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RESOLUTION OF AN INVERSE PARABOLIC PROBLEM USING SINC-GALERKIN METHOD

Year 2013, Volume 3, Issue 2, 160 - 181, 01.12.2013

Abstract

In this paper, a numerical method is proposed to solve an Inverse Heat Conduction Problem IHCP using noisy data based on Sinc-Galerkin method. A stable numerical solution is determined for the problem. To do this, we use a sensor located at a point inside the body and measure u x, t at a point x = a, where 0 < a < 1. We also show that the rate of convergence of the method is as exponential. The numerical results show the efficiency of our approach to estimate the unknown functions of the inverse problem. The function can be computed within a couple of minutes CPU time at pentium IV-2.7 GHz PC.

References

  • [1] M. Abtahi, R. Pourgholi, A. Shidfar, Existence and uniqueness of solution for a two dimensional nonlinear inverse diffusion problem , Nonlinear Analysis: Theory, Methods & Applications, 74 (2011) 2462–2467
  • [2] O. M. Alifanov, Inverse Heat Transfer Problems, Springer, NewYork, 1994.
  • [3] P. Amore, A variational Sinc collocation method for strong-coupling problems, J. Phys. A: Math. Gen.39 (22) (2006) L349–L355.
  • [4] J. V. Beck, B. Blackwell, C.R.St. Clair, Inverse Heat Conduction: IllPosed Problems, WileyInterscience, NewYork, 1985.
  • [5] J.V. Beck, B. Blackwell, A. Haji-sheikh, Comparison of some inverse heat conduction methods using experimental data, Internat. J. Heat Mass Transfer 3 (1996) 3649–3657.
  • [6] J. V. Beck, Murio D. C., Combined function specification-regularization procedure for solution of inverse heat condition problem, AIAA J. 24 (1986) 180–185.
  • [7] J.M.G. Cabeza, J.A.M Garcia, A.C. Rodriguez, A Sequential Algorithm of Inverse Heat Conduction Problems Using Singular Value Decomposition, International Journal of Thermal Sciences 44 (2005) 235–244.
  • [8] J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math 21 (1963) 155–160.
  • [9] J.R. Cannon, S.P. Eteva, J. Van de Hoek, A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal. 24 (1987) 499–515.
  • [10] J.R. Cannon, J. Van de Hoek, The one phase stefan problem subject to energy, J. Math. Anal. Appl. 86 (1982) 281–292.
  • [11] V. Capasso, K. Kunisch, A reaction-diffusion system arising in modeling manevironment diseases, Quart. Appl. Math 46 (1988) 431–450.
  • [12] M. Dehghan, An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model. 25 (2001) 743–754.
  • [13] M. Dehghan, Numerical techniques for a parabolic equation subject to an overspecified boundary condition, Appl. Math. Comput. 132 (2002) 299-313.
  • [14] M. Dehghan, Numerical solution of one-dimensional parabolic inverse problem, Appl. Math. Comput. 136 (2003) 333–344.
  • [15] L. Elden, A Note on the Computation of the Generalized Cross-validation Function for Ill-conditioned Least Squares Problems, BIT, 24 (1984) 467–472.
  • [16] G. H. Golub, M. Heath, G.Wahba, Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter, Technometrics, 21 (2) (1979) 215–223.
  • [17] S. Koonprasert, K. Bowers, The Fully Sinc-Galerkin Method for Time-Dependent Boundary Conditions, Numerical Method for Partial Differential Equations, 20 (4) (2004) 494–526.
  • [18] P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev, 34 (1992) 561–80.
  • [19] C.-H. Huang, Y.-L. Tsai, A transient 3-D inverse problem in imaging the time- dependentlocal heat transfer coefficients for plate fin, Applied Thermal Engineering 25 (2005) 2478–2495.
  • [20] C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Philadelphia, PA: SIAM, (1995).
