Year 2013,
Volume 3,
Issue 1,
10 - 32,
01.06.2013
Reza POURGHOLİ
Saedeh FOADİAN
Amin ESFAHANİ
References
- [1] Abtahi, M., Pourgholi, R. and Shidfar, A., (2011), Existence and uniqueness of solution for a two dimensional nonlinear inverse diffusion problem, Nonlinear Analysis: Theory, Methods & Applications, 74, 2462-2467.
- [2] Alifanov, O. M., (1994), Inverse Heat Transfer Problems, Springer, NewYork.
- [3] Beck, J. V., Blackwell, B. and Clair, C. R. St., (1985), Inverse Heat Conduction: IllPosed Problems, Wiley-Interscience, NewYork.
- [4] Beck, J. V., Blackwell, B. and Haji-sheikh, A., (1996), Comparison of some inverse heat conduction methods using experimental data, Internat. J. Heat Mass Transfer, 3, 3649-3657.
- [5] Beck, J. V. and Murio, D. C., (1986), Combined function specification-regularization procedure for solution of inverse heat condition problem, AIAA J., 24, 180-185.
- [6] Cabeza, J. M. G, Garcia, J. A. M and Rodriguez, A. C., (2005), A Sequential Algorithm of Inverse Heat Conduction Problems Using Singular Value Decomposition, International Journal of Thermal Sciences, 44, 235-244.
- [7] Chen, C. F. and Hsiao, C. H., (1997), Haar wavelet method for solving lumped and distributedparameter systems, IEE Proc.: part D, 144 (1), 87-94.
- [8] Chen, H. T. and Lin, J. Y., (1993, 1994), Analysis of two-dimensional hyperbolic heat conduction problems, lnt, J. Heat Mass Transfer, 37 (1), 153164.
- [9] Ching-yu, Y., (2009), Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems, Appl. Math. Model., 33, 2907-2918.
- [10] Elden, L., (1984), A Note on the Computation of the Generalized Cross-validation Function for Illconditioned Least Squares Problems, BIT, 24, 467-472.
- [11] Golub, G. H., Heath, M. and Wahba, G., (1979), Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter, Technometrics, 21 (2), 215-223.
- [12] Haar, A., (1910), Zur theorie der orthogonalen Funktionsysteme, Math. Annal., 69, 331-371.
- [13] Hansen, P. C., (1992), Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev, 34, 561-580.
- [14] Hariharan, G., Kannan, K. and Sharma, K. R., (2009), Haar wavelet method for solving Fisher’s equation, Applied Mathematics and Computation, 211, 284-292.
- [15] Hsiao, C. H. and Wang, W. J., (2001), Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simul., 57, 347-353.
- [16] Huang, C.-H. and Tsai, Y.-L., (2005), A transient 3-D inverse problem in imaging the time- dependentlocal heat transfer coefficients for plate fin, Applied Therma Engineering, 25, 2478-2495.
- [17] Huang, C.-H., Yeha, C.-Y. and Orlande, H. R. B., (2003), A nonlinear inverse problem in simultaneously estimating the heat and mass production rates for a chemically reacting fluid, Chemical Engineering Science, 58 (16), 3741-3752.
- [18] Kalpana, R. and Raja Balachandar, S., (2007), Haar wavelet method for the analysis of transistor circuits, Int. J.Electron. Commun. (AEU), 61, 589-594.
- [19] Lawson, C. L. and Hanson, R. J., (1995), Solving Least Squares Problems, Philadelphia, PA: SIAM.
- [20] Martin, L., Elliott, L., Heggs, P. J., Ingham, D. B., Lesnic, D. and Wen, X., (2006), Dual Reciprocity Boundary Element Method Solution of the Cauchy Problem for Helmholtz-type Equations with Variable Coefficients, Journal of sound and vibration, 297, 89-105.
- [21] Molhem, H. and Pourgholi, R., (2008), A numerical algorithm for solving a one-dimensional inverse heat conduction problem, Journal of Mathematics and Statistics, 4 (1), 60-63.
- [22] Murio, D. A., (1993), The Mollification Method and the Numerical Solution of Ill-Posed Problems, Wiley-Interscience, New York.
- [23] Murio, D. C. and Paloschi, J. R., (1988), Combined mollification-future temperature procedure for solution of inverse heat conduction problem, J. Comput. Appl. Math., 23, 235-244.
