TWO-SIDED APPROXIMATIONS METHOD BASED ON THE GREEN’S FUNCTIONS USE FOR CONSTRUCTION OF A POSITIVE SOLUTION OF THE DIRICHLE PROBLEM FOR A SEMILINEAR ELLIPTIC EQUATION

Context. The question of constructing a method of two-sided 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 second-order semilinear elliptic equation. Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order 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 semi-ordered 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 two-sided 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 two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).

w * is a boundary of the sequence of upper approximations; Δ is the Laplace operator; 0 κ > is a parameter in the Helmholtz operator 2 u Δ − κ ; θ is zero element of the Banach space;

INTRODUCTION
The problem of mathematical modeling of many stationary processes considered in chemical kinetics, biology, combustion theory, etc. [1][2][3][4], leads to the necessity for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation. Due to this, the problem of developing new and improving existing methods of numerical analysis of this class of problems is relevant.
The object of the study is the Dirichlet problem for a second-order semilinear elliptic equation.
The subject of the research is the method of twosided approximations for solving the boundary value problems for the second-order semilinear elliptic equations.
Currently there are many methods for the numerical analysis of the boundary value problems for semilinear elliptic equations. Among them, one can single out, in particular, the methods of finite differences, finite elements, boundary integral equations, artificial neural network technique [1,[5][6][7][8][9][10][11] or successive approximations with two-sided convergence [12][13][14]. The methods of the last group allow to construct two sequences of functions that approximate the desired solution of the problem from below and from above, respectively. Due to this, when implementing these methods, one has a convenient a posteriori estimate of the approximation error, and, consequently, a convenient criterion for the termination of iterations. This makes the methods of two-sided approximations more attractive in comparison with other methods that are used to solve boundary value problems for semilinear elliptic equations.
The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green's functions method and to study its work in solving test problems.

PROBLEM STATEMENT
The problem of finding a positive solution of a semilinear elliptic equation with the homogeneous Dirichlet condition is considered in the paper: where x x x Let us assume that the function ( , ) f u x is continuous and positive for ∈ Ω x , 0 u > . The operator Δ is the Laplace operator and the operator 2 Δ − κ is the Helmholtz operator. The problem (1)-(3) often appears as a mathematical model of the nonlinear stationary processes considered in thermal physics, electromagnetism, biology, chemical kinetics, etc. [1,2,4]. In this case, the positivity condition (2) naturally arises from the meaning of the function u in a particular field.

REVIEW OF THE LITERATURE
The construction of two-sided approximation methods for solving the boundary value problems for partial differential equations is based on the use of the theory of nonlinear operators in semi-ordered spaces.
The theory of linear semi-ordered spaces was constructed by L. V. Kantorovich in the second half of the 30s of the XX century [15,16]. Further development of the methods of this theory is associated with the works of M. A. Krasnoselsky [17], H. Amann [18], V. I. Opoitsev [19], N. S. Kurpel, B. A. Shuvar [20,21], A. I. Kolosov [22].
In [17,18,23,24], the existence of the positive solutions of the equations with monotone operators was investigated, and in [19,25], the solvability of the equations with operators that have the generalized property of monotonicity (the so-called heterotone or mixed monotone operators) was explored. When proving the corresponding theorems of existence, the sequences of functions, which on both sides converged to the solution of the investigated problem, were constructed. As examples of applications of this theory, the questions of the existence of positive solutions of the boundary value problems for nonlinear ordinary differential equations, boundary value problems for nonlinear partial differential equations and integral equations were considered. In these works, the theoretical foundations for the development of two-sided iterative schemes were laid, but the iterations themselves were considered by the authors as an auxiliary means of proving the existence theorems for fixed points of operators, and there were no computational results.
In [20,21], the equations and inequalities in which the operators do not have the monotonicity property are considered, and for them two-sided monotonic iterative processes are constructed. In [22], it was obtained a generalization of the theory of the heterotone operators, which were applied, in particular, to finding the approximate solutions of the boundary value problems with a free boundary for nonlinear ordinary differential equations.
The works [12][13][14] are devoted to the development of two-sided iterative schemes for solving the boundary value problems for partial differential equations as means of applied mathematics with bringing them to computational implementation. But only problem (1) which has a power-law or exponential monotonic nonlinearity was investigated.
The boundary value problems for the equation (1) in This work continues the studies begun in [12][13][14]26] and is aimed at their generalization and extension to the equation with the Helmholtz operator.