  • [21] J. Lund, K. Bowers, Sinc Methods for Quadrature and Differential Equations, Siam, Philadelphia, PA, 1992.
  • [22] J. Lund, C. Vogel, A Fully-Galerkin method for the solution of an inverse problem in a parabolic partial differential equation, Inverse Prole. 6 (1990) 205–217.
  • [23] L. Martin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, Wen X., Dual Reciprocity Boundary Element Method Solution of the Cauchy Problem for Helmholtz-type Equations with Variable Coefficients, Journal of sound and vibration, 297 (2006) 89–105.
  • [24] H. Molhem, R. Pourgholi, A numerical algorithm for solving a one-dimensional inverse heat conduction problem, Journal of Mathematics and Statistics 4 (1) (2008) 60–63.
  • [25] D. A. Murio, The Mollification Method and the Numerical Solution of Ill-Posed Problems, WileyInterscience, NewYork, 1993.
  • [26] D. A. Murio, J. R. Paloschi, Combined mollification-future temperature procedure for solution of inverse heat conduction problem, J. comput. Appl. Math., 23 (1988) 235–244.
  • [27] R. Pourgholi, N. Azizi, Y.S. Gasimov, F. Aliev, H.K. Khalafi, Removal of Numerical Instability in the Solution of an Inverse Heat Conduction Problem, Communications in Nonlinear Science and Numerical Simulation, 14 (6) (2009) 2664–2669.
  • [28] R. Pourgholi, M. Rostamian, A numerical technique for solving IHCPs using Tikhonov regularization method, Applied Mathematical Modelling, 34 (8) (2010) 2102–2110.
  • [29] R. Pourgholi, M. Rostamian, M. Emamjome, A numerical method for solving a nonlinear inverse parabolic problem, Inverse Problems in Science and Engineering, 18 (8) (2010), 1151–1164.
  • [30] Reza Pourgholi, Mortaza Abtahi, S. Hashem Tabasi, A Numerical Solution Of An Inverse Parabolic Problem, TWMS J. App. & Eng. Math., 2 (2) (2012), 195–209.
  • [31] A. Shidfar, R. Pourgholi, M. Ebrahimi, A numerical method for solving of a nonlinear inverse diffusion problem, Comput. Math. Appl. 52 (2006) 1021–1030.
  • [32] A. Shidfar, R. Zolfaghari, J. Damirchi, Application of Sinc-collocation method for solving an inverse problem,Journal of computational and Applied Mathematics, 233 (2009) 545–554.
  • [33] R. Smith, K. Bowers, A Sinc-Galerkin estimation of diffusivity in parabolic problems, Inverse Problem 9 (1993).
  • [34] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York,1993.
  • [35] F. Stenger, A Sinc-Galerkin method of solution of boundary-value problems, Math. Comp. 33 (1979) 85-109.
  • [36] K.K. Sun, B.S Jung and W.l. Lee, An inverse estimation of surface temperature using the maximum entropy method , International Communication of Heat and Mass Transfer, 34 (2007) 37–44.
  • [37] M. Tadi, Inverse Heat Conduction Based on Boundary Measurement, Inverse Problems, 13 (1997) 1585–1605.
  • [38] M. Tatari, M. Dehghan, Identifying a control function in parabolic partial differential equations from overspecified boundary data, Computers and Mathematics with Applications, 53 (2007) 1933–1942.
  • [39] A.N. Tikhonov, V.Y. Arsenin, Solution of Ill-Posed Problems, V. H. Winston and Sons, Washington, DC, 1977.
  • [40] A.N. Tikhonov, V.Y. Arsenin, On the solution of ill-posed problems, New York, Wiley, 1977.
  • [41] G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59, SIAM, Philadelphia, 1990.
  • [42] E.T. Whittaker, On the functions which are represented by the expansions of the interpolation theory, Proc. Roy. Soc. Edinburg, 35 (1915) 181–194.
  • [43] J.M. Whittaker, Interpolation Function Theory, in: Cambridge Tracts in Mathematics and Mathematical Physics, vol. 33, Cambridge University Press, London, 1935.

