Rational Wavelets and Their Application for Solving the Heat Transfer Equations in Porous Medium

  • P. Rahimkhani
  • Y. OrdokhaniEmail author
Original Paper


In this paper, a new approach is presented for solving natural convection heat transfer equations embedded in porous medium. These equations are non-linear, three-point boundary value equations on semi-infinite interval. Our approach is based upon the rational Bernoulli wavelets; these wavelets are first introduced. Then the derivative operational matrix of rational Bernoulli wavelets is given. This matrix is utilized to reduce the solution of the under study problem to a system of algebraic equations. Error estimation of our approximation is provided. We also present the comparison of this work with Runge–Kutta method and other methods, moreover, in the figure of the relative errors, we show that our results are accurate and applicable.


Operational matrix Rational Bernoulli wavelets Numerical solution Porous media 

Mathematics Subject Classification

34K28 42C40 65L10 



The first author is supported by the national elites foundation. Authors are very grateful to one of the reviewers for carefully reading the paper and for his(her) comments and suggestions which have improved the paper.


  1. 1.
    Cheng, P., Chang, I.D.: On buoyancy induced flows in a saturated porous medium adjacent to impermeable horizontal surfaces. Int. J. Heat Mass Transf. 19, 1267–1272 (1976)CrossRefGoogle Scholar
  2. 2.
    Merkin, J.H.: Free convection boundary layers in a saturated porous medium with lateralmassflux. Int. J. Heat Mass Transf. 21, 1499–1504 (1978)CrossRefGoogle Scholar
  3. 3.
    Nilson, R.H.: Natural convective boundary layer on two-dimensional and axisymmetric surfaces in high-Pr fluids or in fluid saturated porous media. ASME J. Heat Transf. 103, 803–807 (1981)CrossRefGoogle Scholar
  4. 4.
    Cheng, P., Le, T.T., Pop, I.: Natural convection of a Darcian fluid about a cone. Int. Commun. Heat Mass Transf. 12, 705–717 (1985)CrossRefGoogle Scholar
  5. 5.
    Parand, K., Abbasbandy, S., Kazem, S., Rezaei, A.R.: Comparison between two common collocation approaches based on radial basis functions for the case of heat transfer equations arising in porous medium. Commun. Nonlinear Sci. Numer. Simul. 16, 1396–1407 (2011)CrossRefGoogle Scholar
  6. 6.
    Rotem, Z., Claussen, L.: Natural convection above unconfined horizontal surfaces. J. Fluid Mech. 38(1), 173–192 (1969)CrossRefGoogle Scholar
  7. 7.
    Nakayama, A., Hossain, M.A.: An integral treatment for combined heat and mass transfer by natural convection in a porous medium. Int. J. Heat Mass Transf. 38, 761–765 (1995)CrossRefGoogle Scholar
  8. 8.
    Parand, K., Baharifard, F., Bayat Babolghani, F.: Comparison between rational Gegenbauer and modiffied generalized Laguerre functions collocation methods for solving the case of Heat transfer equations arising in porous medium. Int. J. Ind. Math. 4(2), 107–122 (2012)Google Scholar
  9. 9.
    Sohouli, A.R., Famouri, M., Kimiaeifar, A., Domairry, G.: Application of homotopy analysis method for natural convection of Darcian fluid about a vertical full cone embedded in pours media prescribed surface heat flux. Commun. Nonlinear Sci. Numer. Simul. 15, 1691–1699 (2010)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sohouli, A.R., Domairry, D., Famouri, M., Mohsenzadeh, A.: Analytical solution of natural convection of Darcian fluid about a vertical full cone embedded in porous media prescribed wall temperature by means of HAM. Int. Commun. Heat Mass Transf. 35, 1380–1384 (2008)CrossRefGoogle Scholar
  11. 11.
    Tajvidi, T., Razzaghi, M., Dehghan, M.: Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate. Chaos Solitons Fractals 35, 59–66 (2008)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Shen, J.: Stable and efficient spectral methods in unbounded domains using Laguerre functions. SIAM J. Numer. Anal. 38, 1113–1133 (2000)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Siyyam, H.I.: Laguerre tau methods for solving higher order ordinary differential equations. J. Comput. Anal. Appl. 3, 173–182 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Guo, B.Y.: Gegenbauer approximation and its applications to differential equations with rough asymptotic behaviors at infinity. Appl. Numer. Math. 38, 403–425 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Guo, B.Y.: Gegenbauer approximation and its applications to differential equations on the whole line. J. Math. Anal. Appl. 226, 180–206 (1998)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Boyd, J.P.: A comparison of numerical algorithms for fourier extension of the first, second, and third kinds. J. Comput. Phys. 178(1), 118–160 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Parand, K., Razzaghi, M.: Rational Legendre approximation for solving some physical problems on semi-infinite intervals. Phys. Scripta 69(5), 353–357 (2004)CrossRefGoogle Scholar
  18. 18.
    Parand, K., Taghavi, A.: Rational scaled generalized Laguerre function collocation method for solving the Blasius equation. J. Comput. Appl. Math. 233, 980–989 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Calvert, V., Mashayekhi, S., Razzaghi, M.: Solution of Lane–Emden type equations using rational Bernoulli functions. Math. Methods Appl. Sci. 39, 1268–1284 (2016)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Calvert, V., Razzaghi, M.: Solutions of the Blasius and MHD Falkner–Skan boundary-layer equations by modified rational Bernoulli functions. Int. J. Numer. Methods Heat Fluid Flow 27, 1687–1705 (2017). CrossRefGoogle Scholar
  21. 21.
    Jena, M.K., Sahu, K.S.: Haar wavelet operational matrix method to solve initial value problems: a short survey. Int. J. Appl. Comput. Math. 3, 3961–3975 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Shah, F.A., Abass, R., Debnath, L.: Numerical solution of fractional differential equations using Haar wavelet operational matrix method. Int. J. Appl. Comput. Math. 3, 2423–2445 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rahimkhani, P., Ordokhani, Y., Babolian, E.: Fractional-order Bernoulli wavelets and their applications. Appl. Math. Model. 40, 8087–8107 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Rahimkhani, P., Ordokhani, Y., Babolian, E.: Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. J. Comput. Appl. Math. 309, 493–510 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Yi, M., Wang, L., Jun, H.: Legendre wavelets method for the numerical solution of fractional integro-differential equations with weakly singular kernel. Appl. Math. Model. 40, 3422–3437 (2016)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Gupta, A.K., Ray, S.S.: Numerical treatment for the solution of fractional fifth-order Sawada–Kotera equation using second kind Chebyshev wavelet method. Appl. Math. Model. 39, 5121–5130 (2015)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Rahimkhani, P., Ordokhani, Y., Babolian, E.: Müntz-Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numer. Algorithms 77(4), 1283–1305 (2018)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Rahimkhani, P., Ordokhani, Y., Babolian, E.: An efficient approximate method for solving delay fractional optimal control problems. Nonlinear Dyn. 86, 1649–1661 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Keshavarz, E., Ordokhani, Y., Razzaghi, M.: Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations. Appl. Math. Model. 38, 6038–6051 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesAlzahra UniversityTehranIran

Personalised recommendations