SeMA Journal

pp 1–9

# The Legendre wavelet method for solving the steady flow of a third-grade fluid in a porous half space

• Simin Shekarpaz
• Kourosh Parand
• Hossein Azari
Article

## Abstract

In this study, based on the Legendre wavelet method, we will present a new efficient method to obtain the numerical solutions of a nonlinear ordinary differential equation arising in fluid dynamics. The Legendre wavelet collocation method (LWCM) will be used and problem is converted into a system of algebraic equations, where the solutions of obtained system are computed by using the Newton’s method. Finally some numerical examples are given to check the accuracy of proposed method and a comparison is made with the other methods which reflects the efficiency and capability of method.

## Keywords

Wavelet analysis Legendre wavelet Third-grade fluid Porous half space

## Mathematics Subject Classification

65T60 34B15 34B40

## Notes

### Acknowledgements

This research was supported by Shahid Beheshti University and the research group of Scientific Computations.

## References

1. 1.
Ahmad, F.: A simple analytical solution for the steady flow of a third grade fluid in a porous half space. Commun. Nonlinear Sci. Numer. Simul. 14, 2848–2852 (2009).
2. 2.
Alam, J.M., Kevlahan, N.K.R., Vasilyev, O.V.: Simultaneous space-time adaptive wavelet solution of nonlinear parabolic differential equations. J. Comput. Phys. 214(2), 829–857 (2006)
3. 3.
Aziz, I., ul Islam, S., Sarler, B.: Wavelets collocation methods for the numerical solution of elliptic bv problems. Appl. Math. Model. 37(1), 676–694 (2013)
4. 4.
Aziz, I., ul Islam, S., Asif, M.: Haar wavelet collocation method for three-dimensional elliptic partial differential equations. Comput. Math. Appl. 73(9), 2023–2034 (2017)
5. 5.
Banifatemi, E., Razzaghi, M., Yousefi, S.: Two-dimensional legendre wavelets method for the mixed Volterra–Fredholm integral equations. J. Vib. Control 13(11), 1667–1675 (2007)
6. 6.
Beylkin, G., Coifman, R., Rokhlin, V.: Fast wavelet transforms and numerical algorithms I. Commun. Pure Appl. Math. 44(2), 141–183 (1991).
7. 7.
Boyd, J.P.: Orthogonal rational functions on a semi-infinite interval. J. Comput. Phys. 70(1), 63–88 (1987). , http://www.sciencedirect.com/science/article/pii/0021999187900027
8. 8.
Boyd, J.P.: Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys. 69(1), 112–142 (1987).
9. 9.
Chui, C.K.: Wavelets: A Mathematical Tool for Signal Analysis. SIAM e-books, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1997). https://books.google.com/books?id=362JiezGzR0C
10. 10.
Constantinides, A.: Applied Numerical Methods with Personal Computers. McGraw-Hill, New York (1987)
11. 11.
Funaro, D.: Computational aspects of pseudospectral Laguerre approximations. Appl. Numer. Math. 6(6), 447–457 (1990). , http://www.sciencedirect.com/science/article/pii/016892749090003X
12. 12.
Goedecker, S., Ivanov, O.: Solution of multiscale partial differential equations using wavelets. Comput. Phys. 12(6), 548 (1998)Google Scholar
13. 13.
Guf, J.S., Jiang, W.S.: The Haar wavelets operational matrix of integration. Int. J. Syst. Sci. 27(7), 623–628 (1996).
14. 14.
Guo, B-Y.: Gegenbauer approximation and its applications to differential equations on the whole line. J. Math. Anal. Appl. 226(1), 180–206 (1998). , http://www.sciencedirect.com/science/article/pii/S0022247X98960255
15. 15.
Hayat, T., Shahzad, F., Ayub, M.: Analytical solution for the steady flow of the third grade fluid in a porous half space. Appl. Math. Model. 31(11), 2424–2432 (2007). , http://www.sciencedirect.com/science/article/pii/S0307904X06002216
