Identification of the Solely Time-Dependent Zero-Order Coefficient in a Linear Bi-Flux Diffusion Equation from an Integral Measurement

Identification of the Solely Time-Dependent Zero-Order Coefficient in a Linear Bi-Flux Diffusion Equation from an Integral Measurement

Bi-flux diffusion equation, can be easily affected by the existence of external factors, is known as an anomalous diffusion. In this paper, the inverse problem (IP) of determining the solely time-dependent zero-order coefficient in a linear Bi-flux diffusion equation with initial and homogeneous boundary conditions from an integral additional specification of the energy is considered. The unique solvability of the inverse problem is demonstrated by using the contraction principle for sufficiently small times.

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  • [1] S. Guenneau, D. Petiteau, M. Zerrad, C. Amra, T. Puvirajesinghe, Transformed Fourier and Fick equations for the control of heat and mass diffusion, AIP Adv., 5(5) (2015), 053404.
  • [2] D. Shi, S. Zhang, Unconditional superconvergence of the fully-discrete schemes for nonlinear prey-predator model, Appl. Numer. Math., 172 (2022), 118-132.
  • [3] D. Shi, C. Li, Superconvergence analysis of two-grid methods for bacteria equations, Numer. Algorithms, 86(1) (2021), 123-152.
  • [4] L. Bevilacqua, M. Jiang, A. Silva Neto, A.C.R. Gale˜ao, An evolutionary model of bi-flux diffusion processes, J. Braz. Soc. Mech. Sci. Eng., 38(5) (2016), 1421-1432.
  • [5] M. Jiang, L. Bevilacqua, A.J.S. Neto, A.C.R. Gale˜ao, J. Zhu, Bi-flux theory applied to the dispersion of particles in anisotropic substratum, Appl. Math. Model., 64 (2018), 121-134.
  • [6] M. Jiang, L. Bevilacqua, J. Zhu, X. Yu, Nonlinear Galerkin finite element methods for fourth-order Bi-flux diffusion model with nonlinear reaction term, Comput. Appl. Math., 39(3) (2020), 1-16.
  • [7] D. Shi, S. Zhang, Unconditional superconvergence analysis for the nonlinear Bi-flux diffusion equation, Appl. Math. Comput., 442 (2023), 127771.
  • [8] S. Guenneau, C. Amra, D. Veynante, Transformation thermodynamics: cloaking and concentrating heat flux, Opt. Express, 20(7) (2012), 8207-8218.
  • [9] E.P. Scott, P.S. Robinson, T.E. Diller, Development of methodologies for the estimation of blood perfusion using a minimally invasive thermal probe, Meas. Sci. Technol., 9(6) (1998), 888.
  • [10] L. Marin, L. Elliott, P.J. Heggs, D.B. Ingham, D. Lesnic, X. Wen, Analysis of polygonal fins using the boundary element method, Appl. Therm. Eng., 24(8-9) (2004), 1321-1339.
  • [11] E.A. Sudicky, The Laplace transform Galerkin technique: A time-continuous finite element theory and application to mass transport in groundwater, Water Resour. Res., 25(8) (1989), 1833-1846.
  • [12] M.I. Ismailov, Direct and inverse problems for thermal grooving by surface diffusion with time dependent mullins coefficient, Math. Model. Anal., 26(1) (2021), 135-147.
  • [13] J.R. Cannon, Determination of an unknown heat source from overspecified boundary data, SIAM J. Numer. Anal., 5(2) (1968), 275–286.
  • [14] A. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: weak solution approach, J. Math. Anal. Appl., 330 (2007), 766-779
  • [15] A. Hazanee, D. Lesnic, M.I. Ismailov, N.B. Kerimov, An inverse time-dependent source problem for the heat equation with a non-classical boundary condition, Appl. Math. Model., 39(20) (2015), 6258-6272.
  • [16] M.I. Ismailov, F. Kanca, D. Lesnic, Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions, Appl. Math. Comput., 218(8) (2011), 4138-4146.
  • [17] J.R. Cannon, Y. Lin, S. Wang, Determination of a control parameter in a parabolic partial differential equation, J. Aust. Math. Soc. Series B, 33 (1991), 149-163.
  • [18] M.I. Ismailov, I. Tekin, S. Erkovan, An inverse problem for finding the lowest term of a heat equation with Wentzell–Neumann boundary condition, Inverse Probl. Sci. Eng., 27(11) (2019), 1608-1634.
  • [19] M.I. Ismailov, F. Kanca, An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions, Math. Methods Appl. Sci., 34(6) (2011), 692-702.
  • [20] N.B. Kerimov, M.I. Ismailov, An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions, J. Math. Anal. Appl., 396(2) (2012), 546-554.
  • [21] L. Zhuo, D. Lesnic, M.I. Ismailov, ˙I. Tekin, S. Meng, Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem, Int. J. Comput. Math., 96(10) (2019), 2079-2099.