Exploring Solutions of Geometry Problems for Inverse Cauchy Problems in Helmholtz and Modified Helmholtz Equations
DOI:
https://doi.org/10.24237/Keywords:
: Inverse Cauchy problem, Helmholtz equation, modified Helmholtz equation, polynomial expansion, Newton methodAbstract
Abstract
In the present paper explores a reverse Cauchy problem for a heat transfer issue described by the Helmholtz and modified Helmholtz equation. Our goal is to identify an unknown defect within a simply connected bounded domain , given the Dirichlet data (temperature) on the boundary , and Neumann data (heat flux) on the boundary . We assume that the temperature satisfies the Helmholtz equation (or modified Helmholtz equation) that governs the heat condition in the fin. To solve this problem, we propose a method that involves two steps. First, we solve a Cauchy problem using the Helmholtz equation (or modified Helmholtz equation) to determine the temperature Then, in the second phase, we solve a system of nonlinear scalar equations to determine the coordinates of the points defining the boundary . This can be achieved using an iterative method, such as Newton's method.
References
[1] F. Aboud, I. T. Jameel, A. F. Hasan, B. K. Mostafa, A. Nachaoui, Polynomial approximation of an inverse Cauchy problem for Helmholtz type equations, Adv. Math. Models Appl., 7(3), 306–322(2022)
[2] F. Aboud, A. Nachaoui, M. Nachaoui, On the approximation of a Cauchy problem in a non- homogeneous medium, J. Phys.: Conf. Ser., 1743(1), 012003(2021), DOI(10.1088/1742-6596/1743/1/012003)
[3] F. Aboud, S. Ibrahim, Numerical Method for Solving an inverse Cauchy problem for Helmholtz equation using CGLS, BICGSTAB and PCG, Academic Science Journal, 2(3), 232-253(2024), DOI(https://doi.org/10.24237/ASJ.02.03.776B)
[4] F. Aboud, L. Qasim, Resolving an Inverse Cauchy Problem of Mixed Boundary Value of Bi-harmonic using a meshless collocation method, Academic Science Journal, 2(3), 273-289(2024), DOI(https://doi.org/10.24237/ASJ.02.03.772B)
[5] K. A. Berdawood, A. Nachaoui, R. Saeed, M. Nachaoui, F. Aboud, An alternating procedure with dynamic relaxation for Cauchy problems governed by the modified Helmholtz equation, Adv. Math. Models Appl., 5(1), 131–139(2020)
[6] K. Berdawood, A. Nachaoui, M. Nachaoui, An accelerated alternating iterative algorithm for data completion problems connected with Helmholtz equation, Stat. Optim. Inf. Comput., 11(1), 2–21 (2023), DOI(https://doi.org/10.1002/num.22793)
[7] K. Berdawood, A. Nachaoui, R. Saeed, M. Nachaoui, F. Aboud, An efficient D-N alternating algorithm for solving an inverse problem for Helmholtz equation, Discrete Contin. Dyn. Syst. Ser. S, (2022), DOI(https://doi.org/10.3934/dcdss.2021013)
[8] F. Berntsson, V. A. Kozlov, L. Mpinganzima, B. O. Turesson, Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation, Comput. Math. Appl., 73, 163–172(2017), DOI(https://doi.org/10.1016/j.camwa.2016.11.004)
[9] B. Bin-Mohsin, The method of fundamental solutions for Helmholtz-type problems, Ph.D. thesis at the University of Leeds,(2013)
[10] K. Boumzough, A. Azzouzi, A. Bouidi, The incomplete LU preconditioner using both CSR and CSC formats, Adv. Math. Models Appl., 7(2), 156–167(2022)
[11] A. F. Hasan, D. Jasim, I. Th. Jameel ,B. Kh. Mostafa, F. M. Aboud, Polynomial Approximation of an Inverse Cauchy Problem for Modified Helmholtz Equations, Academic Science Journal, 1(2), 153-170(2023), DOI(https://doi.org/10.24237/ASJ.01.02.650D)
[12] C. H. Huang, W. C. Chen, A three-dimensional inverse forced convection problem in estimating surface heat flux by conjugate gradient method, Int. J. Heat Mass Transf, 43(17), 3171-3181(2000), DOI(https://doi.org/10.1016/S0017-9310(99)00330-0)
[13] V. Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences, vol. 127 (Springer, Cham, 2017).
