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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
     

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

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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