| Summary: | 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.
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