MATERIALS AND METHODS
To study the solvability of the problem (1)-(3) and numerically finding its solution, let us construct a method of two-sided approximations using the methods of the theory of nonlinear operators in semi-ordered spaces [17,19,27].
If ( , ) G x s is the Green's function of the problem (1)-(3), then the problem (1)-(3) is equivalent to the Hammerstein's integral equation Let us consider equation (4) in the Banach space of nonnegative functions. The cone + K in ( ) C Ω is normal (and even sharp). Using the cone + K in the space ( ) C Ω , let us introduce the semi-ordering by the rule: If there exists a classical solution of the problem (1)- Also, the operator T of the form (5) is 0 u -positive operator, where the function 0 ( ) u x belongs to \{ } + θ K , is defined by the equality x xs s (6) and is the solution of the problem The property of 0 u -positivity follows from the fact [17]: if 0 Ω is some subdomain of the domain Ω , moreover, 0 ( ) 0 μ Ω > , then there is such 0 ( ) 0 γ = γ Ω > , that the following inequality holds , which is the definition of the 0 u -positivity of the operator T . A constructive study of the equation (4) (and, consequently, of the problem (1)-(3)) with the opportunity of constructing two-sided approximations to its positive solution is possible if the function ( , ) f u x has the monotonicity property.
Let the function ( , ) f u x allows a diagonal representa- , where the nonnegative function ˆ( , , ) f v w x , continuous in the set of variables x , v , w , monotonically increases with respect to v and monotonically decreases with respect to w for all ∈ Ω x . Then the operator T of the form (5) will be heterotone one with the companion operator Obviously, the operators T and T are completely continuous.
If the function ( , ) f u x monotonically increases with respect to u for all ∈ Ω x , then one can choose ˆ( , , ) ( x and the companion operator is defined by the equality For a function ( , ) f u x monotonically decreasing with respect to u , one can put ˆ( , , ) , and then the companion operator will have the form If for any positive numbers v , w , for any then the heterotone operator T of the form (5) for which the operator T of the form (8) is companion one will be pseudo-concave and, moreover, 0 u -pseudo-concave with the function 0 ( ) u x of the form (6).
Actually, for any , , the inequality (7) implies that Then it follows from the inequality (11) and the condition of continuity of the function ˆ( , , ) f , which means the pseudoconcavity of the operator T . Further, from the inequalities (11), (12) it follows that for all ∈ Ω x Then, applying inequality (12) again, one gets that for , which means 0 u -pseudoconcavity of the operator T . Thus, the following statement holds. Lemma. The operator T of the form (5) f u x allows a diagonal representation where the function ˆ( , , ) f v w x , continuous in the set of variables x , v , w , monotonically increases with respect to v and monotonically decreases with respect to w for all ∈ Ω x ; d) if inequality (11) holds, then it is a pseudo-concave and even 0 u -pseudo-concave operator, where the function 0 ( ) u x has the form (6). Further one will assume that the operator T of the form (5) is heterotone one with the companion operator of the form (8). Let us construct a method of two-sided approximations for finding a positive solution of the integral equation (4) (and, therefore, of the boundary value problem (1)-(3)).
In the cone + K one distinguishes a strongly invariant w , which for the operator T , that is determined by the equality (8), take the form: According to the scheme Due to the strong invariance of the cone segment There , which in this case has the form: The equality v w * * = will be satisfied if the system (18), (19) does not have on Thus, the following theorem holds.
> be a strongly invariant cone segment for a heterotone operator T of the form (5) with the companion operator T of the form (8) and the system of equations (18), (19) has no solutions on Then the iterative process As one sees, it follows from the chain of inequalities (20) that each of the cone segments , is a strongly invariant cone segment for the heterotone operator T of the form (5) with the companion operator T of the form (8).
The conditions for the existence of a unique positive solution of the boundary value problem (1)-(3) and the two-sided convergence of successive approximations (15)- (17) to it can be refined by clarifying the conditions under which the system of equations (18), (19) does not have solutions such that v w ≠ on some of the strongly invariant cone segments One of the conditions that will ensure the implementation of the equality v w * * = is the condition of the existence of such Suppose there exists such number 0 L > that the func-

and for all
∈ Ω x satisfies the inequality Then Therefore, It is clear that inequality (22) implies an estimate Thus, the equality v w * * = will hold if 1 LM γ = < , and then the following theorem is valid.  Let us now consider the partial cases when the function ( , ) f u x only monotonically increases or only monotonically decreases with respect to u .
If the function ( , ) f u x monotonically increases with respect to u and ˆ( , , ) x is chosen, then the companion operator T is given by the equality (9), and the conditions (13), (14), which distinguish a strongly invariant cone segment 0 0 , v w < > (in this case, the strong invariance coincides with the common invariance of the operator T ), look like: for all ∈ Ω x 0 0 As one can see, each of the inequalities (24), (25) independently of the other ones distinguishes its end of the For the function ( , ) f u x monotonic in u , the system of equations (18) for all ∈ Ω x satisfies the inequality and condition (11) of 0 u -pseudo-concavity takes the following form: for any positive number u and for any (0, 1) τ∈ The system of the equations (18), (19) for the antitone with respect to u function ( , ) f u x has the form and the condition (11) of 0 u -pseudo-concavity takes the following form: for any positive number u and for any (0, 1) converge bilaterally to a unique on 0 0 , v w < > continuous positive solution u * of the boundary value problem (1)- . At the k -th iteration, as an approximate solution of the boundary value problem (1)-(3), the following function is taken Then one will have a convenient a posteriori error estimate for the approximate solution (35): which is an undoubted advantage of the constructed twoway iterative process. So, if the accuracy 0 ε > is given, then the iterative process should be carried out until the inequality is satisfied and with an accuracy of ε it can be assumed that x .
In addition, from the conditions of Theorem 3 an a priori estimate of the error can be written: Then from the inequality one finds that to achieve the accuracy of ε it is necessary to make iterations, where square brackets denote an integer part of the number.
The strongly invariant cone segment which is distinguished by the conditions (13), (14), is an a priori estimate for the unknown exact solution u * . We further it will be given the general recommendations for , the ends of a strongly invariant cone

