The Journal of the Indian Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

Heat Potentials Method in the Treatment of One-Dimensional Free Boundry Problems Applied in Cryomedicine


Affiliations
1 FSBEI of Higher Education, "Kabardino Balkarian State University named after H.M. Berbekov", Institute of Physics and Mathematics, Russian Federation
2 FSBEI of Higher Education "Kabardino Balkarian State University named after H.M. Berbekov", Institute of Physics and Mathematics, Russian Federation
     

   Subscribe/Renew Journal


Free boundary problems are considered to be the most difcult and the least researched in the eld of mathematical physics. The present article is concerned with the research of the following issue: treatment of one-dimensional free boundary problems. The treated problem contains a nonlinear evolutionary equation, which occurs within the context of mathematical modeling of cryosurgery problems. In the course of the research, an integral expression has been obtained. The obtained integral expression presents a general solution to the non-homogeneous evolutionary equation which contains the functions that represent simple-layer and double-layer heat potential density. In order to determine the free boundary and the density of potential a system of nonlinear, the second kind of Fredholm integral equations was obtained within the framework of the given work. The treated problem has been reduced to the system of integral equations. In order to reduce the problem to the integral equation system, a method of heat potentials has been used. In the obtained system of integral equations instead of K(ξ; x; τ - t) in case of Dirichlet or Neumann conditions the corresponding Greens functions G(ξ; x; τ - t) or N(ξ; x; τ - t) have been applied. Herewith the integral expression contains fewer densities, but the selection of arbitrary functions is reserved. The article contains a number of results in terms of building a mathematical model of cooling and freezing processes of biological tissue, as well as their effective solution development.

Keywords

Free Boundary, Evolutionary Equation, Heat Potential, Mathematical Modeling, Continuously Differentiable Function, Integral Equation, Phase Transition, Applicable Surface, Temperature Pattern, Greens Function, Tissue Destruction, Stationary Problem.
Subscription Login to verify subscription
User
Notifications
Font Size


  • Alperovich B.I., Merzlikin N.V., Komkova T.B. [et al.], Cryosurgical Operations in Hepatic and Pancreatic Disorders [under the editorship of Professor B.I. Alperovich]. -M.: GEOTAR-Media, 2015. - 240 p.

  • Vasiliev F.P., The Method of Straight Lines for the Solution of a One-phase Stefan-type Problem // Journal of Computational Mathematics and Mathematical Physics. - M.: Science, 2015. - V. 8. -1. - P. 64-78.

  • Ignatiev A.O., Period Boundaries in Solving Ordinary Dierential Equations // Ukr.Mat. Journal - 2015. { V. 67. - 11. - P. 1569-1573 [Electronic resource]. - URL: http://umj.imath.kiev.ua/volumes/issues/?lang=ua&year=2015&number=11

  • S.A. Vasiliyev, S.B. Pesnya-Prasolov, M.N. Aslanukov, R.S. Levin, Intraoperative Sonographic Imaging of Cryodestruction of Brain Tumors.. Materials of the 1st All-Russian Conference with International Participation "Cryosurgery and New Technologies in Medicine" (Saint-Petersburg, 15.05.2015). - P. 1 - 2.

  • S.A.Vasiliyev, S.B. Pesnya-Prasolov, S.V. Kungurtsev, V.N. Pavlov, Cryodestruction of Brain Tumors (cryogenic equipment and methods).// Clinical and Experimental Surgery: Journal named after the Member of the Academy of Science B.V. Petrovskiy - 2015. - 1. - P. 15 - 21.

  • Kudayeva F.H., Kaigermazov A.A., Two-dimensional Free Boundary Problems // Southern-Siberian Scientic Bulletin: Scientic and Technical Journal - 2014. 3(7), December - P. 16-19.

  • Kudayeva F.H., Kaigermazov A.A., Nakhusheva F.M., Dolova M.H., Mambetov M.J., Two-dimensional Free Boundary Problems // Modern Problems of Science and Education: Scientic journal ISSN 2070-7428. - 2015. - 2 [Electronic resource]. - URL: www.science-education.ru/129-22182 (accessed date: 14.03.2017).

  • Kudayeva F.H., Kaigermazov A.A., Karmokov M.M., Nakhusheva F.M., Mathematical Model of Flat Chyodestruction of Biological Tissue // Modern Problems of Science and Education: Scientic journal ISSN 2070-7428. - 2015. - 2 [Electronic resource]. URL: https://www.science-education.ru/ru/article/view?id=21683 (accessed date: 11.03.2017).

