Research Article
Reliable Iterative Methods for Solving Convective Straight and Radial Fins with Temperature-Dependent Thermal Conductivity Problems
Abstract
In our article, three iterative methods are performed to solve the nonlinear differential equations that represent the straight and radial fins affected by thermal conductivity. The iterative methods are the Daftardar-Jafari method namely (DJM), Temimi-Ansari method namely (TAM) and Banach contraction method namely (BCM) to get the approximate solutions. For comparison purposes, the numerical solutions were further achieved by using the fourth Runge-Kutta (RK4) method, Euler method and previous analytical methods that available in the literature. Moreover, the convergence of the proposed methods was discussed and proved. In addition, the maximum error remainder values are also evaluated which indicates that the proposed methods are efficient and reliable. Our computational works have been done by using the computer algebra system MATHEMATICA®10 to evaluate the terms in the iterative processes.
Keywords
References
- 1. D. Q. Kem, D. A. Kraus, Extended surface heat transfer, McGraw-Hill, New York, (1972).2. S. O. Akindeinde, Parker-Sochacki method for the solution of convective straight fins problem with temperature-dependent thermal conductivity, International Journal of Nonlinear Science 20 (2018) 1-11.3. E. M. A. Mokheimer, Performance of annular fins with different profiles subject to variable heat transfer coefficient, International Journal of Heat and Mass Transfer 45, (17) (2002) 3631-3642.4. D. D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and perturbation methods in heat radiation equation, International Communications in Heat and Mass Transfer 33 (2006) 391-400.5. D. D. Ganji, The application of he's homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A 355 (2006) 337-341.6. B. T. F. Chung, B. X. Zhang, Optimization of radiating fin array including mutual irradiations between radiator elements, ASME Journal of Heat Transfer 113, (4) (1991) 814-822.7. D. Lesnic, P. Heggs, A decomposition method for power-law fin-type problems, International Communications in Heat and Mass Transfer 31, (5) (2004) 673-682.8. J. G. Bartas, W. H. Sellers, Radiation fin effectiveness, ASME Journal of Heat Transfer 82, (1) (1960) 73-75.9. S. B. Coskun, M. T. Atay, Analysis of convective straight and radial fins with temperature-dependent thermal conductivity using variational iteration method with comparison with respect to finite element analysis, Mathematical Problems in Engineering (2007), Article ID 42072, 15.10. E. Cuce, P.M. Cuce, Homotopy perturbation method for temperature distribution, fin efficiency and fin effectiveness of convective straight fins with temperature-dependent thermal conductivity, Institution of Mechanical Engineers (2012) 1-7.11. C. H. Chiu, C. K. Chen, A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, International Journal of Heat and Mass Transfer 45, (10) (2002) 2067-2075.12. C. H. Chiu, C. K. Chen, Application of Adomian’s decomposition procedure to the analysis of convective-radiative fins, ASME Journal of Heat Transfer 125, (2) (2003) 312-316.13. A. Patra, S. S. Ray, Homotopy perturbation sumudu transform method for solving convective radial fins with temperature-dependent thermal conductivity of fractional order energy balance equation, International Journal of Heat and Mass Transfer 76 (2014) 162-170.14. J. E. Wilkins Jr, Minimizing the mass of thin radiating fins, Journal of Aerospace Science 27, (2) (1960) 145-146. 15. R. D. Cockfield, Structural optimization of a space radiator, Journal of Spacecraft Rockets, 5, (10) (1968) 1240-1241.16. V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications 316 (2006) 753-763. 17. H. Temimi, A. R. Ansari, A semi-analytical iterative technique for solving nonlinear problems, Computers and Mathematics with Applications 61 (2011) 203-210.18. M. A. AL-Jawary, R. K. Raham, A semi-analytical iterative technique for solving chemistry problems, Journal of King Saud University 29, (3) (2017) 320-332.19. I. Tabet, M. Kezzar, K. Touafeka, N. Bellelb, S. Gheriebc, A. Khelifa, M. Adouane, Adomian decomposition method and pad´e approximation to determine fin efficiency of convective straight fins in solar air collector, International Journal of Mathematical Modelling and Computations 5, (4) (2015) 335-346.20. G. Sevilgen, A numerical analysis of a convective straight fin with temperature-dependent thermal conductivity, Thermal Science 21, (2) (2017) 939-952.21. