Kyungpook Mathematical Journal

  • 제64권2호
  • /
  • Pages.353-369
  • /
  • 2024
  • /
  • 1225-6951(pISSN)
  • /
  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
권호 표지
DOI QR코드

DOI QR Code

Determination of Unknown Time-Dependent Heat Source in Inverse Problems under Nonlocal Boundary Conditions by Finite Integration Method

  • Areena Hazanee (Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus) ;
  • Nifatamah Makaje (Department of Mathematics and Computer Science, Faculty of Science and Technology, Prince of Songkla University, Pattani Campus)
  • 투고 : 2022.08.15
  • 심사 : 2024.01.02
  • 발행 : 2024.06.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this study, we investigate the unknown time-dependent heat source function in inverse problems. We consider three general nonlocal conditions; two classical boundary conditions and one nonlocal over-determination, condition, these genereate six different cases. The finite integration method (FIM), based on numerical integration, has been adapted to solve PDEs, and we use it to discretize the spatial domain; we use backward differences for the time variable. Since the inverse problem is ill-posed with instability, we apply regularization to reduce the instability. We use the first-order Tikhonov's regularization together with the minimization process to solve the inverse source problem. Test examples in all six cases are presented in order to illustrate the accuracy and stability of the numerical solutions.

키워드

과제정보

This research was fully financially supported by the Research Career Development Grant 2017 under the Office of Research Affairs, Faculty of Science and Technology, Prince of Songkla University, Thailand. Authors would like to thank Assoc. Prof. Dr. Seppo Karrila for valuable comments/suggestions on this paper.