- [24] Pourgholi, R., Azizi, N., Gasimov, Y. S., Aliev, F. and Khalafi, H. K., (2009), Removal of Numerical Instability in the Solution of an Inverse Heat Conduction Problem, Communications in Nonlinear Science and Numerical Simulation, 14 (6), 2664-2669.
- [25] Pourgholi, R. and Rostamian, M., (2010), A numerical technique for solving IHCPs using Tikhonov regularization method, Applied Mathematical Modelling, 34 (8), 2102-2110.
- [26] Pourgholi, R., Rostamian, M. and Emamjome, M., (2010), A numerical method for solving a nonlinear inverse parabolic problem, Inverse Problems in Science and Engineering, 18 (8), 1151-1164.
- [27] Sun, K. K., Jung, B. S. and Lee, W. l., (2007), An inverse estimation of surface temperature using the maximum entropy method, International Communication of Heat and Mass Transfer, 34, 37-44.
- [28] Su, J. and Silva Neto, A. J., (2001), Two-dimensional inverse heat conduction problems of source strength estimation in cylindrical rods, Applied Mathematical Modelling, 25, 861-872.
- [29] Tadi, M., (1997), Inverse Heat Conduction Based on Boundary Measurement, Inverse Problems, 13, 1585-1605.
- [30] Tikhonov, A. N. and Arsenin, V. Y., (1977), On the solution of ill-posed problems, New York, Wiley.
- [31] Tikhonov, A. N. and Arsenin, V. Y., (1977), Solution of Ill-Posed Problems, V. H. Winston and Sons, Washington, DC.
- [32] Zhou, J., Zhang, Y., Chen, J. K. and Feng, Z. C., (2010), Inverse Heat Conduction in a Composite Slab With Pyrolysis Effect and Temperature-Dependent Thermophysical Properties, J. Heat Transfer, 132 (3), 034502 (3 pages).
HAAR BASIS METHOD TO SOLVE SOME INVERSE PROBLEMS FOR TWO-DIMENSIONAL PARABOLIC AND HYPERBOLIC EQUATIONS
Year 2013,
Volume 3,
Issue 1,
10 - 32,
01.06.2013
Reza POURGHOLİ
Saedeh FOADİAN
Amin ESFAHANİ
Abstract
A numerical method consists of combining Haar basis method and Tikhonov regularization method. We apply the method to solve some inverse problems for twodimensional parabolic and hyperbolic equations using noisy data. In this paper, a stable numerical solution of these problems is presented. This method uses a sensor located at a point inside the body and measures the u x, y, t at a point x = a, 0 < a < 1. We also show that the rate of convergence of the method is as exponential. Numerical results show that a good estimation on the unknown functions of the inverse problems can be obtained within a couple of minutes CPU time at Pentium IV-2.53 GHz PC.
References
- [1] Abtahi, M., Pourgholi, R. and Shidfar, A., (2011), Existence and uniqueness of solution for a two dimensional nonlinear inverse diffusion problem, Nonlinear Analysis: Theory, Methods & Applications, 74, 2462-2467.
- [2] Alifanov, O. M., (1994), Inverse Heat Transfer Problems, Springer, NewYork.
- [3] Beck, J. V., Blackwell, B. and Clair, C. R. St., (1985), Inverse Heat Conduction: IllPosed Problems, Wiley-Interscience, NewYork.
- [4] Beck, J. V., Blackwell, B. and Haji-sheikh, A., (1996), Comparison of some inverse heat conduction methods using experimental data, Internat. J. Heat Mass Transfer, 3, 3649-3657.
- [5] Beck, J. V. and Murio, D. C., (1986), Combined function specification-regularization procedure for solution of inverse heat condition problem, AIAA J., 24, 180-185.
- [6] Cabeza, J. M. G, Garcia, J. A. M and Rodriguez, A. C., (2005), A Sequential Algorithm of Inverse Heat Conduction Problems Using Singular Value Decomposition, International Journal of Thermal Sciences, 44, 235-244.
- [7] Chen, C. F. and Hsiao, C. H., (1997), Haar wavelet method for solving lumped and distributedparameter systems, IEE Proc.: part D, 144 (1), 87-94.
- [8] Chen, H. T. and Lin, J. Y., (1993, 1994), Analysis of two-dimensional hyperbolic heat conduction problems, lnt, J. Heat Mass Transfer, 37 (1), 153164.
- [9] Ching-yu, Y., (2009), Direct and inverse solutions of the two-dimensional hyperbolic heat conduction problems, Appl. Math. Model., 33, 2907-2918.
