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Sergii Kovalenko

Publications and source records attributed to Sergii Kovalenko.

8 recordsLinked to original sources
arXiv2018

Symmetry reduction and exact solutions of the non-linear Black--Scholes equation

In this paper, we investigate the non-linear Black--Scholes equation: $$u_t+ax^2u_{xx}+bx^3u_{xx}^2+c(xu_x-u)=0,\quad a,b>0,\ c\geq0.$$ and show that the one can be reduced to the equation $$u_t+(u_{xx}+u_x)^2=0$$ by an appropriate point transformation of variables. For the resulting equation, we study the group-theoretic properties, namely, we find the maximal algebra of invariance of its in Lie sense, carry out the symmetry reduction and seek for a number of exact group-invariant solutions of the equation. Using the results obtained, we get a number of exact solutions of the Black--Scholes equation under study and apply the ones to resolving several boundary value problems with appropriate from the economic point of view terminal and boundary conditions.

q-fin.MF↗
arXiv2014

Kinetics of the elementary act of electrochemical reactions at the semiconductor--electrolyte solution interface

In the framework of the quantum-mechanical theory of elementary act of non-adiabatic electrochemical reactions, it is carried out the calculation of the discharge current of ions at the semiconductor--electrolyte solution interface using the model of isotropic spherically symmetric band. It is shown that our results generalize the well-known formulae for the current density obtained by Dogonadze, Kuznetsov, and Chizmadzhev [R.R. Dogonadze, A.M. Kuznetsov, and Yu.A. Chizmadzhev, The kinetics of some heterogeneous reactions at semiconductor--electrolyte interface, Zhur. Fiz. Khim. 38 (1964) 1195--1202]. The average densities of states in the valence band and the conduction band of the semiconductor electrode in the heterogeneous charge transfer are found.

physics.chem-ph↗
arXiv2012

Lie symmetries of nonlinear boundary value problems

Nonlinear boundary value problems (BVPs) by means of the classical Lie symmetry method are studied. A new definition of Lie invariance for BVPs is proposed by the generalization of existing those on much wider class of BVPs. A class of two-dimensional nonlinear boundary value problems, modeling the process of melting and evaporation of metals, is studied in details. Using the definition proposed, all possible Lie symmetries and the relevant reductions (with physical meaning) to BVPs for ordinary differential equations are constructed. An example how to construct exact solution of the problem with correctly-specified coefficients is presented and compared with the results of numerical simulations published earlier.

math-ph↗
arXiv2012

Exact solutions of nonlinear boundary value problems of the Stefan type

The (1+1)-dimensional nonlinear boundary value problem, modeling the process of melting and evaporation of metals, is studied by means of the classical Lie symmetry method. All possible Lie operators of the nonlinear heat equation, which allow us to reduce the problem to the boundary value problem for the system of ordinary differential equations, are found. The forms of heat conductivity coefficients are established when the given problem can be analytically solved in an explicit form.

math-ph↗
arXiv2012

Lie symmetries and reductions of multi-dimensional boundary value problems of the Stefan type

A new definition of Lie invariance for nonlinear multi-dimensional boundary value problems (BVPs) is proposed by the generalization of known definitions to much wider classes of BVPs. The class of (1+3)-dimensional nonlinear BVPs of the Stefan type, modeling the process of melting and evaporation of metals, is studied in detail. Using the definition proposed, the group classification problem for this class of BVPs is solved and some reductions (with physical meaning) to BVPs of lower dimensionality are made. Examples of how to construct exact solutions of the (1+3)-dimensional nonlinear BVP with the correctly-specified coefficients are presented.

math-ph↗
arXiv2012

Group theoretic analysis of a class of boundary value problems for a nonlinear heat equation

A definition of invariance in Lie's sense for a boundary value problem (BVP) with the basic evolution differential equations is proposed. A problem of group classification at a wide class of BVPs parameterized by arbitrary elements is formulated and an algorithm for its solution is also proposed. The group classification of a class of BVPs for an (1+1)--dimensional nonlinear heat equation arising from mathematical modeling of heat conduction in semi-infinite body is carried out. An example of invariant solution of a BVP from the class under study is presented.

math-ph↗