A mollified solution of a nonlinear inverse heat conduction problem

Authors

Faculty of Mathematics, K. N. Toosi University of Technology, Tehran, Iran

Abstract

In this paper a nonlinear inverse heat conduction problem in one dimensional space is considered. This
inverse problem reformulate as an auxiliary inverse problem. Ill-posedness is identified as one of the main
characteristics of the inverse problems. So, a numerical algorithm based on the combination of discrete
mollification and space marching method is applied to conquer ill-posedness of the auxiliary inverse problem.
Moreover, a proof of stability and convergence of the aforementioned algorithm is provided. Eventually, the
efficiency of this method is illustrated by a numerical example.

Keywords


[1] C. D. Acosta, C. E. Mejia, Stabilization of explicit methods for convection diffusion equations by discrete mollifi-
cation, Comput. Math. Appl., 55 (2008), 368–380. 

[2] J. V. Beck, B. Blackwell, C.R. St Clair Jr., Inverse Heat Conduction: Ill-Posed Problems, John Wiley and Sons,
New York, (1985). 

[3] J. V. Beck, Surface heat flux determination using an integral method, Nucl. Eng. Des., 7 (1968), 170–178.

[4] J. R. Cannon, The One-Dimensional Heat Equation, Addison-Wesley, Menlo Park, CA, (1984). 

[5] J. R. Cannon, P. Duchateau, An inverse problem for a nonlinear diffusion equation, SIAM J. Appl. Math., 39
(2) (1980), 272–289. 

[6] C. Coles, D. A. Murio, Identification of Parameters in the 2-D IHCP, Comput. Math. Appl., 40 (2000) 939–956.

[7] G. A. Doral, D. A. Tortorellit, Transient inverse heat conduction problem solutions via Newton’s method, Int. J.
Heat Mass Transfer., 40 (17) (1997), 4115–4127. 

[8] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J., (1964). 

[9] M. Garshasbi, H. Dastour, Estimation of unknown boundary functions in an inverse heat conduction problem
using a molified marching scheme, Numer. Algorithms., 68 (2015), 769–790. 

[10] J. M. Gutierrez, J. A. Martin, A. Corz, A sequential algorithm of inverse heat conduction problems using singular
value decomposition, Int. J. Therm. Sci., 44 (2005), 235–244. 

[11] P. C. Hansen, The truncated SVD as a method for regularization, BIT Numer. Math., 27 (4) (1987), 534–553. 

[12] P. C. Hansen, Rank-deficient and Discrete Ill-posed Problems: Numerical Aspects of Linear Inversion, SIAM
Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics
(SIAM), Philadelphia, PA, (1998). 

[13] D. N. Hao, A mollification method for ill-posed problems, Numer. Math., 68 (1994), 469–506. 

[14] D. N. Hao, N. T. Thanh, H. Sahli , Splitting-based conjugate gradient method for a multi-dimensional linear
inverse heat conduction problem, J. Comput. Appl. Math., 232 (2009), 361–377. 

[15] C. C. Ji, P. C. Tuan, H. Y. Jang, A recursive least-squares algorithm for on-line 1-D inverse heat conduction
estimation, Int. J. Heat Mass Transfer., 40 (9) (1997), 2081–2096. 

[16] C. E. Mejia, D. A. Murio, Numerical solution of generalized IHCP by discrete mollification, Comput. Math. Appl.,
32 (2) (1996), 33–50. 

[17] C. E. Mejia, D. A. Murio, Mollified hyperbolic method for coefficient identification problems, Comput. Math.
Appl., 26 (5) (1993), 1–12.

[18] C. E. Mejia, C. D. Acosta, K. I. Saleme, Numerical identification of a nonlinear diffusion coeficient by discrete
mollification, Comput. Math. Appl., 62 (2011), 2187–2199. 

[19] D. A. Murio, Mollification and space marching, In: Woodbury, K (ed.) Inverse Engineering Handbook. CRC
Press (2002). 

[20] M. N. Ozisik, Heat Conduction, Wiley, NewYork, (1980). 

[21] M. Prudhomme, T. H. Nguyen, On the iterative regularization of inverse heat conduction problems by conjugate
gradient method, Int. Comm. Heat Mass Transfer., 25 (7) (1998), 999–1108. 

[22] J. R. Shenefelt, R. Luck, R. P. Taylor, J. T. Berry, Solution to inverse heat conduction problems employing
singular value decomposition and model reduction, Int. J. Heat Mass Transfer., 45 (2002), 67–74. 

[23] A. N. Tikhonov, On the regularization of ill-posed problems, Dokl. Akad. Nauk SSSR, 153 (1963), 49–52. 

[24] A. N. Tikhonov, On the solution of ill-posed problems and the method of regularization, Dokl. Akad. Nauk SSSR,
151 (1963), 501–504. 

[25] A. N. Tikhonov, V. Arsenin, F. John, Solutions of ill-posed problems, Wiley, (1977).