Finite Integration Method Based on a Trapezoid Rule for Solving Inverse Source Problem for the Heat Equation with a Non-Local Boundary Condition
Abstract
This research aims to study an identification of a time-dependent heat source function for an inverse problem with a non-local boundary condition by using a finite integration method based on combining of trapezoid rule and backward differences. This method is a numerical method using an approximation integration technique for solving the -order differential equation. By using the trapezoid rule, the integration matrix obtained by using this method is a lower triangular matrix which is an advantage of applying this method to solve the complicated problem such as inverse problems. Since the inverse problem is ill-posed which causes to the instability solution, i.e. the small perturbations in the input data result in large perturbation in the solution, then the regularization is used to solve the ill-pose problem in order to stabilize the solution. Keywords : heat equation, ill-pose problem, inverse problem, finite integration method, regularizationReferences
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Beck, J.V., Blackwell, B. & St Clair Jr., C. R. (1985). Inverse Heat Conduction. New York: Wiley.
Blackwell, B. (1981). Efficient technique for the numerical solution of the one-dimensional inverse problem of heat conduction. Numerical Heat Transfer, 4, 229–238.
Dehghan, M. (2001). An inverse problem of finding a source parameter in a semilinear parabolic equation. Applied Mathematical Modelling, 25(9), 743-754.
Farcas, A. & Lesnic, D. (2006). The boundary element method for the determination of a heat source dependent on one variable. Journal of Engineering Mathematics, 54(4), 375–388.
Hansen, P. C. (2001). The L-curve and its use in the numerical treatment of inverse problems. In Johnston, P. (Eds), Computational Inverse Problems in Electrocardiology. (pp. 119–142). Southampton: WIT Press.
Hasanov, A. (2007). Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach. Journal of Mathermatical Analysis with Applications, 330(2), 766–779.
Hazanee, A. (2017). Finite integration method for the time-dependent heat source determination of inverse problem. In Proceeding of the 6th Burapha University International Conference 2017 (BUU2017).
(pp. 391–401). Thailand: Chonburi.
Hazanee, A. (2018). A note on finite integration method for solving differential equations. Burapha Science Journal, 23(1), 288–303. (in Thai).
Hazanee, A., Ismailov, M. I., Lesnic, D. & Kerimov, N. B. (2013). An inverse time-dependent source problem for the heat equation. Applied Numerical Mathematics, 69, 13–33.
Hazanee, A. & Lesnic, D. (2014). Determination of a time-dependent coefficient in the bioheat equation. International Journal of Mechanical Sciences, 88, 259–266.
Hazanee, A., Lesnic, D., Ismailov, M. I. & Kerimov, N. B. (2015). An inverse time-dependent source problem for the heat equation with a non-classical boundary condition. Applied Mathematical Modelling, 39(20), 6258–6272.
Hussein, M.S.; Kinash, N.; Lesnic, D. & Ivanchov, M. (2017). Retrieving the time-dependent thermal conductivity of an orthotropic rectangular conductor. Applicable Analysis, 96(15), 2604-2618.
Ismailov, M.I. & Çiçek, M. (2016). Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions. Applied Mathematical Modelling, 40(7-8), 4891-4899.
Jin, B. & Marin, L. (2006). The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. International Journal for Numerical methods in Engineering, 69(8),
1570–1589.
Johansson, B. T. & Lesnic, D. (2007). A variational method for identifying a spacewise dependent heat source. IMA Journal of Applied Mathematics, 72(6), 748–760.
Kabanikhin, S. I. (2008). Definitions and examples of inverse and ill-posed problems. Journal of Inverse and Ill-posed Problems, 16(4), 317–357.
Lesmana, R., Hazanee, A., Phon-On, A. & Saelee, J. (2017). A finite integration method for a time-dependent heat source identification of inverse problem. In Proceeding of the 5th Asian Academic Society International Conference (AASIC). (pp. 444–451). Thailand: Khon Kaen.
Li, M., Chen, C. S., Hon, Y. C. & Wen, P. H. (2015). Finite integration method for solving multi-dimensional partial differential equations. Applied Mathematical Modelling, 39(1), 4979-4994.
Li, M., Tian, Z. L., Hon, Y. C., Chen, C. S. & Wen, P. H. (2016). Improved finite integration method for partial differential equations. Engineering Analysis with Boundary Elements, 64, 230-236.
Morozov, A. (1966). On the solution of functional equations by the method of regularization. Soviet Mathematics Doklady, 7, 414–417.
Petrov, Y. P. & Sizikov, V. S. (2005). Well-posed, Ill-posed and Intermediate Problems. Netherlands: VSP.
Shamsi, M. & Dehghan, M. (2010). Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method. Numerical Methods for Partial Differential Equations, 28(1), 74–93.
Tian, N., Sun, J., Xu, W. & Lai, C. H. (2011). Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization. International Journal of Heat and Mass Transfer, 54(17-18), 4110-4116.
Tikhonov, A.N. & Arsenin, V.Y. (1977). Solution of Ill-posed Problems. Washington, DC: Winston.
Wen, P. H., Hon, Y. C., Li, M. & Korakianitis, T. (2013). Finite integration method for partial differential equations. Applied Mathematical Modelling, 37(24), 10092–10106.
Xiong, X., Yan, Y. & Wang, J. (2011). A direct numerical method for solving inverse heat source problems. Journal of Physics: Conference Series, 290, 012017 (10 pages).
Yan, L., Fu, C. L., & Yang, F. L. (2008). The method of fundamental solutions for the inverse heat source problem. Engineering Analysis with Boundary Elements, 32(3), 216–222.