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dodrecht, (1994).22. C. Arslanturk, A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, International Communications in Heat and Mass Transfer, 32 (2005) 831-841.23. A. A. Joneidi, D. D. Ganji, M. Babaelahi, Differential transformation method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity, International Communications in Heat and Mass Transfer 36, (7) (2009) 757-762.24. K. Saravanakumar, V. Ananthaswamy, M. Subha, L. Rajendran, Analytical solution of nonlinear boundary value problem for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, ISRN Thermodynamics (2013), Article ID 282481, 8.25. M. Torabi, H. Yaghoobi, A. Colantoni, P. Biondi, K. Boubaker, Analysis of radiative radial fin with temperature-dependent thermal conductivity using nonlinear differential transformation methods, Chinese Journal of Engineering (2013), Article ID 470696, 12. 26. D. Kumar, J. Singh, D. Baleanu, A fractional model of convective radial fins with temperature-dependent thermal conductivity, Romanian Reports in Physics 103(2017) 69.27. S. B. Coskun, M. T. Atay, Analysis of convective straight and radial fins with temperature-dependent thermal conductivity using variational iteration method with comparison with respect to finite element analysis, Mathematical Problems in Engineering (2007), Article ID 42072, 15.28. G. Domairry, M. Fazeli, 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) (2009) 489–499. 29. C. Arslanturk, Correlation equations for optimum design of annular fins with temperature dependent thermal conductivity, Heat and Mass Transfer 45, (4) (2009) 519–525. 30. B. Kundu, Analysis of thermal performance and optimization of concentric circular fins under dehumidifying conditions, International Journal of Heat and Mass Transfer 52, (11-12) (2009) 2646–2659. 31. V. Daftardar-Gejji, S. Bhalekar, Solving fractional boundary value problems with dirichlet boundary conditions using a new iterative method, Computers and Mathematics with Applications 59, (5) (2010) 1801-1809.32. M. A. AL-Jawary, G. H. Radhi, J. Ravnik, Daftardar-Jafari method for solving nonlinear thin film flow problem, Arab Journal of Basic and Applied Sciences 25, (1) (2018) 20–27.33. M. A. AL-Jawary, A semi-analytical iterative method for solving nonlinear thin film flow problems, Chaos, Solitons and Fractals 99 (2017) 52-56.34. M.A. Al-Jawary, An efficient iterative method for solving the Fokker–Planck equation, Results in Physics 6 (2016) 985-991.35. V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications 316 (2006) 753-763.36. S. Bhalekar, V. Daftardar-Gejji, Convergence of the new iterative method, International Journal of Differential Equations (2011), Article ID 989065, 10. 37. M. A. AL-Jawary, S. Hatif, A semi-analytical iterative method for solving differential algebraic equations, Ain Shams Engineering Journal (2017) (In press).38. M. A. AL-Jawary, G. H. Radhi, J. Ravnik, Semi-analytical method for solving Fokker-Planck’s equations, Journal of the Association of Arab Universities for Basic and Applied Sciences 24 (2017) 254-262.39. M. A. AL-Jawary, M. M. Azeez, G. H. Radhi, Analytical and numerical solutions for the nonlinear Burgers and advection–diffusion equations by using a semi-analytical iterative method, Computers and Mathematics with Applications, 76 (2018) 155-171.40. V. Daftardar-Gejji, S. Bhalekar, Solving nonlinear functional equation using Banach contraction principle, Far East Journal of Applied Mathematics 34, (3) (2009) 303-314.41. M.A. Al-Jawary, M.I. Adwan, G.H. Radhi, Three iterative methods for solving second order nonlinear ODEs arising in physics, Journal of King Saud University – Science, (2018) (In press).42. A. M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press, Beijing and Springer-Varlag Berlin Heidelberg (2011).43. Z. M. Odibat, A study on the convergence of variational iteration method, Mathematical and Computer Modelling 51 (2010) 1181-1192.
