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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Reproducing Kernel Method to Detect the Temperature Distribution for Annular Fins with Temperature-Dependent Thermal Conductivity

Journal of Applied Nonlinear Dynamics 11(2) (2022) 283--295 | DOI:10.5890/JAND.2022.06.002

M. Fardi$^{1}$, J. Tenreiro Machado$^{2}$

$^{1}$ Department of Mathematics, Faculty of Mathematical Science, Shahrekord University, Shahrekord, P. O. Box

115, Iran

$^{2}$ Department of Electrical Engineering, Institute of Engineering, Polytechnic of Porto, Porto 4249-015, Portugal

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Abstract

The efficiency of convective straight fins with temperature dependent thermal conductivity is determined by means of the reproducing kernel (RK) method. The RK space $W^3 [0,\lambda-1]$ is constructed so that every function satisfies the boundary conditions of the problem. The representation of the exact solution is given in the form of a series and the approximation is obtained by its truncation. The paper (i) derives the error estimates, (ii) proves the convergence and (iii) develops an iterative algorithm for obtaining the solution in the space $W^3 [0,\lambda -1]$. The results obtained by the proposed method are compared with those given by schemes in previous works demonstrating a fast convergence and high precision.

References

  1. [1]  Kraus, A.D., Aziz, A., and Welty, J.R. (2013), Extended surface heat transfer, New Delhi: Wiley India.
  2. [2]  Aziz, A. and Bouaziz, M. (2011), A least squares method for a longitudinal fin with temperature dependent internal heat generation and thermal conductivity, Energy Conversion and Management, 52(8-9), 2876-2882.
  3. [3]  Sun, Y. and Ma, J. (2015), Application of Collocation Spectral Method to Solve a Convective-Radiative Longitudinal Fin with Temperature Dependent Internal Heat Generation, Thermal Conductivity and Heat Transfer Coefficient, Journal of Computational and Theoretical Nanoscience, 12(9), 2851-2860.
  4. [4]  Coskun, S.B. and Atay, M.T. (2008), Fin efficiency analysis of convective straight fins with temperature dependent thermal conductivity using variational iteration method, Applied Thermal Engineering, 28(17-18), 2345-2352.
  5. [5]  Dulkin, I. and Garasko, G. (2008), Analysis of the $1$-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient: Part I-Extended inverse and direct solutions, International Journal of Heat and Mass Transfer, 51(13-14), 3309-3324.
  6. [6]  Domairry, G. and Fazeli, M. (2009), Homotopy analysis method to determine the fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Communications in Nonlinear Science and Numerical Simulation, 14(2), 489-499.
  7. [7]  Aksoy, I.G. (2013), Thermal analysis of annular fins with temperature-dependent thermal properties, Applied Mathematics and Mechanics, 34(11), 1349-1360.
  8. [8]  Ganji, D.D., Ganji, Z.Z., and Ganji, D.H. (2011), Determination of temperature distribution for annular fins with temperature dependent thermal conductivity by HPM, Thermal Science, 15, 111-115.
  9. [9]  Chiu, C. and Chen, C. (2002), A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, International Journal of Heat and Mass Transfer, 45(10), 2067-2075.
  10. [10]  Rajabi, A. (2007), Homotopy perturbation method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, Physics Letters A, 364(1), 33-37.
  11. [11]  Torabi, M. and Zhang, Q.B. (2013), Analytical solution for evaluating the thermal performance and efficiency of convectiveradiative straight fins with various profiles and considering all non-linearities, Energy Conversion and Management, 66, 199-210.
  12. [12]  Sun, Y. and Ma, J. (2015), Application of Collocation Spectral Method to Solve a Convective-Radiative Longitudinal Fin with Temperature Dependent Internal Heat Generation, Thermal Conductivity and Heat Transfer Coefficient, Journal of Computational and Theoretical Nanoscience, 12(9), 2851-2860.
  13. [13] Aziz, A. and Bouaziz, M. (2011), A least squares method for a longitudinal fin with temperature dependent internal heat generation and thermal conductivity, Energy Conversion and Management, 52(8-9), 2876-2882.
  14. [14]  EL-Zahar, E.R., Rashad, A.M., and Seddek, L.F. (2019), The Impact of Sinusoidal Surface Temperature on the Natural Convective Flow of a Ferrofluid along a Vertical Plate, Mathematics, 7(11), 1014.
  15. [15]  El-Zahar, E.R., Algelany, A.M., and Rashad, A.M. (2020), Sinusoidal Natural Convective Flow of Non-Newtonian Nanoliquid Over a Radiative Vertical Plate in a Saturated Porous Medium, IEEE Access, 8, 136131-136140.
  16. [16]  Faheem, M., Khan, A., and El-Zahar, E.R. (2020), On some wavelet solutions of singular differential equations arising in the modeling of chemical and biochemical phenomena, Advances in Difference Equations, 526(1).
  17. [17]  Aronszajn, N. (1950), Theory of Reproducing Kernels, American Mathematical Society, 68(3), 337-404
  18. [18]  Berlinet, A. and Thomas-Agnan, C. (2004), Reproducing Kernel Hilbert Spaces in Probability and Statistics, Springer.
  19. [19]  Daniel, A. (2003), Reproducing Kernel Spaces and Applications, Springer.
  20. [20]  Ghasemi, M., Fardi, M., and Ghaziani, R.K. (2015), Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space, Applied Mathematics and Computation, 268, 815-831.
  21. [21]  Geng, F. and Qian, S. (2014), Piecewise reproducing kernel method for singularly perturbed delay initial value problems, Applied Mathematics Letters, 37, 67-71.
  22. [22]  Geng, F.Z. and Qian, S.P. (2018), An optimal reproducing kernel method for linear nonlocal boundary value problems, Applied Mathematics Letters, 77, 49-56.
  23. [23]  Xu, M., Zhao, Z., Niu, J., Guo, L. and Lin, Y. (2019), A simplified reproducing kernel method for 1-D elliptic type interface problems, Journal of Computational and Applied Mathematics, 351, 29-40.
  24. [24]  Fardi, M., Ghaziani, R.K., and Ghasemi, M. (2016), The Reproducing Kernel Method for Some Variational Problems Depending on Indefinite Integrals, Mathematical Modelling and Analysis, 21(3), 412-429.
  25. [25]  Ghasemi, M., Fardi, M., and Ghaziani, R.K. (2015), Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space, Applied Mathematics and Computation, 268, 815-831.
  26. [26]  Fardi, M. and Ghasemi, M. (2016), Solving nonlocal initial-boundary value problems for parabolic and hyperbolic integro-differential equations in reproducing kernel Hilbert space, Numerical Methods for Partial Differential Equations, 33(1), 174-198.
  27. [27]  Mei, L., Jia, Y., and Lin, Y. (2018), Simplified reproducing kernel method for impulsive delay differential equations, Applied Mathematics Letters, 83, 123-129.
  28. [28]  Xu, M. and Lin, Y. (2016), Simplified reproducing kernel method for fractional differential equations with delay, Applied Mathematics Letters, 52, 156-161.
  29. [29]  Cui, M. and Lin, Y. (2009), Nonlinear numerical analysis in the reproducing kernel space, New York: Nova Science.

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