- [10] Elden, L., (1984), A Note on the Computation of the Generalized Cross-validation Function for Illconditioned Least Squares Problems, BIT, 24, 467-472.
- [11] Golub, G. H., Heath, M. and Wahba, G., (1979), Generalized Cross-validation as a Method for Choosing a Good Ridge Parameter, Technometrics, 21 (2), 215-223.
- [12] Haar, A., (1910), Zur theorie der orthogonalen Funktionsysteme, Math. Annal., 69, 331-371.
- [13] Hansen, P. C., (1992), Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev, 34, 561-580.
- [14] Hariharan, G., Kannan, K. and Sharma, K. R., (2009), Haar wavelet method for solving Fisher’s equation, Applied Mathematics and Computation, 211, 284-292.
- [15] Hsiao, C. H. and Wang, W. J., (2001), Haar wavelet approach to nonlinear stiff systems, Math. Comput. Simul., 57, 347-353.
- [16] Huang, C.-H. and Tsai, Y.-L., (2005), A transient 3-D inverse problem in imaging the time- dependentlocal heat transfer coefficients for plate fin, Applied Therma Engineering, 25, 2478-2495.
- [17] Huang, C.-H., Yeha, C.-Y. and Orlande, H. R. B., (2003), A nonlinear inverse problem in simultaneously estimating the heat and mass production rates for a chemically reacting fluid, Chemical Engineering Science, 58 (16), 3741-3752.
- [18] Kalpana, R. and Raja Balachandar, S., (2007), Haar wavelet method for the analysis of transistor circuits, Int. J.Electron. Commun. (AEU), 61, 589-594.
- [19] Lawson, C. L. and Hanson, R. J., (1995), Solving Least Squares Problems, Philadelphia, PA: SIAM.
- [20] Martin, L., Elliott, L., Heggs, P. J., Ingham, D. B., Lesnic, D. and Wen, X., (2006), Dual Reciprocity Boundary Element Method Solution of the Cauchy Problem for Helmholtz-type Equations with Variable Coefficients, Journal of sound and vibration, 297, 89-105.
- [21] Molhem, H. and Pourgholi, R., (2008), A numerical algorithm for solving a one-dimensional inverse heat conduction problem, Journal of Mathematics and Statistics, 4 (1), 60-63.
- [22] Murio, D. A., (1993), The Mollification Method and the Numerical Solution of Ill-Posed Problems, Wiley-Interscience, New York.
- [23] Murio, D. C. and Paloschi, J. R., (1988), Combined mollification-future temperature procedure for solution of inverse heat conduction problem, J. Comput. Appl. Math., 23, 235-244.
- [24] Pourgholi, R., Azizi, N., Gasimov, Y. S., Aliev, F. and Khalafi, H. K., (2009), Removal of Numerical Instability in the Solution of an Inverse Heat Conduction Problem, Communications in Nonlinear Science and Numerical Simulation, 14 (6), 2664-2669.
- [25] Pourgholi, R. and Rostamian, M., (2010), A numerical technique for solving IHCPs using Tikhonov regularization method, Applied Mathematical Modelling, 34 (8), 2102-2110.
- [26] Pourgholi, R., Rostamian, M. and Emamjome, M., (2010), A numerical method for solving a nonlinear inverse parabolic problem, Inverse Problems in Science and Engineering, 18 (8), 1151-1164.
- [27] Sun, K. K., Jung, B. S. and Lee, W. l., (2007), An inverse estimation of surface temperature using the maximum entropy method, International Communication of Heat and Mass Transfer, 34, 37-44.
- [28] Su, J. and Silva Neto, A. J., (2001), Two-dimensional inverse heat conduction problems of source strength estimation in cylindrical rods, Applied Mathematical Modelling, 25, 861-872.
- [29] Tadi, M., (1997), Inverse Heat Conduction Based on Boundary Measurement, Inverse Problems, 13, 1585-1605.
- [30] Tikhonov, A. N. and Arsenin, V. Y., (1977), On the solution of ill-posed problems, New York, Wiley.
- [31] Tikhonov, A. N. and Arsenin, V. Y., (1977), Solution of Ill-Posed Problems, V. H. Winston and Sons, Washington, DC.
- [32] Zhou, J., Zhang, Y., Chen, J. K. and Feng, Z. C., (2010), Inverse Heat Conduction in a Composite Slab With Pyrolysis Effect and Temperature-Dependent Thermophysical Properties, J. Heat Transfer, 132 (3), 034502 (3 pages).