EXPERIMENTS
The computational experiment was performed for the problems (1)-(3) with the right part of the form where , 0 p q > , , 0 λ μ > .

For a function ( , )
f u x of the form (38) it was chosen ˆ( , , ) and the corresponding operators (5), (8) where ( , ) G x s is the Green's function for the operator The pseudo-concavity condition (11), written for the function ( , ) f u x of the form (38), leads to the inequality which will be performed for all The ends of a strongly invariant cone segment 0 0 , v w < > will be sought in the form 0 Then inequalities (36) for finding α , β , take the form 1 2 p q m m − α ≤ λ α + μ β , 1 2 The iterative process (15)- (17) in this case will take the form Thus, if 0 1 p < < , 0 1 q < < , then the problem (1)- (3) with Lu u ≡ −Δ or 2 Lu u u ≡ −Δ + κ and a function ( , ) f u x , which has the form (38), for any , 0 λ μ > has a unique positive solution ( ) u * x , to which the iterative process (40)-(42) converges bilaterally.
In order to construct an approximate solution of the problem (1)-(3) it is necessary to find α , β ( 0 < α < β ) as a solution of the system of inequalities (39) (the highest value α and the lowest value β will be the solution of the corresponding system) and taking some 0 ε > (calculation accuracy), implement the iterative process (40)-(42) to perform the inequality (36). The approximate solution of the problem will be further determined by the formula (35).
The numerical implementation of the process (40)-(42) was performed using the PYTHON language. For computational experiments it was chosen 4 10 − ε = , the integrals in (40), (41) were calculated with the accuracy 6 10 − by an adaptive procedure based on Gaussian quadrature with previous piecewise linear interpolation with the same accuracy of functions ( ) ( )

RESULTS
The results of solving the problem (1)-(3) in a unit and a unit sphere with Lu u ≡ −Δ and 2 Lu u u ≡ −Δ + κ for the function ( , ) f u x of the form (38) for 1 2 p q = = and 1 λ = μ = are given in Tables 1-4 and in Fig. 1-10.
Let us consider the results of a computational experiment for the problem , cos cos cos sin sin cos( ) γ = θ ϑ+ θ ϑ ϕ−ψ , 1 sin cos x r = θ ϕ, 2 sin sin x r = θ ϕ, 3 cos x r = θ, 1 sin cos s = ρ ϑ ψ , 2 sin sin s = ρ ϑ ψ , 3 cos s = ρ ϑ , For the problem (1)-(3) with the right-hand side of the form (38) and the Laplace operator considered in a unit circle, it is found that 0.83292 α = , 1.97354 β = , and the accuracy 4 10 − ε = was achieved on the 12-th iteration, and (12) 0.5636 u = .  Table 1 shows the values of the approximate solution (12)  Each iteration was performed for an average of 185 sec., the total operating time of the program was 37 minutes.  Table 2 shows the values of the approximate solution (12)  Each iteration was performed for an average of 207 sec., the total operating time of the program was 41,4 minutes.  sin cos s = ρ ϑ ψ , 2 sin sin s = ρ ϑ ψ , 3 cos s = ρ ϑ , For the problem (1) Table 3 shows the values of the approximate solution (12) Table 4 shows the values of the approximate solution (12)   Each iteration was performed for an average of 223 sec., the total operating time of the program was 44,6 minutes. Figure 10 shows, respectively, the level surfaces of the approximate solution (12) ( ) u x .

DISCUSSION
The analysis of the results shows that the method of two-sided approximations is an effective numerical method for solving the boundary value problems of the form (1)-(3) with the Laplace and Helmholtz operators. Its advantages include convenient a posteriori error estimation, iteration completion criterion and easy to implement algorithm. Analyzing the results of the computational experiment, one can see that for the test problem in the approximate solution, the correct sign after the comma is set in about two or three iterations. Considering the x , k is the number of iteration, it can be seen that the iterative sequence has a geometric rate of convergence. At the same time, when switching to a three-dimensional problem, the program runtime increased, but the convergence rate remained almost unchanged. It can also be noted that in the transition to the case of three-dimensional space, the length of the initial strongly invariant cone segment decreased, which indicates a better (compared to the twodimensional case) choice of initial approximations. The solution norm in the three-dimensional problem also turned out to be less than for the two-dimensional one. Figures 1, 4, 6, and 9 clearly demonstrate the twosided nature of the convergence of the constructed itera-