625
Views
1
CrossRef citations to date
0
Altmetric
Research Article

Time-dependent lowest term estimation in a 2D bioheat transfer problem with nonlocal and convective boundary conditions

, &
Pages 1282-1307 | Received 07 Jan 2020, Accepted 28 Oct 2020, Published online: 11 Nov 2020

References

  • Bedin L, Bazán FSV. On the 2D bioheat equation with convective boundary conditions and its numerical realization via a highly accurate approach. Appl Math Comp. 2014;236:422–436.
  • Fan J, Wang L. Analytical theory of bioheat transport. J Appl Phys. 2011;109:104702.
  • Pennes HH. Analysis of tissue and arterial blood temperatures in the resting human forearm. J Appl Physiol. 1948;1:93–122.
  • Lakhssassi A, Kengne E, Semmaoui H. Modified Pennes' equation modelling bioheat transfer in living tissues: analytical and numerical analysis. Nat Sci. 2010;2:1375–1385.
  • Cao L, Qin QH, Zhao N. An RBMFS model for analysing thermal behavior of skin tissues. Int J Heat Mass Transfer. 2010;53:1298–1307.
  • Azizbavov EI. The nonlocal inverse problem of the identification of the lowest coefficient and the right-hand side in a second-order parabolic equation with integral conditions. Bound Value Probl. 2019;2019:1–19.
  • Ionkin NI. Solution of a boundary-value problem in heat conduction with a non-classical boundary condition. Differ Equ. 1977;13:294–304.
  • Hazanee A, Lesnic D. Determination of a time-dependent coefficient in the bioheat equation. Int J Mech Sci. 2014;88:259–266.
  • Hochmuth R. Homogenization for a non-local coupling model. Appl Anal. 2008;87:1311–1323.
  • Ionkin NI, Morozova VA. The two-dimensional heat equation with nonlocal boundary conditions. Differ Equ. 2000;36:982–987.
  • Ismailov MI, Erkovana S. Inverse problem of finding the coefficient of the lowest term in two-dimensional heat equation with Ionkin-type boundary condition. Comput Math Math Phys. 2019;59:791–808.
  • Ciesielski M, Duda M, Mochnacki B. Comparison of bioheat transfer numerical models based on the Pennes and Cattaneo–Vernotte equations. J Appl Math Comput Mech. 2016;15:33–38.
  • Grysa K, Maciag A. Trefftz method in solving the Pennes' and single-phase-lag heat conduction problems with perfusion in the skin. Int J Numer Methods Heat Fluid Flow. 2019;30:3231–3246.
  • Grysa K, Maciag A. Identifying heat source intensity in treatment of cancerous tumor using therapy based on local hyperthermia – The Trefftz method approachs. J Therm Biol. 2019;84:16–25.
  • Iljaz J, Skerget L. Blood perfusion estimation in heterogeneous tissue using BEM based algorithm. Eng Anal Bound Elem. 2014;39:75–87.
  • Majchrzak E, Turchan L. The general boundary element method for 3D dual-phase lag model of bioheat transfer. Eng Anal Bound Elem. 2015;50:76–82.
  • Ismailov MI, Bazán FSV, Bedin L. Time-dependent perfusion coefficient estimation in a bioheat transfer problem. Comput Phys Commun. 2018;230:50–58.
  • Grabski JK, Lesnic D, Johansson BT. Identification of a time-dependent bioheat blood perfusion coefficient. Int Commun Heat Mass Transfer. 2016;75:18–22.
  • Lesnic D, Ivanchov MI. Determination of the time-dependent perfusion coefficient in the bioheat equation. Appl Math Lett. 2015;39:96–100.
  • Lin Y. An inverse problem for a class of quasi-linear parabolic equations. SIAM J Math Anal. 1991;22:146–156.
  • Prilepko AI, Solov'ev VV. On the solvability of inverse boundary value problems for the determination of the coefficient preceding the lower derivative in a parabolic equation. Differ Equ. 1987;23:136–143.
  • Yousefi SA. Finding a control parameter in a one-dimensional parabolic inverse problem by using the Bernstein Galerkin method. Inverse Probl Sci Eng. 2009;17:821–828.
  • Ismailov MI, Kanca F. An inverse coefficient problem for a parabolic equation in the case of nonlocal boundary and overdetermination conditions. Math Methods Appl Sci. 2011;34:692–702.
  • Ivanchov MI, Pabyrivska NV. Simultaneous determination of two coefficients of a parabolic equation in the case of nonlocal and integral conditions. Ukr Math J. 2001;53:674–684.
  • Kerimov NB, Ismailov MI. Direct and inverse problems for the heat equation with a dynamic-type boundary condition. IMA J Appl Math. 2015;80:1519–1533.
  • Kerimov NB, Ismailov MI. An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions. J Math Anal Appl. 2012;396:546–554.
  • Cannon JR, Lin Y, Wang S. Determination of source parameter in parabolic equation. Meccanica. 1992;27:85–94.
  • Daoud DS, Subasi D. A splitting up algorithm for the determination of the control parameter in multi-dimensional parabolic problem. Appl Math Comput. 2005;166:584–595.
  • Dehghan M. Numerical methods for two-dimensional parabolic inverse problem with energy overspecification cation. Int J Comput Math. 2000;77:441–455.
  • Pyatkov SG. Solvability of some inverse problems for parabolic equations. J Inv Ill-Posed Probl. 2004;12:397–412.
  • Bedin L, Bazán FSV. A note on existence and uniqueness of solutions for a 2D bioheat problem. Appl Anal. 2017;96:590–605.
  • Bazán FSV, Bedin L, Borges LS. Space-dependent perfusion coefficient estimation in a 2D bioheat transfer problem. Comput Phys Commun. 2017;214:18–30.
  • Fornberg B. A practical guide to pseudospectral methods. Cambridge: Cambridge University Press; 1996.
  • Trefethen LN. Spectral methods in matlab. Philadelphia (PA): SIAM; 2000.
  • Levenberg K.A. Method for the solution of certain non-linear problems in least squares. Quart Appl Math. 1944;2:164–168.
  • Marquardt DW. An algorithm for least squares estimation of nonlinear parameters. J Soc Ind Appl Math. 1963;11:431–441.
  • Naimark MA. Linear differential operators: elementary theory of linear differential operators. New York (NY): Frederick Ungar Publishing Co.; 1967.
  • Lo CY. Boundary value problems. Singapore: World Scientfic; 2000.
  • Yu. Kapustin N, Moiseev EI. A spectral problem for the Laplace operator in the square with a spectral parameter in the boundary condition. Differ Equ. 1998;34:663–668.
  • Beck JV, Arnold KJ. Parameter estimation in engineering and science. New York (NY): Wiley; 1977.
  • Dennis J, Schnabel R. Numerical methods for unconstrained optimization and nonlinear equations. Englewood Cliffs (NJ): Prentice Hall; 1983.
  • Yamashita N, Fukushima M. On the rate of convergence of the Levenberg–Marquardt method. Comput (Suppl). 2001;15:239–249.
  • Trefethen LN. Is gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 2008;50:67–87.
  • Golub HH, Ortega JM. Scientific computing and differential equations-an introduction to numerical methods. San Diego (CA): Academic Press; 1992.
  • Pao CV. Nonlinear parabolic and elliptic equations. New York (NY): Plenum Press; 1992.
  • Yu. Kapustin N. Basis properties of a problem for the Laplace operator on the square with spectral parameter in a boundary condition. Differ Equ. 2017;53:563–565.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.