TWOSIDED APPROXIMATIONS METHOD BASED ON THE GREEN’S FUNCTIONS USE FOR CONSTRUCTION OF A POSITIVE SOLUTION OF THE DIRICHLE PROBLEM FOR A SEMILINEAR ELLIPTIC EQUATION
DOI:
https://doi.org/10.15588/16073274202133Keywords:
dirichlet problem for a semilinear elliptic equation, positive solution, strongly invariant conic segment, heterotone operator, method of twosided approximations, Green’s function.Abstract
Context. The question of constructing a method of twosided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a secondorder semilinear elliptic equation.
Objective. The purpose of the research is to develop a method of twosided approximations for solving the Dirichlet problem for secondorder semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems.
Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semiordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and twosided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice.
Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations.
Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of twosided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semidiscrete methods (for example, the Rothe’s method of lines).
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