THERMAL SCIENCE

International Scientific Journal

SOLUTION OF THE INVERSE HEAT CONDUCTION PROBLEM WITH NEUMANN BOUNDARY CONDITION BY USING THE HOMOTOPY PERTURBATION METHOD

ABSTRACT
In the paper a solution of the inverse heat conduction problem with the Neumann boundary condition is presented. For finding this solution the homotopy perturbation method is applied. Investigated problem consists in calculation of the temperature distribution in considered domain, as well as in reconstruction of the functions describing the temperature and the heat flux on the boundary, in case when the temperature measurements in some points of the domain are known. An example confirming usefulness of the homotopy perturbation method for solving problems of this kind are also included.
KEYWORDS
PAPER SUBMITTED: 2012-08-26
PAPER REVISED: 2013-01-08
PAPER ACCEPTED: 2013-04-24
PUBLISHED ONLINE: 2013-06-01
DOI REFERENCE: https://doi.org/10.2298/TSCI120826051H
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2013, VOLUME 17, ISSUE 3, PAGES [643 - 650]
REFERENCES
  1. Beck, J.V., Blackwell, B., St.Clair, C.R., Inverse Heat Conduction: Ill Posed Problems, Wiley Intersc., New York, 1985
  2. ¨Ozisik, M.N., Orlande, H.R.B., Inverse Heat Transfer: Fundamentals and Applications, Taylor & Francis, New York, 2000
  3. Hebi, Y., Man, Y., Inverse Problem-Based Analysis on Non-Uniform Profiles of Thermal Resistance Between Strand and Mould for Continuous Round Billets Casting, J. Mater. Proc. Tech., 413 (2007), pp. 49-56
  4. Fakhraie, M., Shidfar, A., Garshasbi, M., A Computational Procedure for Estimation of an Unknown Coefficient in an Inverse Boundary Value Problem, Appl. Math. Comput., 187 (2007), pp. 1120-1125
  5. Le Bideau, P., Ploteau, J.P., Glouannec, P., Heat Flux Estimation in an Infrared Experi- mental Furnace Using an Inverse Method, Appl. Therm. Eng., 27 (2009), pp. 2463-2472
  6. Mera, N.S. Elliott, L., Ingham, D.B., A Multi-Population Genetic Algorithm Approach for Solving Ill-Posed Problems, Comput. Mech., 33 (2004), pp. 254-262
  7. Hetmaniok, E., S lota, D., Zielonka, A., Solution of the Inverse Heat Conduction Problem by Using the ABC Algorithm, Lecture Notes in Comput. Sci., 6086 (2010), pp. 659-668
  8. S lota, D., Restoring Boundary Conditions in the Solidification of Pure Metals, Comput. & Structures, 89 (2011), pp. 48-54
  9. S lota, D., Reconstruction of the Boundary Condition in the Problem of the Binary Alloy Solidification, Arch. Metall. Mater., 56 (2011), pp. 279-285
  10. Hetmaniok, E., S lota, D., Zielonka, A., Application of the Ant Colony Optimization Algo- rithm for Reconstruction of the Thermal Conductivity Coefficient, Lecture Notes in Comput. Sci., 7269 (2012), pp. 240-248
  11. Hetmaniok, E., Nowak, I., S lota, D., Zielonka, A., Determination of Optimal Parameters for the Immune Algorithm Used for Solving the Inverse Heat Conduction Problems With and Without the Phase Change, Numer. Heat Transfer B, 62 (2012), pp. 462-478
  12. Nowak, I., Smolka, J., Nowak, A.J., An Effective 3-D Inverse Procedure to Retrieve Cooling Conditions in an Aluminium Alloy Continuous Casting Problem, Appl. Therm. Eng., 30 (2010), pp. 1140-1151
  13. Nowak, I., Smolka, J., Nowak, A.J., Application of Bezier Surfaces to the 3-D Inverse Geometry Problem in Continuous Casting, Inverse Probl. Sci. Eng., 19 (2011), pp. 78-86
  14. Monde, M., Arima, H., Liu, W., Mitutake, Y., Hammad, J.A., An Analytical Solution for Two-Dimensional Inverse Heat Conduction Problems Using Laplace Transform, Int. J. Heat Mass Transfer, 46 (2003), pp. 2135-2148
  15. Woodfielda, P.L., Mondeb, M., Mitsutakeb, Y., Implementation of an Analytical Two- Dimensional Inverse Heat Conduction Technique to Practical Problems, Int. J. Heat Mass Transfer, 49 (2006), pp. 187-197
  16. Woodfielda, P.L., Mondeb, M., Estimation of Uncertainty in an Analytical Inverse Heat Conduction Solution, Experimental Heat Transfer, 22 (2009), pp. 129-143
  17. Fernandes, A.P., Sousa, P.F.B., Borges, V.L., Guimaraes, G., Use of 3D-Transient Analyt- ical Solution Based on Greens Function to Reduce Computational Time in Inverse Heat Conduction Problems, Appl. Math. Modelling, 34 (2010), pp. 4040-4049
  18. He, J.-H., Homotopy Perturbation Technique, Comput. Methods Appl. Mech. Engrg., 178 (1999), pp. 257-262
  19. He, J.-H., A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems, Int. J. Non-Linear Mech., 35 (2000), pp. 37-43
