ON APPLYING THE CONTINUOUS OPERATOR METHOD TO SOLVE THE DIRECT PROBLEM FOR NONLINEAR PARABOLIC EQUATIONS

Abstract

Background. Parabolic differential equations of mathematical physics play very important role in mathematical modeling of the wide range of phenomena in physical and technical sciences. In particular, parabolic equations are widely used for modeling diffusion processes, processes of fluid dynamics as well as biological and ecological phenomena. These equations are also occur in the problems of heat and mass transfer, combustion theory, filter theory etc. Besides, in spite of sufficiently wide amount of known results in the field of approximate solution of parabolic equations, there is important need for developing effective numerical methods for solving nonlinear parabolic equations. Such methods must be quite simple and in the same time be resistant to initial data disturbances as well as be applicable to a wide range of equations. Materials and methods. The main subject of this paper is Cauchy problem for one-dimensional parabolic equations that is nonlinear in unknown function. We consider the problem of constructing the numerical method for solving the mentioned equation. In order to do that we change from the Cauchy problem for parabolic differential equation to a nonlinear integral equation. The integral equation is then solved by means of continuous operator method for nonlinear equations: an auxiliary system of integro-differential equations of special type is constructed and then it solved with one of the numerical methods for solving differential equations. The result of the method is a set of approximate values of unknown function in the nodes of the uniform mesh, which is constructed in a finite domain. Results. A numerical method for solving the Cauchy problem for nonlinear onedimensional parabolic differential equation is proposed in the paper. A high potential of this method is primarily due to its simplicity and also its universality that allows to apply the same algorithm for very wide range of nonlinearities. Conclusions. An effective iterative method for solving the Cauchy problem for nonlinear one-dimensional parabolic differential equation is proposed. Extending the method to boundary problems as well as to multidimensional equations is of considerable theoretical and practical interest.

Keywords

• nonlinear parabolic equations
• cauchy problem
• continuous operator method
• nonlinear integral equations
• logarithmic norm