Skip to main content
SpringerLink
Log in

An efficient wavelet method for nonlinear problems arising in heat transfer

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this article, we study the heat transfer problems which typically occur in nonlinear models. Since nonlinearity causes time-consuming and difficulty in finding analytical solutions, we focus on the Chebyshev wavelets method which is a powerful computational scheme for approximating solutions. In the proposed method, we apply the Chebyshev wavelets to expand the solution through the corresponding operational matrix of integration. Moreover, the efficiency of this approach is experimentally compared with the homotopy perturbation method, differential transformation method and variational iteration method which approves the efficiency of our method rather than the analytical methods in overcoming the nonlinearity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

Abbreviations

A :

Surface area, \(\mathrm{m}^2\)

\(A_{c}\) :

Cross-sectional area, \(\mathrm{m}^2\)

c :

Specific heat, \(\mathrm{J}/\mathrm{kg\, K}\)

\(c_a\) :

Specific heat at temperature \({T}_a, \mathrm{J/kg\, K}\)

c(T):

Specific heat at temperature \({T}, \mathrm{J/kg\, K}\)

E :

Surface emissivity, W

h :

Natural convection coefficient, \(\mathrm{W/m}^2\, \mathrm{K}\)

\(k_{a}\) :

Thermal conductivity at temperature \(T_{a}\), \(\mathrm{J/smK}\)

k(T):

Thermal conductivity at temperature T, \(\mathrm{J}/\mathrm{smK}\)

T :

Temperature, \(\mathrm{K}\)

\(T_a\) :

Environment temperature, \(\mathrm{K}\)

\(T_i\) :

Initial temperature, \(\mathrm{K}\)

\(T_s\) :

Effective sink temperature, \(\mathrm{K}\)

V :

Volume, \(\mathrm{m}^3\)

L :

Later heat length, \(\mathrm{m}\)

x :

Distance variable, \(\mathrm{m}\)

a, b :

Constants

Bi :

Biot number, \(hr_{i}/k_{\infty }\)

\(\theta\) :

Dimensionless temperature

\(\beta\) :

Constant, volumetric thermal expansion coefficient, \(1/\mathrm{K}\)

\(\varepsilon\) :

Small parameter

\(\rho\) :

Mass density, \(\mathrm{kg}/\mathrm{m}^3\)

\(\sigma\) :

Stefan–Boltzmann constant

\(\tau\) :

Dimensionless temporal coordinate, s

\(\xi\) :

Dimensionless distance

\(\delta\) :

Dimensionless thickness of the fin, \(t/r_{i}\)

\(\lambda\) :

The radii ratio, \(r_{o}/r_{i}\)

References

  1. Ganji DD, Hosseini MJ, Shayegh J (2007) Some nonlinear heat transfer equations solved by three approximate methods. Int Commun Heat Mass Transf 34(8):1003–1016

    Article  Google Scholar 

  2. Motsa SS (2012) Application of the new spectral homotopy analysis method (SHAM) in the non-linear heat conduction and convective fin problem with variable thermal conductivity. Int J Comput Methods 9(03):1250039

    Article  MathSciNet  MATH  Google Scholar 

  3. Yaghoobi H, Torabi M (2011) The application of differential transformation method to nonlinear equations arising in heat transfer. Int Commun Heat Mass Transf 38(6):815–820

    Article  Google Scholar 

  4. Abbasbandy S (2006) The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 360(1):109–113

    Article  MathSciNet  MATH  Google Scholar 

  5. Rajabi A, Ganji DD, Taherian H (2007) Application of homotopy perturbation method in nonlinear heat conduction and convection equations. Phys Lett A 360(4):570–573

    Article  MathSciNet  MATH  Google Scholar 

  6. Sajid M, Hayat T (2008) Comparison of ham and hpm methods in nonlinear heat conduction and convection equations. Nonlinear Anal Real World Appl 9(5):2296–2301

    Article  MathSciNet  MATH  Google Scholar 

  7. Domairry G, Nadim N (2008) Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation. Int Commun Heat Mass Transf 35(1):93–102

    Article  Google Scholar 

  8. Mohammadi F, Hosseini MM (2011) A comparative study of numerical methods for solving quadratic Riccati differential equations. J Franklin Inst 348(2):156–164

    Article  MathSciNet  MATH  Google Scholar 

  9. Hosseini MM, Mohyud-Din ST, Ghaneai H (2011) On the coupling of auxiliary parameter, Adomains’s polynomials and correction functional. Math Comput Appl 16(4):959–968

    MathSciNet  MATH  Google Scholar 

  10. Abbasbandy S (2009) Application of the variational iteration method for system of nonlinear Volterra’s integro-differential equations. Math Comput Appl 14(2):147–158

    MathSciNet  MATH  Google Scholar 

  11. Abbasbandy S (2009) Application of variational iteration method for \(n\) th-order integro-differential equations. Zeitschrift fur Naturforschung A 64:439–444

    Article  Google Scholar 

  12. Heydari MH, Hooshmandasl MR, Cattani C (2018) A new operational matrix of fractional order integration for the Chebyshev wavelets and its application for nonlinear fractional Van der Pol oscillator equation. Proc Math Sci 128:26

