6] K. Diethelm. The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer. Heidelberg (2010)

 

[7] F. Mainardi. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Imperial College Press. London. Hackensack NJ (2010)

 

[8] C. Fetecau. The Rayleigh-Stokes problem for heated second grade fuids. International Non-Linear Mechanics 37: 1011–1015 (2002)

 

[9] J. Zierep, C. Fetecau. Energetic balance for the Rayleigh-Stokes problem of a second grade fluid. International Engineering Science 45: 155–162 (2007)

 

[10] Chang-Ming Chen a, F. Liu, V. Anh. Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Applied Mathematics and Computation 204: 340–351 (2008)

 

[11] P. h. Zhuang, Q. Liu. Numerical method of Rayleigh-Stokes problem for heated generalized second grade fluid with fractional derivative. Applied Mathematics and Mechanics -Engl. Ed. 30(12): 1533–1546 (2009)

 

[12] C. Fetecau, J. Zierep. The Rayleigh-Stokes problem for a Maxwell fluid. Z. angew. Math. Phys. 54(6): 1086–1093 (2003)

 

[13] C. Wu. Numerical solution for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. Applied Numerical Mathematics 59: 2571–2583 (2009)

 

[14] C. M Chen, F. Liu, V. Anh. A Fourier method and an extrapolation technique for Stokes’ first problem for a heated generalized second grade fluid with fractional derivative. Computational and Applied Mathematics. 223: 777–789 (2009)

 

[15] C. Xue, J. Nie. Exact solutions of the Rayleigh–Stokes problem for a heated generalized second grade fluid in a porous half-space. Applied Mathematical Modelling. 33: 524–531 (2009)

 

[16] A. Mohebbi, M. Abbaszadeh, M. Dehghan. Compact finite difference scheme and RBF meshless approach for solving 2D Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives. Computer Methods in Applied Mechanics and Engineering 264: 163–177 (2013)

 

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

 

[18] H. Wendland. Scattered Data Approximation. Cambridg University Press. New York (2005)

 

[19] A. Fedoseyer, M. J. Friedman, E. J. Kansa. Improved multiquadrics method for elliptic partial differential equations via PDE collocation on the boundary. Computers and Mathematics with Applications 43: 439–455 (2002)

 

[20] B. Fornberg, G. Wright, E. Larsson. Some observation regarding interpolants in the limit of flat radial basis functions. Computers and Mathematics with Applications 47: 37–55 (2004)

 

[21] T. Okayama, T. Matsuo, M. Sugihara. Approximate Formulae for Fractional Derivatives by Means of Sinc Methods. Concrete and Applicable Mathematics 8: 470–488 (2010)

 

[22] T. Okayama, T. Matsuo, M. Sugihara. Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind. Computational and Applied Mathematics 234: 1211–1227 (2010)

 

[23] G. A. Zakeri, M. Navab. Sinc collocation approximation of non-smooth solution of a nonlinear weakly singular Volterra integral equation. Computational Physics 229: 6548–6557 (2010)

 

[24] B. V. Riley. The numerical solution of Volterra integral equations with nonsmooth solutions based on sinc approximation. Applied Numerical Mathematics 9: 249–257 (1992)

 

[25] G. Baumann, F. Stenger. Fractional calculus and Sinc methods. Fractional Calculus and Applied Analysis 14: 568–622 (2011)

 

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

 

[27] F. Stenger. Handbook of Sinc Numerical Methods. CRC Press. Boca Raton (2011)

 

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

 

[29] K. Tanaka, M. Sugihara, K. Murota. Function Classes for Successful DE-Sinc Approximations. Mathematics of Computation 78: 1553–1571 (2009)

 

[30] K. Tanaka, M. Sugihara, K. Murota, M. Mori. Function classes for double exponential integration formulas. Numerische Mathematik 111: 631–655 (2009)

 

[31] M. Sugihara, T. Matsuo. Recent developments of the Sinc numerical methods. Computational and Applied Mathematics 164–165: 673–689 (2004)

 

[32] M. Mori, M. Sugihara. The double-exponential transformation in numerical analysis. Computational and Applied Mathematics 127: 287–296 (2001)