Year 2019,
Volume: 32 Issue: 3, 967 - 989, 01.09.2019
Abstract
References
- 1. D. Q. Kem, D. A. Kraus, Extended surface heat transfer, McGraw-Hill, New York, (1972).2. S. O. Akindeinde, Parker-Sochacki method for the solution of convective straight fins problem with temperature-dependent thermal conductivity, International Journal of Nonlinear Science 20 (2018) 1-11.3. E. M. A. Mokheimer, Performance of annular fins with different profiles subject to variable heat transfer coefficient, International Journal of Heat and Mass Transfer 45, (17) (2002) 3631-3642.4. D. D. Ganji, A. Rajabi, Assessment of homotopy-perturbation and perturbation methods in heat radiation equation, International Communications in Heat and Mass Transfer 33 (2006) 391-400.5. D. D. Ganji, The application of he's homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letters A 355 (2006) 337-341.6. B. T. F. Chung, B. X. Zhang, Optimization of radiating fin array including mutual irradiations between radiator elements, ASME Journal of Heat Transfer 113, (4) (1991) 814-822.7. D. Lesnic, P. Heggs, A decomposition method for power-law fin-type problems, International Communications in Heat and Mass Transfer 31, (5) (2004) 673-682.8. J. G. Bartas, W. H. Sellers, Radiation fin effectiveness, ASME Journal of Heat Transfer 82, (1) (1960) 73-75.9. S. B. Coskun, M. T. Atay, Analysis of convective straight and radial fins with temperature-dependent thermal conductivity using variational iteration method with comparison with respect to finite element analysis, Mathematical Problems in Engineering (2007), Article ID 42072, 15.10. E. Cuce, P.M. Cuce, Homotopy perturbation method for temperature distribution, fin efficiency and fin effectiveness of convective straight fins with temperature-dependent thermal conductivity, Institution of Mechanical Engineers (2012) 1-7.11. C. H. Chiu, C. K. Chen, A decomposition method for solving the convective longitudinal fins with variable thermal conductivity, International Journal of Heat and Mass Transfer 45, (10) (2002) 2067-2075.12. C. H. Chiu, C. K. Chen, Application of Adomian’s decomposition procedure to the analysis of convective-radiative fins, ASME Journal of Heat Transfer 125, (2) (2003) 312-316.13. A. Patra, S. S. Ray, Homotopy perturbation sumudu transform method for solving convective radial fins with temperature-dependent thermal conductivity of fractional order energy balance equation, International Journal of Heat and Mass Transfer 76 (2014) 162-170.14. J. E. Wilkins Jr, Minimizing the mass of thin radiating fins, Journal of Aerospace Science 27, (2) (1960) 145-146. 15. R. D. Cockfield, Structural optimization of a space radiator, Journal of Spacecraft Rockets, 5, (10) (1968) 1240-1241.16. V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications 316 (2006) 753-763. 17. H. Temimi, A. R. Ansari, A semi-analytical iterative technique for solving nonlinear problems, Computers and Mathematics with Applications 61 (2011) 203-210.18. M. A. AL-Jawary, R. K. Raham, A semi-analytical iterative technique for solving chemistry problems, Journal of King Saud University 29, (3) (2017) 320-332.19. I. Tabet, M. Kezzar, K. Touafeka, N. Bellelb, S. Gheriebc, A. Khelifa, M. Adouane, Adomian decomposition method and pad´e approximation to determine fin efficiency of convective straight fins in solar air collector, International Journal of Mathematical Modelling and Computations 5, (4) (2015) 335-346.20. G. Sevilgen, A numerical analysis of a convective straight fin with temperature-dependent thermal conductivity, Thermal Science 21, (2) (2017) 939-952.21. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Dodrecht, (1994).22. C. Arslanturk, A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, International Communications in Heat and Mass Transfer, 32 (2005) 831-841.23. A. A. Joneidi, D. D. Ganji, M. Babaelahi, Differential transformation method to determine fin efficiency of convective straight fins with temperature dependent thermal conductivity, International Communications in Heat and Mass Transfer 36, (7) (2009) 757-762.24. K. Saravanakumar, V. Ananthaswamy, M. Subha, L. Rajendran, Analytical solution of nonlinear boundary value problem for fin efficiency of convective straight fins with temperature-dependent thermal conductivity, ISRN Thermodynamics (2013), Article ID 282481, 8.25. M. Torabi, H. Yaghoobi, A. Colantoni, P. Biondi, K. Boubaker, Analysis of radiative radial fin with temperature-dependent thermal conductivity using nonlinear differential transformation methods, Chinese Journal of Engineering (2013), Article ID 470696, 12. 26. D. Kumar, J. Singh, D. Baleanu, A fractional model of convective radial fins with temperature-dependent thermal conductivity, Romanian Reports in Physics 103(2017) 69.27. S. B. Coskun, M. T. Atay, Analysis of convective straight and radial fins with temperature-dependent thermal conductivity using variational iteration method with comparison with respect to finite element analysis, Mathematical Problems in Engineering (2007), Article ID 42072, 15.28. G. Domairry, M. Fazeli, 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) (2009) 489–499. 29. C. Arslanturk, Correlation equations for optimum design of annular fins with temperature dependent thermal conductivity, Heat and Mass Transfer 45, (4) (2009) 519–525. 30. B. Kundu, Analysis of thermal performance and optimization of concentric circular fins under dehumidifying conditions, International Journal of Heat and Mass Transfer 52, (11-12) (2009) 2646–2659. 31. V. Daftardar-Gejji, S. Bhalekar, Solving fractional boundary value problems with dirichlet boundary conditions using a new iterative method, Computers and Mathematics with Applications 59, (5) (2010) 1801-1809.32. M. A. AL-Jawary, G. H. Radhi, J. Ravnik, Daftardar-Jafari method for solving nonlinear thin film flow problem, Arab Journal of Basic and Applied Sciences 25, (1) (2018) 20–27.33. M. A. AL-Jawary, A semi-analytical iterative method for solving nonlinear thin film flow problems, Chaos, Solitons and Fractals 99 (2017) 52-56.34. M.A. Al-Jawary, An efficient iterative method for solving the Fokker–Planck equation, Results in Physics 6 (2016) 985-991.35. V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications 316 (2006) 753-763.36. S. Bhalekar, V. Daftardar-Gejji, Convergence of the new iterative method, International Journal of Differential Equations (2011), Article ID 989065, 10. 37. M. A. AL-Jawary, S. Hatif, A semi-analytical iterative method for solving differential algebraic equations, Ain Shams Engineering Journal (2017) (In press).38. M. A. AL-Jawary, G. H. Radhi, J. Ravnik, Semi-analytical method for solving Fokker-Planck’s equations, Journal of the Association of Arab Universities for Basic and Applied Sciences 24 (2017) 254-262.39. M. A. AL-Jawary, M. M. Azeez, G. H. Radhi, Analytical and numerical solutions for the nonlinear Burgers and advection–diffusion equations by using a semi-analytical iterative method, Computers and Mathematics with Applications, 76 (2018) 155-171.40. V. Daftardar-Gejji, S. Bhalekar, Solving nonlinear functional equation using Banach contraction principle, Far East Journal of Applied Mathematics 34, (3) (2009) 303-314.41. M.A. Al-Jawary, M.I. Adwan, G.H. Radhi, Three iterative methods for solving second order nonlinear ODEs arising in physics, Journal of King Saud University – Science, (2018) (In press).42. A. M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press, Beijing and Springer-Varlag Berlin Heidelberg (2011).43. Z. M. Odibat, A study on the convergence of variational iteration method, Mathematical and Computer Modelling 51 (2010) 1181-1192.
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Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Mathematics |
| Authors | |
| Publication Date | September 1, 2019 |
| Published in Issue | Year 2019 Volume: 32 Issue: 3 |