Yang, C.Y. (1999). The determination of two heat sources in an inverse heat conduction problem. International Journal of Heat and Mass Transfer, 42(2), 345-356.
Yang, F. & Fu, C. L. (2010). A simplified Tikhonov regularization method for determining the heat source. Applied Mathematical Modelling, 34(11), 3286–3299.
Beck, J.V., Blackwell, B. & St Clair Jr., C. R. (1985). Inverse Heat Conduction. New York: Wiley.
Blackwell, B. (1981). Efficient technique for the numerical solution of the one-dimensional inverse problem of heat conduction. Numerical Heat Transfer, 4, 229–238.
Dehghan, M. (2001). An inverse problem of finding a source parameter in a semilinear parabolic equation. Applied Mathematical Modelling, 25(9), 743-754.
Farcas, A. & Lesnic, D. (2006). The boundary element method for the determination of a heat source dependent on one variable. Journal of Engineering Mathematics, 54(4), 375–388.
Hansen, P. C. (2001). The L-curve and its use in the numerical treatment of inverse problems. In Johnston, P. (Eds), Computational Inverse Problems in Electrocardiology. (pp. 119–142). Southampton: WIT Press.
Hasanov, A. (2007). Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach. Journal of Mathermatical Analysis with Applications, 330(2), 766–779.
Hazanee, A. (2017). Finite integration method for the time-dependent heat source determination of inverse problem. In Proceeding of the 6th Burapha University International Conference 2017 (BUU2017).
(pp. 391–401). Thailand: Chonburi.
Hazanee, A. (2018). A note on finite integration method for solving differential equations. Burapha Science Journal, 23(1), 288–303. (in Thai).
Hazanee, A., Ismailov, M. I., Lesnic, D. & Kerimov, N. B. (2013). An inverse time-dependent source problem for the heat equation. Applied Numerical Mathematics, 69, 13–33.
Hazanee, A. & Lesnic, D. (2014). Determination of a time-dependent coefficient in the bioheat equation. International Journal of Mechanical Sciences, 88, 259–266.
Hazanee, A., Lesnic, D., Ismailov, M. I. & Kerimov, N. B. (2015). An inverse time-dependent source problem for the heat equation with a non-classical boundary condition. Applied Mathematical Modelling, 39(20), 6258–6272.
Hussein, M.S.; Kinash, N.; Lesnic, D. & Ivanchov, M. (2017). Retrieving the time-dependent thermal conductivity of an orthotropic rectangular conductor. Applicable Analysis, 96(15), 2604-2618.
Ismailov, M.I. & Çiçek, M. (2016). Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions. Applied Mathematical Modelling, 40(7-8), 4891-4899.
Jin, B. & Marin, L. (2006). The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction. International Journal for Numerical methods in Engineering, 69(8),
1570–1589.
Johansson, B. T. & Lesnic, D. (2007). A variational method for identifying a spacewise dependent heat source. IMA Journal of Applied Mathematics, 72(6), 748–760.
Kabanikhin, S. I. (2008). Definitions and examples of inverse and ill-posed problems. Journal of Inverse and Ill-posed Problems, 16(4), 317–357.
Lesmana, R., Hazanee, A., Phon-On, A. & Saelee, J. (2017). A finite integration method for a time-dependent heat source identification of inverse problem. In Proceeding of the 5th Asian Academic Society International Conference (AASIC). (pp. 444–451). Thailand: Khon Kaen.
Li, M., Chen, C. S., Hon, Y. C. & Wen, P. H. (2015). Finite integration method for solving multi-dimensional partial differential equations. Applied Mathematical Modelling, 39(1), 4979-4994.
Li, M., Tian, Z. L., Hon, Y. C., Chen, C. S. & Wen, P. H. (2016). Improved finite integration method for partial differential equations. Engineering Analysis with Boundary Elements, 64, 230-236.
Morozov, A. (1966). On the solution of functional equations by the method of regularization. Soviet Mathematics Doklady, 7, 414–417.
Petrov, Y. P. & Sizikov, V. S. (2005). Well-posed, Ill-posed and Intermediate Problems. Netherlands: VSP.
Shamsi, M. & Dehghan, M. (2010). Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method. Numerical Methods for Partial Differential Equations, 28(1), 74–93.
Tian, N., Sun, J., Xu, W. & Lai, C. H. (2011). Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization. International Journal of Heat and Mass Transfer, 54(17-18), 4110-4116.
Tikhonov, A.N. & Arsenin, V.Y. (1977). Solution of Ill-posed Problems. Washington, DC: Winston.
Wen, P. H., Hon, Y. C., Li, M. & Korakianitis, T. (2013). Finite integration method for partial differential equations. Applied Mathematical Modelling, 37(24), 10092–10106.
Xiong, X., Yan, Y. & Wang, J. (2011). A direct numerical method for solving inverse heat source problems. Journal of Physics: Conference Series, 290, 012017 (10 pages).
Yan, L., Fu, C. L., & Yang, F. L. (2008). The method of fundamental solutions for the inverse heat source problem. Engineering Analysis with Boundary Elements, 32(3), 216–222.
Yang, C.Y. (1999). The determination of two heat sources in an inverse heat conduction problem. International Journal of Heat and Mass Transfer, 42(2), 345-356.
Yang, F. & Fu, C. L. (2010). A simplified Tikhonov regularization method for determining the heat source. Applied Mathematical Modelling, 34(11), 3286–3299.
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2018-09-17
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