    Article  MathSciNet  MATH  Google Scholar 

  13. Jiwari R (2015) A hybrid numerical scheme for the numerical solution of the Burgers’ equation. Comput Phys Commun 188:59–67

    Article  MathSciNet  MATH  Google Scholar 

  14. Jiwari R (2012) A Haar wavelet quasilinearization approach for numerical simulation of Burgers’ equation. Comput Phys Commun 183:2413–2423

    Article  MathSciNet  MATH  Google Scholar 

  15. Mittal RC, Pandit S (2019) A numerical algorithm to capture spin patterns of fractional Bloch NMR flow models. J Comput Nonlinear Dyn 89(4):799–808

    Google Scholar 

  16. Mittal RC, Pandit S (2017) Numerical simulation of unsteady squeezing nanofluid and heat flow between two parallel plates using wavelets. Int J Thermal Sci 118:410–422

    Article  Google Scholar 

  17. Pandit S, Jiwari R, Bedi K, Koksal ME (2017) Haar wavelets operational matrix based algorithm for computational modelling of hyperbolic type wave equations. Eng Comput 34(8):793–814

    Article  Google Scholar 

  18. Kumar M, Pandit S (2015) An efficient algorithm based on Haar wavelet for numerical simulation of Fokker-Planck equation with constant and variable coefficient. Int J Numer Methods Heat Fluid Flow 25:41–56

    Article  MathSciNet  MATH  Google Scholar 

  19. Heydari MH (2020) Chebyshev cardinal wavelets for nonlinear variable-order fractional quadratic integral equations. Appl Numer Math 144:190–203

    Article  MathSciNet  MATH  Google Scholar 

  20. Heydari MH, Avazzadeh Z (2018) A new wavelet method for variable-order fractional optimal control problems. Asian J Control 20(5):1–14

    Article  MathSciNet  MATH  Google Scholar 

  21. Hosseininia M, Heydari MH (2019) Legendre wavelets for the numerical solution of nonlinear variable-order time fractional 2D reaction–diffusion equation involving Mittag–Leffler non-singular kernel. Chaos Solitons Fract 127:400–407

    Article  MathSciNet  MATH  Google Scholar 

  22. Heydari MH, Mahmoudi MR, Shakiba A, Avazzadeh Z (2018) Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion. Commun Nonlinear Sci Numer Simul 64:98–121

    Article  MathSciNet  MATH  Google Scholar 

  23. Moradi L, Mohammadi F, Baleanu D (2019) A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. J Vib Control 25(2):310–324

    Article  MathSciNet  Google Scholar 

  24. Heydari MH, Hooshmandasl MR, Barid Loghmani Gh, Cattani C (2016) Wavelets Galerkin method for solving stochastic heat equation. Int J Comput Math 93(9):1579–1596

    Article  MathSciNet  MATH  Google Scholar 

  25. Hosseininia M, Heydari MH, Avazzadeh Z, Maalek Ghaini FM (2018) Two-dimensional Legendre wavelets for solving variable-order fractional nonlinear advection-diffusion equation with variable coefficients. Int J Nonlinear Sci Numer Simul 19(7–8):793–802

    Article  MathSciNet  MATH  Google Scholar 

  26. Hosseininia M, Heydari MH, Avazzadeh Z (2020) Numerical study of the variable-order fractional version of the nonlinear fourth-order 2D diffusion-wave equation via 2D Chebyshev wavelets. Eng Comput. https://doi.org/10.1007/s00366-020-00995-z

    Article  MATH  Google Scholar 

  27. Fattahzadeh F, Babolian E (2007) Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration. Appl Math Comput 188(1):1016–1022

    MathSciNet  MATH  Google Scholar 

  28. Fattahzadeh F, Babolian E (2007) Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration. Appl Math Comput 188(1):417–426

    MathSciNet  MATH  Google Scholar 

  29. Danish M, Kumar S, Kumar S (2011) Exact analytical solutions of three nonlinear heat transfer models. Proc World Cong Eng 3:2011

    Google Scholar 

  30. Arslanturk C (2009) Correlation equations for optimum design of annular fins with temperature dependent thermal conductivity. Heat Mass Transf 45:519–525

    Article  Google Scholar 

  31. Ganji DD, Afrouzi GA, Talarposhti RA (2007) Application of variational iteration method and homotopy-perturbation method for nonlinear heat diffusion and heat transfer equations. Phys Lett A 368(6):450–457

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Shiraz University of Technology, Shiraz, Iran

    M. H. Heydari

  2. Department of Mechanical and Aerospace Engineering, Shiraz University of Technology, Shiraz, Iran

    O. Bavi

Authors
  1. M. H. Heydari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. Bavi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. H. Heydari.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Heydari, M.H., Bavi, O. An efficient wavelet method for nonlinear problems arising in heat transfer. Engineering with Computers 38 (Suppl 4), 2867–2878 (2022). https://doi.org/10.1007/s00366-021-01437-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-021-01437-0

Keywords

Search

Navigation