  20. He, J.-H., Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation.de- Verlag, Berlin, 2006
  21. Ganji, D.D., Rajabi, A., Assessment of Homotopy-Perturbation and Perturbation Methods in Heat Radiation Equations, Int. Comm. Heat & Mass Transf., 33 (2006), pp. 391-400
  22. Ganji, D.D., Hosseini, M.J., Shayegh, J., Some Nonlinear Heat Transfer Equations Solved by Three Approximate Methods, Int. Comm. Heat & Mass Transf., 34 (2007), pp. 1003- 1016
  23. Khaleghi, H., Ganji, D.D., Sadighi, A., Application of Variational Iteration and Homotopy- Perturbation Methods to Nonlinear Heat Transfer Equations With Variable Coefficients, Numer. Heat Transfer A, 52 (2007), pp. 25-42
  24. S lota, D., The Application of the Homotopy Perturbation Method to One-Phase Inverse Stefan Problem, Int. Comm. Heat & Mass Transf., 37 (2010), pp. 587-592
  25. S lota, D., Homotopy Perturbation Method for Solving the Two-Phase Inverse Stefan Prob- lem, Numer. Heat Transfer A, 59 (2011), pp. 755-768
  26. Hetmaniok, E., Nowak, I., S lota, D., Witu la, R., Application of the Homotopy Perturbation Method for the Solution of Inverse Heat Conduction Problem, Int. Comm. Heat & Mass Transf., 39 (2012), pp. 30-35
  27. Dehghan, M., Shakeri, F., Solution of a Partial Differential Equation Subject to Temper- ature Overspecification by He's Homotopy Perturbation Method, Phys. Scr., 75 (2007), pp. 778-787
  28. Shakeri, F., Dehghan, M., Inverse Problem of Diffusion Equation by He's Homotopy Per- turbation Method, Phys. Scr., 75 (2007), pp. 551-556
  29. Grzymkowski, R., Hetmaniok, E., S lota, D., Application of the Homotopy Perturbation Method for Calculation of the Temperature Distribution in the Cast-Mould Heterogeneous Domain, J. Achiev. Mater. Manuf. Eng., 43 (2010), pp. 299-309
  30. Shakeri, F., Dehghan, M., Solution of Delay Differential Equations via a Homotopy Pertur- bation Method, Math. Comput. Modelling, 48 (2008), pp. 486-498
  31. Biazar, J., Ghazvini, H., Homotopy Perturbation Method for solving Hyperbolic Partial Differential Equations, Comput. Math. Appl., 56 (2008), pp. 453-458
  32. Madani, M., Fathizadeh, M., Khan, Y., Yildirim, A., On the Coupling of the Homotopy Perturbation Method and Laplace Transformation, Math. Comput. Modelling, 53 (2011), pp. 1937-1945
  33. Khan, Y., Akbarzade, M., Kargar, A., Coupling of Homotopy and Variational Approach for Conservative Oscillator With Strong Odd-Nonlinearity, Sci. Iran., 19 (2012), pp. 417-422
  34. Dehghan, M., Heris, J., Study of The Wave-Breaking'S Qualitative Behavior of the Fornberg-Whitham Equation via Quasi-Numeric Approaches, Int. J. Numer. Methods Heat Fluid Flow, 22 (2012), pp. 537-553
  35. Biazar, J., Ghazvini, H., Convergence of the Homotopy Perturbation Method for Partial Differential Equations, Nonlinear Anal.: Real World Appl., 10 (2009), pp. 2633-2640
  36. Biazar, J., Aminikhah, H., Study of Convergence of Homotopy Perturbation Method for Systems of Partial Differential Equations, Comput. Math. Appl., 58 (2009), pp. 2221-2230
  37. Turkyilmazoglu, M., Convergence of the Homotopy Perturbation Method, Int. J. Nonlin. Sci. Numer. Simulat., 12 (2011), pp. 9-14
  38. Alawneh, A., Al-Khaled, K., Al-Towaiq, M., Reliable Algorithms for Solving Integro- Differential Equations With Applications, Int. J. Comput. Math., 87 (2010), pp. 1538-1554
  39. Biazar, J., Ghanbari, B., Porshokouhi, M.G., Porshokouhi, M.G., He's Homotopy Perturba- tion Method: A Strongly Promising Method for Solving Non-Linear Systems of the Mixed Volterra-Fredholm Integral Equations, Comput. Math. Appl., 61 (2011), pp. 1016-1023
  40. Chen, Z., Jiang,W., Piecewise Homotopy Perturbation Method for Solving Linear and Non- linear Weakly Singular VIE of Second Kind, Appl. Math. Comput., 217 (2011), pp. 7790- 7798
  41. Jafari, H., Alipour, M., Tajadodi, H., Convergence of Homotopy Perturbation Method for Solving Integral Equations, Thai J. Math., 8 (2010), pp. 511-520
  42. Hetmaniok, E., S lota, D., Witu la, R., Convergence and Error Estimation of Homotopy Perturbation Method for Fredholm and Volterra Integral Equations, Appl. Math. Comput., 218 (2012), pp. 10717-10725
  43. Hetmaniok, E., Nowak, I., S lota, D., Witu la, R., A Study of the Convergence of and Error Estimation for the Homotopy Perturbation Method for the Volterra-Fredholm Integral Equations, Appl. Math. Lett., 26 (2013), pp. 165-169

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