16. 16.
Jameson, L.: On the wavelet based differentiation matrix. J. Sci. Comput. 8(3), 267–305 (1993).
17. 17.
Jameson, L.: The differentiation matrix for Daubechies-based wavelets on an interval. SIAM J. Sci. Comput. 17(2), 498–516 (1996).
18. 18.
Kazem, S., Rad, J.A., Parand, K., Abbasbandy, S.: A new method for solving steady flow of a third-grade fluid in a porous half space based on radial basis functions. Z. Naturforschung A 66(10–11), 591–598 (2011)Google Scholar
19. 19.
Liu, N., Lin, E.B.: Legendre wavelet method for numerical solutions of partial differential equations. Numer. Methods Partial Differ. Equ. 26(1), 81–94 (2010)
20. 20.
Parand, K., Hajizadeh, E.: Solving steady flow of a third-grade fluid in a porous half Space via normal and modified rational Christov functions collocation method. Z. Naturforschung A 69, 188–194 (2014). Google Scholar
21. 21.
Parand, K., Razzaghi, M.: Rational Legendre approximation for solving some physical problems on semi-infinite intervals. Phys. Scr. 69(5), 353 (2004). http://stacks.iop.org/1402-4896/69/i=5/a=001
22. 22.
Parand, K., Shahini, M., Dehghan, M.: Rational legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type. J. Comput. Phys. 228(23), 8830–8840 (2009).
23. 23.
Razzaghi, M., Yousefi, S.: Legendre wavelets direct method for variational problems. Math. Comput. Simul. (MATCOM) 53(3), 185–192 (2000). https://ideas.repec.org/a/eee/matcom/v53y2000i3p185-192.html
24. 24.
Razzaghi, M., Yousefi, S.: Legendre wavelets method for the solution of nonlinear problems in the calculus of variations. Math. Comput. Model. 34(1), 45–54 (2001). , http://www.sciencedirect.com/science/article/pii/S0895717701000486
25. 25.
Saadatmandi, A., Sanatkar, Z., Toufighi, S.P.: Computational methods for solving the steady flow of a third grade fluid in a porous half space. Appl. Math. Comput. 298, 133–140 (2017). , http://www.sciencedirect.com/science/article/pii/S0096300316306865
26. 26.
Shen, J.: Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal. 38(4), 1113–1133 (2000).
27. 27.
ul Islam, S., Aziz, I., Sarler, B.: The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets. Math. Comput. Model. 52(1), 1577–1590 (2010)
28. 28.
ul Islam, S., Sarler, B., Aziz, I., i Haq, F.: Haar wavelet collocation method for the numerical solution of boundary layer fluid flow problems. Int. J. Therm. Sci. 50(1), 686–697 (2011)Google Scholar
29. 29.
ul Islam, S., Aziz, I., Al-Fhaid, A.S., Shah, A.: A numerical assessment of parabolic partial differential equations using Haar and Legendre wavelets. Appl. Math. Model. 37(23), 9455–9481 (2013)
30. 30.
Yousefi, S.A.: Legendre wavelets method for solving differential equations of Lane–Emden type. Appl. Math. Comput. 181(2), 1417–1422 (2006).
31. 31.
Yousefi, S.A.: Legendre multiwavelet galerkin method for solving the hyperbolic telegraph equation. Numer. Methods Partial Differ. Equ. 26(3), 535–543 (2010)
32. 32.
Yousefi, S., Razzaghi, M.: Legendre wavelets method for the nonlinear Volterra–Fredholm integral equations. Math. Comput. Simul. 70(1), 1–8 (2005).

## Authors and Affiliations

• Simin Shekarpaz
• 1
Email author
• Kourosh Parand
• 2
• 3
• Hossein Azari
• 1
1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesShahid Beheshti UniversityTehranIran
2. 2.Department of Computer Sciences, Faculty of Mathematical SciencesShahid Beheshti UniversityTehranIran
3. 3.Department of Cognitive Modeling, Institute for Cognitive and Brain SciencesShahid Behehsti UniversityTehranIran