  • Kudayeva F.H., Kaigermazov A.A., Karmokov M.M., Mambetov M.J., Dolova M.H., Mathematical Model of Flat Chyodestruction of Biological Tissue // Modern Problems of Science and Education: Scientic journal ISSN 2070-7428. - 2015. - 2 [Electronic resource]. - URL: https://science-education.ru/ru/article/view?id=22558 (accessed date: 1.03.2017).

  • Kudayeva F.H., Kaigermazov A.A., Mathematical Model of Spherically Symmetrical Hypothermia of Biological Tissue // Modern Problems of Science and Education: Scientic journal ISSN 2070-7428. - 2015. - 2 [Electronic resource] - URL: www.scienceeducation.ru/129-22001 (accessed date: 10.03.2017).

  • Formalev V.F., Rabinskiy L.N., Stefan-type Problem with Two Non-stationary Moving Boundaries of Phase Transitions // Proceedings of the Russian Academy of Sciences. Power Engineering. -.: Science, 2014. - 4. - P. 74-81.

  • Chernikova A.S., Heat Distribution on a Plane which Consists of Two Dierent Nonhomogeneous Materials with a Semi-bounded Interphase Crack // Saint-Petersburg University Bulletin: Ser. 10. Applied Mathematics, Informatics. Administrative Process. - 2014. - Issue 3. -P. 66{81.

  • Johansson B.T., Chapko R., An Alternating Boundary Integral Based Method for a Cauchy Problem for the Laplace Equation in Semi-innite Regions // Ukr. math. Journal - 2016. - V. 68. - 12. - P. 1665-1682 [Electronic resource]. - URL: http://umj.imath.kiev.ua/volumes/issues/?lang=ua&year=2016&number=12

  • Osipchuk M.M., Portenko M.I., On Simple Layer Potentials for One Class of Pseudodierential Equations // Ukr. math. journal - 2015. - V. 67. - 11. { P. 1512-1524 [Electronic resource]. - URL: http://umj.imath.kiev.ua/volumes/issues/?lang=ua&year=2015&number=11

  • Alphonse A., Elliott C.M., A Stefan problem on an evolving surface [Electronic resource].- URL: www.ncbi.nlm.nih.gov/pmc/articles/PMC4535267 (accessed date: 14.03.2017).

  • Antontsev S., Meirmanov A., Yurinsky B.V., Free A., Boundary Problem for Stokes Equations: Classical Solutions. Interfaces and Free Boundaries. - 2 (2000). - pp. 413- 424.

  • Balasubramanian S.K., Wolkers W.F., Bischof J.C., Membrane Hydration Correlates to Cellular Biophysics During Freezing in Mammalian Cells // Biochem. Biophys. Acta. - 2009. - 1788. - . 945{953.

  • Briozzo A.C., Natale M.F., One-Dimensional Nonlinear Stefan Problems in Storms Materials // Mathematics. - 2014. - 2(1). - 1-11; doi:10.3390/math2010001.

  • S.V. Osipov, V.V. Khovrin, T.N. Galyan, S.A. Vasiliev, S.B. Pesnya-Prasolov, Capabilities of MRI in Assessment of Results of Focal Cryodestruction of the Mammalian Brain (experimental work) : Conference materials, 37 European Society of Neuroradiology (Frankfurt, Germany, 28.09.-01.10.2013). - P. 63.

  • S. Pesnya-Prasolov, S. Vasiliev, V. Krylov, S. Kungurtcev, M. Aslanucov, A. Vyatkin, ryodestruction of Intracerebral Brain Tumors : conference materials, EANS Annual Meeting (Tel Aviv, Israel, 11-14.11.2013). - P. 104-105.

  • S. Vasiliev, S. Pesnya-Prasolov, V. Krylov, S. Kungurtsev, Cryosurgery in Neuroepithelial Brain Tumors : Conference materials, XV WFNS World Congress of Neurosurgery (Seoul, Korea, 08-13.09.2013). FA1184.

  • S. Vasiliev, V. Krylov, S. Pesnya-Prasolov, A. Zuev, A. Vyatkin, T.Galyan, S.Kungurcev, V.Pavlov, ryodestruction of Brain Tumors : Materials of the 39th Annual Meeting of Japan Society for Low Temperature Medicine "Cryomedicine 2012" (Japan, Tokyo, 2122.11.2012). - P. 43-44.

  • S. Vasiliev, V. Sandrikov, V. Krylov, S. Pesnya-Prasolov, A. Zuev, V. Hovrin, S. Kungurtsev, V. Pavlov, Cryosurgery of Brain Tumors by Means of Ultrasound Neuronavigation : materials of the Second International Applied Science Conference "Cryosurgery. Modern Methods and Innovative Technologies" (Saint-Petersburg, 29.06.2012). - P. 11-12.

