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Reproducing kernel method to detect the temperature distribution for annular fins with temperature-dependent thermal conductivity. (English) Zbl 1490.74023

Summary: 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.

MSC:

74F05 Thermal effects in solid mechanics
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[1] 1. Kraus, A.D., Aziz, A., and Welty, J.R. (2013), Extended surface heat transfer, New Delhi: Wiley India.
[2] 1. Kraus, A.D., Aziz, A., and Welty, J.R. (2013), Extended surface heat transfer, New Delhi: Wiley India.
[3] 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.
[4] 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.
[5] 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.
[6] 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.
[7] 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.
[8] 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.
[9] 5. Dulkin, I. and Garasko, G. (2008), Analysis of the \(111\)-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. · Zbl 1148.80312
[10] 5. Dulkin, I. and Garasko, G. (2008), Analysis of the \(111\)-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. · Zbl 1148.80312
[11] 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.
[12] 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.
[13] 7. Aksoy, I.G. (2013), Thermal analysis of annular fins with temperature-dependent thermal properties, Applied Mathematics and Mechanics, 34(11), 1349-1360.
[14] 7. Aksoy, I.G. (2013), Thermal analysis of annular fins with temperature-dependent thermal properties, Applied Mathematics and Mechanics, 34(11), 1349-1360.
[15] 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.
[16] 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.
[17] 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. · Zbl 1011.80011
[18] 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. · Zbl 1011.80011
[19] 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. · Zbl 1203.74148
[20] 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. · Zbl 1203.74148
[21] 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.
[22] 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.
[23] 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.
[24] 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.
[25] 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.
[26] 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.
[27] 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.
[28] 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.
[29] 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.
[30] 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.
[31] 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). · Zbl 1486.65305
[32] 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). · Zbl 1486.65305
[33] 17. Aronszajn, N. (1950), Theory of Reproducing Kernels, American Mathematical Society, 68(3), 337-404 · Zbl 0037.20701
[34] 17. Aronszajn, N. (1950), Theory of Reproducing Kernels, American Mathematical Society, 68(3), 337-404 · Zbl 0037.20701
[35] 18. Berlinet, A. and Thomas-Agnan, C. (2004), Reproducing Kernel Hilbert Spaces in Probability and Statistics, Springer. · Zbl 1145.62002
[36] 18. Berlinet, A. and Thomas-Agnan, C. (2004), Reproducing Kernel Hilbert Spaces in Probability and Statistics, Springer. · Zbl 1145.62002
[37] 19. Daniel, A. (2003), Reproducing Kernel Spaces and Applications, Springer. · Zbl 1021.00005
[38] 19. Daniel, A. (2003), Reproducing Kernel Spaces and Applications, Springer. · Zbl 1021.00005
[39] 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. · Zbl 1410.34185
[40] Ghasemi, M., Fardi, M., and Ghaziani, R.K. (2015), Numerical solution of nonlinear delay differential equa-tions of fractional order in reproducing kernel Hilbert space, Applied Mathematics and Computation, 268, 815-831. · Zbl 1410.34185
[41] 21. Geng, F. and Qian, S. (2014), Piecewise reproducing kernel method for singularly perturbed delay initial value problems, Applied Mathematics Letters, 37, 67-71. · Zbl 1321.34086
[42] 21. Geng, F. and Qian, S. (2014), Piecewise reproducing kernel method for singularly perturbed delay initial value problems, Applied Mathematics Letters, 37, 67-71. · Zbl 1321.34086
[43] 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. · Zbl 1380.65129
[44] 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. · Zbl 1380.65129
[45] 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. · Zbl 1409.65048
[46] 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. · Zbl 1409.65048
[47] 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. · Zbl 1488.49015
[48] 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. · Zbl 1488.49015
[49] Ghasemi, M., Fardi, M., and Ghaziani, R.K. (2015), Numerical solution of nonlinear delay differential equa-tions of fractional order in reproducing kernel Hilbert space, Applied Mathematics and Computation, 268, 815-831. · Zbl 1410.34185
[50] 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. · Zbl 1410.34185
[51] Fardi, M. and Ghasemi, M. (2016), Solving nonlocal initial-boundary value problems for parabolic and hy-perbolic integro-differential equations in reproducing kernel Hilbert space, Numerical Methods for Partial Differential Equations, 33(1), 174-198. · Zbl 1357.65317
[52] 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. · Zbl 1357.65317
[53] 27. Mei, L., Jia, Y., and Lin, Y. (2018), Simplified reproducing kernel method for impulsive delay differential equations, Applied Mathematics Letters, 83, 123-129. · Zbl 1503.65138
[54] 27. Mei, L., Jia, Y., and Lin, Y. (2018), Simplified reproducing kernel method for impulsive delay differential equations, Applied Mathematics Letters, 83, 123-129. · Zbl 1503.65138
[55] Xu, M. and Lin, Y. (2016), Simplified reproducing kernel method for fractional differential equations with delay, Applied Mathematics Letters, 52, 156-161. · Zbl 1330.65102
[56] 28. Xu, M. and Lin, Y. (2016), Simplified reproducing kernel method for fractional differential equations with delay, Applied Mathematics Letters, 52, 156-161. · Zbl 1330.65102
[57] Cui, M. and Lin, Y. (2009), Nonlinear numerical analysis in the reproducing kernel space, New York: Nova Science. · Zbl 1165.65300
[58] 29. Cui, M. and Lin, Y. (2009), Nonlinear numerical analysis in the reproducing kernel space, New York: Nova Science. · Zbl 1165.65300
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