Year 2013, Volume 3, Issue 2, 160 - 181, 01.12.2013

Abstract

References

  • [1] M. Abtahi, R. Pourgholi, A. Shidfar, Existence and uniqueness of solution for a two dimensional nonlinear inverse diffusion problem , Nonlinear Analysis: Theory, Methods & Applications, 74 (2011) 2462–2467
  • [2] O. M. Alifanov, Inverse Heat Transfer Problems, Springer, NewYork, 1994.
  • [3] P. Amore, A variational Sinc collocation method for strong-coupling problems, J. Phys. A: Math. Gen.39 (22) (2006) L349–L355.
  • [4] J. V. Beck, B. Blackwell, C.R.St. Clair, Inverse Heat Conduction: IllPosed Problems, WileyInterscience, NewYork, 1985.
  • [5] J.V. Beck, B. Blackwell, A. Haji-sheikh, Comparison of some inverse heat conduction methods using experimental data, Internat. J. Heat Mass Transfer 3 (1996) 3649–3657.
  • [6] J. V. Beck, Murio D. C., Combined function specification-regularization procedure for solution of inverse heat condition problem, AIAA J. 24 (1986) 180–185.
  • [7] J.M.G. Cabeza, J.A.M Garcia, A.C. Rodriguez, A Sequential Algorithm of Inverse Heat Conduction Problems Using Singular Value Decomposition, International Journal of Thermal Sciences 44 (2005) 235–244.
  • [8] J.R. Cannon, The solution of the heat equation subject to the specification of energy, Quart. Appl. Math 21 (1963) 155–160.
  • [9] J.R. Cannon, S.P. Eteva, J. Van de Hoek, A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. Numer. Anal. 24 (1987) 499–515.
  • [10] J.R. Cannon, J. Van de Hoek, The one phase stefan problem subject to energy, J. Math. Anal. Appl. 86 (1982) 281–292.
  • [11] V. Capasso, K. Kunisch, A reaction-diffusion system arising in modeling manevironment diseases, Quart. Appl. Math 46 (1988) 431–450.
  • [12] M. Dehghan, An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl. Math. Model. 25 (2001) 743–754.
  • [13] M. Dehghan, Numerical techniques for a parabolic equation subject to an overspecified boundary condition, Appl. Math. Comput. 132 (2002) 299-313.
  • [14] M. Dehghan, Numerical solution of one-dimensional parabolic inverse problem, Appl. Math. Comput. 136 (2003) 333–344.
  • [15] L. Elden, A Note on the Computation of the Generalized Cross-validation Function for Ill-conditioned Least Squares Problems, BIT, 24 (1984) 467–472.
  • [16] G. H. Golub, M. Heath, G.Wahba, Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter, Technometrics, 21 (2) (1979) 215–223.
  • [17] S. Koonprasert, K. Bowers, The Fully Sinc-Galerkin Method for Time-Dependent Boundary Conditions, Numerical Method for Partial Differential Equations, 20 (4) (2004) 494–526.
  • [18] P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev, 34 (1992) 561–80.
  • [19] C.-H. Huang, Y.-L. Tsai, A transient 3-D inverse problem in imaging the time- dependentlocal heat transfer coefficients for plate fin, Applied Thermal Engineering 25 (2005) 2478–2495.
  • [20] C. L. Lawson, R. J. Hanson, Solving Least Squares Problems, Philadelphia, PA: SIAM, (1995).
  • [21] J. Lund, K. Bowers, Sinc Methods for Quadrature and Differential Equations, Siam, Philadelphia, PA, 1992.
  • [22] J. Lund, C. Vogel, A Fully-Galerkin method for the solution of an inverse problem in a parabolic partial differential equation, Inverse Prole. 6 (1990) 205–217.
  • [23] L. Martin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, Wen X., Dual Reciprocity Boundary Element Method Solution of the Cauchy Problem for Helmholtz-type Equations with Variable Coefficients, Journal of sound and vibration, 297 (2006) 89–105.
  • [24] H. Molhem, R. Pourgholi, A numerical algorithm for solving a one-dimensional inverse heat conduction problem, Journal of Mathematics and Statistics 4 (1) (2008) 60–63.