참고문헌

  1. B. Bin-Mohsin and D. Lesnic, Determination of inner boundaries in modified Helmholtz inverse geometric problems using the method of fundamental solutions, Math. Comput. Simul., 82(8)(2012), 1445-1458.  https://doi.org/10.1016/j.matcom.2012.02.002
  2. K. V. Bockstal and M. Slodicka, Recovery of a time-dependent heat source in one-dimensional thermoelasticity of type-III, Inverse Probl. Sci. Eng., 25(5)(2017), 749-770.  https://doi.org/10.1080/17415977.2016.1199696
  3. H. Egger, and B. Hofmann, Tikhonov regularization in Hilbert scales under conditional stability assumptions, Inverse Probl.,34(11)(2018), 115015. 
  4. A. Hazanee, Finite integration method for the time-dependent heat source determination of inverse problem, Proc 6th BUU Conf. (2017), 391-401. 
  5. A. Hazanee and D. Lesnic, Determination of a time-dependent heat source from nonlocal boundary conditions, Eng. Anal. Bound. Elem., 37(2013), 936-956.  https://doi.org/10.1016/j.enganabound.2013.03.003
  6. M. S. Hussein, N. Kinash, D. Lesnic and M. Ivanchov, Retrieving the timedependent thermal conductivity of an orthotropic rectangular conductor, Appl. Anal., 96(15)(2017), 2604-2618.  https://doi.org/10.1080/00036811.2016.1232401
  7. M. I. Ismailov, F. Kanca and D. Lesnic, Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions, Appl. Math. Comput., 218(8)(2011), 4138-4146.  https://doi.org/10.1016/j.amc.2011.09.044
  8. N. I. Ivanchov, On the determination of unknown source in the heat equation with nonlocal boundary conditions, Ukr. Math. J., 47(10)(1995), 1647-1652.  https://doi.org/10.1007/BF01060166
  9. T. Kang, K. V. Bockstal and R. Wang, The reconstruction of a time-dependent source from a surface measurement for full Maxwell's equations by means of the potential field method, Comput. Math. Appl., 75(3)(2018), 764-786.  https://doi.org/10.1016/j.camwa.2017.10.004
  10. A. Karageorghis, D. Lesnic and L. Marin, A survey of applications of the MFS to inverse problems, Inverse Probl. Sci. Eng., 19(3)(2011), 309-336.  https://doi.org/10.1080/17415977.2011.551830
  11. R. Lesmana, A. Hazanee, A. Phon-On, J. Saelee, A finite integration method for a time-dependent heat source identification of inverse problem, Proc 5th AASIC. (2017), 444-451. 
  12. M. Li, C. S. Chen, Y. C. Hon and P. H. Wen, Finite integration method for solving multi-dimensional partial differential equations, Appl. Math. Model., 39(17)(2015), 4979-4994.  https://doi.org/10.1016/j.apm.2015.03.049
  13. M. Li, Z.L. Tian, Y.C. Hon, C.S. Chen and P.H. Wen, Improved finite integration method for partial differential equations, Eng. Anal. Bound. Elem., 64(2016), 230-236.  https://doi.org/10.1016/j.enganabound.2015.12.012
  14. C. Shi, C. Wang and T.Wei, Numerical solution for an inverse heat source problem by an iterative method, Appl. Math. Comput., 244(2014), 577-597. 
  15. M. Slodicka, Determination of a solely time-dependent source in a semilinear parabolic problem by means of boundary measurements, J. Comput. Appl. Math., 289(2015), 433-440.  https://doi.org/10.1016/j.cam.2014.10.004
  16. V. V. Ternovskii, M. M. Khapaev and T. M. Khapaeva, Application of the Variational Method for Solving Inverse Problems of Optimal Control, Dokl. Math., 98(2018), 603-606.  https://doi.org/10.1134/S1064562418070219
  17. N. Tian, J. Sun, W. Xu and C. H. Lai, Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization, Int. J. Heat Mass Transf., 54(2011), 4110-4116.  https://doi.org/10.1016/j.ijheatmasstransfer.2011.03.061
  18. D. Trucu, D. B. Ingham and D. Lesnic, Reconstruction of the space and timedependent blood perfusion coefficient in bio-heat transfer, Heat Transf. Eng., 32(9)(2011), 800-810.  https://doi.org/10.1080/01457632.2011.525430
  19. S. Twomey, On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature, J. ACM., 10(1)(1963), 97-101.  https://doi.org/10.1145/321150.321157
  20. P. N. Vabishchevich, Iterative computational identification of a spacewise dependent the source in a parabolic equations, Inverse Probl. Sci. Eng., 25(8)(2017), 1168-1190.  https://doi.org/10.1080/17415977.2016.1230611
  21. V. I. Vasil'ev and A. M. Kardashevsky, Iterative solution of the retrospective inverse problem for a parabolic equation using the conjugate gradient method, Numerical Analysis and Its Applications, LNCS, I. Dimov, I. Farago and L. Vulkov, (eds), Springer, Cham, 2017, 2698-705,
  22. V. I. Vasil'ev, V. V. Popov and A. M. Kardashevsky, Conjugate Gradient Method for Identification of a Spacewise Heat Source, Large-Scale Scientific Computing. LSSC. Lecture Notes in Computer Science. I. Lirkov, and S. Margenov, (eds), Springer, Cham, 2017, 600-607.
  23. P.H. Wen, Y.C. Hon, M. Li, T. Korakianitis, Finite integration method for partial differential equations, Applied Mathematical Modelling, Appl Math Model. 37(24)(2013), 10092-10106. https://doi.org/10.1016/j.apm.2013.05.054
  24. M. Yazdanparast, B. Voosoghi and F. Mossaiby, The Method of Fundamental Solutions for Solving the Inverse Problem of Magma Source Characterization, Geomat. Nat. Hazards Risk., 10(1)(2019), 797-819. https://doi.org/10.1080/19475705.2018.1547324
  25. D. F. Yun, Z. H. Wen and Y. C. Hon, Adaptive least squares finite integration method for higher-dimensional singular perturbation problems with multiple boundary layers, Appl. Math. Comput., 271(2015), 232-250.  https://doi.org/10.1016/j.amc.2015.08.116