[14] I. Th. JAMEEL, A. F. Hasan, D. Jasim, F. M. Aboud, Solving Inverse Cauchy Problems of the Helmholtz Equation Using Conjugate Gradient Based-Method: BICG, BICGSTAB And PCG, Academic Science Journal, 1(4), 13-31 (2023), DOI(https://doi.org/10.24237/ASJ.01.04.652D)
[15] S. I. Kabanikhin, Inverse and Ill-posed Problems (Walter de Gruyter GmbH & Co. KG, Berlin, 2012)
[16] V. A. Kozlov, V. G. Maz’ya, A.V. Fomin, An iterative method for solving the Cauchy problem for elliptic equations. Zh. Vychisl. Mat. I Mat. Fiz. 31, 64–74(1991)
[17] C. S. Liu, C. L. Kuo, A multiple-scale pascal polynomial triangle solving elliptic equations and inverse Cauchy problems, Eng. Anal. Bound. Elem., 62, 35–43(2016), DOI(https://doi.org/10.1016/j.enganabound.2015.09.003)
[18] L. Marin, B. T. Johansson, A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity, Comput. Methods Appl. Mech. Eng., 199(49– 52), 3179–3196(2010), DOI(https://doi.org/10.1016/j.cma.2010.06.024)
[19] B. Kh. Mostafa, D. Jasim, A. F. Hasan, I. Th. Jameel , F. M. Aboud, Solving Bi-harmonic Cauchy problem using a meshless collocation method, Academic Science Journal, 2(1), 398-416(2024), DOI( https://doi.org/10.24237/ASJ.02.01.694E)
[20] A. Nachaoui, F. Aboud, The International Conference on New Trends in Applied Mathematics, New trends of mathematical inverse problems and applications, In Solving geometric inverse problems with a polynomial based meshless method (pp. 119-Springer International Publishing), essay Laghrib Amine, (2023), DOI(https://doi.org/10.1007/978-3-031-33069-8_8)
[21] A. Nachaoui, Cauchy’s problem for the modified biharmonic equation: ill-posedness and iterative regularizing methods, in New Trends of Mathematical Inverse Problems and Applications, Springer Proceedings in Mathematics & Statistics (Springer, Cham, 2023)
[22] A. Nachaoui, F. Aboud, M. Nachaoui, Acceleration of the KMF algorithm convergence to solve the Cauchy problem for Poisson’s equation, in Mathematical Control and Numerical Applications, Springer Proceedings in Mathematics & Statistics, vol. 372, ed. by A. Nachaoui, A. Hakim, A. Laghrib, (Springer, Cham, 2021), 43–57
[23] A. L. Qian, X. T. Xiong, Y. J. Wu, On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation, J. Comput. Appl. Math., 233(8), 1969–1979(2010), DOI(https://doi.org/10.1016/j.cam.2009.09.031)
[24] S. M. Rasheed, A. Nachaoui, M. F. Hama, A. K. Jabbar, Regularized and preconditioned con479 jugate gradient like-methods methods for polynomial approximation of an inverse Cauchy problem, Adv. Math. Models Appl., 6(2), 89–105(2021)
[25] F. Yang, P. Zhang, X. X. Li, The truncation method for the Cauchy problem of the inhomogeneous Helmholtz equation, Appl. Anal., 98, 991–1004(2019), DOI(https://doi.org/10.1080/00036811.2017.1408080)
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