  • [25] D. A. Murio, The Mollification Method and the Numerical Solution of Ill-Posed Problems, WileyInterscience, NewYork, 1993.
  • [26] D. A. Murio, J. R. Paloschi, Combined mollification-future temperature procedure for solution of inverse heat conduction problem, J. comput. Appl. Math., 23 (1988) 235–244.
  • [27] R. Pourgholi, N. Azizi, Y.S. Gasimov, F. Aliev, H.K. Khalafi, Removal of Numerical Instability in the Solution of an Inverse Heat Conduction Problem, Communications in Nonlinear Science and Numerical Simulation, 14 (6) (2009) 2664–2669.
  • [28] R. Pourgholi, M. Rostamian, A numerical technique for solving IHCPs using Tikhonov regularization method, Applied Mathematical Modelling, 34 (8) (2010) 2102–2110.
  • [29] R. Pourgholi, M. Rostamian, M. Emamjome, A numerical method for solving a nonlinear inverse parabolic problem, Inverse Problems in Science and Engineering, 18 (8) (2010), 1151–1164.
  • [30] Reza Pourgholi, Mortaza Abtahi, S. Hashem Tabasi, A Numerical Solution Of An Inverse Parabolic Problem, TWMS J. App. & Eng. Math., 2 (2) (2012), 195–209.
  • [31] A. Shidfar, R. Pourgholi, M. Ebrahimi, A numerical method for solving of a nonlinear inverse diffusion problem, Comput. Math. Appl. 52 (2006) 1021–1030.
  • [32] A. Shidfar, R. Zolfaghari, J. Damirchi, Application of Sinc-collocation method for solving an inverse problem,Journal of computational and Applied Mathematics, 233 (2009) 545–554.
  • [33] R. Smith, K. Bowers, A Sinc-Galerkin estimation of diffusivity in parabolic problems, Inverse Problem 9 (1993).
  • [34] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer, New York,1993.
  • [35] F. Stenger, A Sinc-Galerkin method of solution of boundary-value problems, Math. Comp. 33 (1979) 85-109.
  • [36] K.K. Sun, B.S Jung and W.l. Lee, An inverse estimation of surface temperature using the maximum entropy method , International Communication of Heat and Mass Transfer, 34 (2007) 37–44.
  • [37] M. Tadi, Inverse Heat Conduction Based on Boundary Measurement, Inverse Problems, 13 (1997) 1585–1605.
  • [38] M. Tatari, M. Dehghan, Identifying a control function in parabolic partial differential equations from overspecified boundary data, Computers and Mathematics with Applications, 53 (2007) 1933–1942.
  • [39] A.N. Tikhonov, V.Y. Arsenin, Solution of Ill-Posed Problems, V. H. Winston and Sons, Washington, DC, 1977.
  • [40] A.N. Tikhonov, V.Y. Arsenin, On the solution of ill-posed problems, New York, Wiley, 1977.
  • [41] G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 59, SIAM, Philadelphia, 1990.
  • [42] E.T. Whittaker, On the functions which are represented by the expansions of the interpolation theory, Proc. Roy. Soc. Edinburg, 35 (1915) 181–194.
  • [43] J.M. Whittaker, Interpolation Function Theory, in: Cambridge Tracts in Mathematics and Mathematical Physics, vol. 33, Cambridge University Press, London, 1935.

Details

Primary Language English
Journal Section Research Article
Authors

Reza POURGHOLİ This is me
School of Mathematics and Computer Sciences, Damghan University, P.O.Box 36715-364, Damghan, Iran


Ali Abbasi MOLAİ This is me
School of Mathematics and Computer Sciences, Damghan University, P.O.Box 36715-364, Damghan, Iran


Tahereh HOULARİ This is me
School of Mathematics and Computer Sciences, Damghan University, P.O.Box 36715-364, Damghan, Iran

Publication Date December 1, 2013
Published in Issue Year 2013, Volume 3, Issue 2

Cite

Bibtex @ { twmsjaem761407, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2013}, volume = {3}, number = {2}, pages = {160 - 181}, title = {RESOLUTION OF AN INVERSE PARABOLIC PROBLEM USING SINC-GALERKIN METHOD}, key = {cite}, author = {Pourgholi, Reza and Molai, Ali Abbasi and Houlari, Tahereh} }