  • S. Vasiliev, S. Pesnya-Prasolov, V. Krylov, R. Levin, M. Aslanucov, A. Vyatkin, S. Osipov, S. Kungurtcev, Cryosurgery of Brain Tumors by Means of Ultrasound Neuronavigation : materials of the 17th World Congress of the International Society of Cryosurgery (Bali, Indonesia, 11-13.12.2013). - P. 145.

  • S. Vasiliev, S. Pesnya-Prasolov, V. Krylov, A. Zuev, S. Kungurtsev, Cryosurgery of Brain Tumors by Means of Ultrasound Neuronavigation Control : materials of the conference "New Horizons for Cryomedicine", ACCryo 2013 (USA, Miami, 02-07.01.2013). - P. 11-12.

  • Etheridge M.L., Choi J., Ramadhyani S., Bischof J.C., Methods for Characterizing Convective Cryoprobe Heat Transfer in Ultrasound Gel Phantoms [Electronic resource]. - URL: www.ncbi.nlm.nih.gov/pmc/articles/PMC4031449 (accessed date: 13.03.2017).

  • Goncharova O.N., Rezanova E.V., Construction of a Mathematical Model of Flows in a Thin Liquid Layer on the Basis of the Classical Convection Equations and Generalized Conditions on an Interface // Proceedings of Altai State University. - 2015. - 1-1 (85). - P.3-5.

  • udaeva F.H., igermazov .., betov .J., Zweidimensionale eine planparallele problem kryochirurgie // European Applied Sciences. - 2014. - 10.- . 28-30.

  • S. Pesnya-Prasolov, S. Vasiliev, A. Zuev, V. Krylov, V. Pavlov, I. Zhidkov, D. Fedorov, T. Galyan, Local Cryodestruction of Mammalian Brain : materials of the 14-th European Congress of Neurosurgery (Rome, Italy, 9=14.10.2011). - P. 540.

  • Meirmanov A., The Free Boundary Muskat Problem: Some Methods of Mathematical Modeling in Porous Media // Submitted to Mathematical Models and Methods in Applied Sciences. - 2012. - Vol. 24. - 4. - P. 59.


Abstract Views: 328

PDF Views: 0




  • Heat Potentials Method in the Treatment of One-Dimensional Free Boundry Problems Applied in Cryomedicine

Abstract Views: 328  |  PDF Views: 0

Authors

Fatimat K. Kudayeva
FSBEI of Higher Education, "Kabardino Balkarian State University named after H.M. Berbekov", Institute of Physics and Mathematics, Russian Federation
Arslan A. Kaigermazov
FSBEI of Higher Education "Kabardino Balkarian State University named after H.M. Berbekov", Institute of Physics and Mathematics, Russian Federation
Elizaveta K. Edgulova
FSBEI of Higher Education "Kabardino Balkarian State University named after H.M. Berbekov", Institute of Physics and Mathematics, Russian Federation
Mariya M. Tkhabisimova
FSBEI of Higher Education "Kabardino Balkarian State University named after H.M. Berbekov", Institute of Physics and Mathematics, Russian Federation
Aminat R. Bechelova
FSBEI of Higher Education "Kabardino Balkarian State University named after H.M. Berbekov", Institute of Physics and Mathematics, Russian Federation

Abstract


Free boundary problems are considered to be the most difcult and the least researched in the eld of mathematical physics. The present article is concerned with the research of the following issue: treatment of one-dimensional free boundary problems. The treated problem contains a nonlinear evolutionary equation, which occurs within the context of mathematical modeling of cryosurgery problems. In the course of the research, an integral expression has been obtained. The obtained integral expression presents a general solution to the non-homogeneous evolutionary equation which contains the functions that represent simple-layer and double-layer heat potential density. In order to determine the free boundary and the density of potential a system of nonlinear, the second kind of Fredholm integral equations was obtained within the framework of the given work. The treated problem has been reduced to the system of integral equations. In order to reduce the problem to the integral equation system, a method of heat potentials has been used. In the obtained system of integral equations instead of K(ξ; x; τ - t) in case of Dirichlet or Neumann conditions the corresponding Greens functions G(ξ; x; τ - t) or N(ξ; x; τ - t) have been applied. Herewith the integral expression contains fewer densities, but the selection of arbitrary functions is reserved. The article contains a number of results in terms of building a mathematical model of cooling and freezing processes of biological tissue, as well as their effective solution development.

Keywords


Free Boundary, Evolutionary Equation, Heat Potential, Mathematical Modeling, Continuously Differentiable Function, Integral Equation, Phase Transition, Applicable Surface, Temperature Pattern, Greens Function, Tissue Destruction, Stationary Problem.

References