Please cite this article as: N.A. Kudryashov, K.E. Shilnikov, Numerical modeling and optimization of the cryosurgery operations, Journal of Computational and Applied Mathematics (2015), http://dx.Abstract Numerical computation of the three dimensional problem of the freezing interface propagation during the cryosurgical impact coupled with the multi-objective optimization mechanism is used in order to improve the efficiency and safety of the cryosurgery operations performing. The heat transfer in soft tissue during the thermal exposure to low temperature is described by the Pennes bioheat model. The finite volume method combined with the control volume approximation of the heat fluxes is applied for the cryosurgery numerical modeling on the tumor tissue of a quite arbitrary shape. The flux relaxation approach is used for the stability improvement of the explicit finite difference schemes. The generalized method of effective traversals is proposed for the searching of the Pareto front segments as the multiobjective optimization problem solutions.
The recent paper ''The tanh-coth method combined with the Riccati equation for solving nonlinear coupled equation in mathematical physics'' (J. King Saud Univ. Sci. 23 (2011) 127-132) is analyzed. We show that the authors of this paper solved equations with the well known solutions. One of these equations is the famous Riccati equation and the other equation is one for the Weierstrass elliptic function. We present the general solutions of these equations. As this takes place, 19 solutions by authors do not satisfy the equation but the other 29 solutions can be obtained from the general solutions.ª 2012 King Saud University. Production and hosting by Elsevier B.V. All rights reserved.
For the numerical solution of two-dimensional (2D) nonlinear heat conduction problem a transition to a moving coordinate system is used. A local speed of the latter is chosen in such a way that the considered process is close to stationary. Such coordinate system guarantees a concentration of grid nodes in a range of the solution features. This allows for a quality improvement of the obtained numerical solution with a small number of computational grid nodes. The developed numerical algorithm is tested on the heat wave propagation problem which has an exact solution.
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