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A weighted algorithm based on the homotopy analysis method: application to inverse heat conduction problems. (English) Zbl 1222.65105
Summary: A weighted algorithm based on the homotopy analysis method (HAM), for solving inverse heat conduction problems, is introduced. Using the HAM, it is possible to find the exact solution or an approximate solution of the problem in the form of a series.

MSC:
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
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[1] Shidfar, A.; Garshasbi, M., A weighted algorithm based on Adomian decomposition method for solving an special class of evolution equations, Commun nonlinear sci numer simulat, 14, 1146-1151, (2009) · Zbl 1221.35202
[2] Liao SJ. The proposed homotopy analysis technique for the solution of nonlinear problems. PhD thesis, Shanghai Jiao Tong University; 1992.
[3] Liao, S.J., An explicit, totally analytic approximate solution for Blasius viscous flow problems, Int J non-linear mech, 34, 759-778, (1999) · Zbl 1342.74180
[4] Liao, S.J., Beyond perturbation: introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Press Boca Raton
[5] Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non- Newtonian fluids over a stretching sheet, J fluid mech, 488, 189-212, (2003) · Zbl 1063.76671
[6] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl math comput, 147, 499-513, (2004) · Zbl 1086.35005
[7] Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int J heat mass transf, 48, 2529-3259, (2005) · Zbl 1189.76142
[8] Liao, S.J., Notes on the homotopy analysis method: some definitions and theorems, Commun nonlinear sci numer simul, 14, 983, (2009) · Zbl 1221.65126
[9] Shidfar, A., A series solution of the nonlinear Volterra and Fredholm integro-differential equations, Commun nonlinear sci numer simulat, 15, 205-215, (2010) · Zbl 1221.65343
[10] Molabahrami, A.; Shidfar, A., A study on the PDEs with power-law nonlinearity, Nonlinear analysis: real world applications, (2009) · Zbl 1189.35012
[11] Molabahrami, A.; Khani, F., The homotopy analysis method to solve the Burgers-Huxley equation, Nonlinear anal-real, 10, 589-600, (2009) · Zbl 1167.35483
[12] Khani, F.; Ahmadzadeh Raji, M.; Hamedi-Nezhad, S., A series solution of the fin problem with a temperature-dependent thermal conductivity, Commun nonlinear sci numer simulat, 14, 3007-3017, (2009)
[13] Khani, F.; Ahmadzadeh Raji, M.; Hamedi-Nezhad, S., Analytical solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient, Commun nonlinear sci numer simulat, 14, 3327-3338, (2009) · Zbl 1221.74083
[14] Yücel, Ugˇur, Homotopy analysis method for the sine-Gordon equation with initial conditions, Appl math comput, 203, 387-395, (2008) · Zbl 1157.65464
[15] Sami Bataineh, A.; Noorani, M.S.M.; Hashim, I., Modified homotopy analysis method for solving systems of second-order BVPs, Commun nonlinear sci numer simulat, 14, 430-442, (2009) · Zbl 1221.65196
[16] Watson, Layne T., Probability-one homotopies in computational science, J comput appl math, 140, 85-807, (2002) · Zbl 0996.65055
[17] Watson, L.T., Globally convergent homotopy methods: a tutorial, Appl math comput, 13BK, 369-396, (1989) · Zbl 0689.65033
[18] Watson, Layne T.; Scott, Melvin R., Solving spline-collocation approximations to nonlinear two-point boundary-value problems by a homotopy method, Appl math comput, 24, 3X-357, (1987) · Zbl 0635.65099
[19] Watson, Layne T., Engineering applications of the chowyorke algorithm, Appl math comput, 9, 111-133, (1981) · Zbl 0481.65029
[20] Watson, Layne T.; Haftka, Raphael T., Modern homotopy methods in optimization, Comput meth appl mech eng, 74, 289-305, (1989) · Zbl 0693.65046
[21] Wang, Y.; Bernstein, D.S.; Watson, L.T., Probability-one homotopy algorithms for solving the coupled Lyapunov equations arising in reduced-order H2=H1 modeling estimation and control, Appl math comput, 123, 155-185, (2001) · Zbl 1028.93011
[22] Burggraf, O.R., An exact solution of the inverse problem in heat conduction theory and applications, J heat transfer, 86C, 373-382, (1964)
[23] Langford, D., New analytic solutions of the one-dimensional heat equation for temperature and heat flow rate both prescribed at the same fixed boundary (with applications to the phase change problem), Quart appl math, 24, 315-322, (1966) · Zbl 0144.14302
[24] Adomian, G., A new approach to the heat equation – an application of the decomposition method, J math anal appl, 113, 202-209, (1986) · Zbl 0606.35037
[25] Lesnic, D.; Elliott, L., The decomposition approach to inverse heat conduction, J math anal appl, 232, 82-98, (1999) · Zbl 0922.35189
[26] Beck, J.V.; Blackwell, B.; St-Clair, C.R., Inverse heat conduction, (1985), Wiley New York · Zbl 0633.73120
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