CALCULATION OF THE GREEN’S FUNCTION OF BOUNDARY VALUE PROBLEMS FOR LINEAR ORDINARY DIFFERENTIAL EQUATIONS Текст научной статьи по специальности «Математика»

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ФункцЫ Грта знаходить широке засто-сування при розв'язку крайових задач для диференщальних рiвнянь, до яких зво-дяться багато математичних i фЬичних задач. Зокрема, розв'язки диференцшльних рiвнянь з частинними похИдними методом Фур'е зводяться до крайових задач для зви-чайних диференцшльних рiвнянь. За допо-могою функцп Грта для однорИдног задачi можна знайти розв'язок неодноргдного диференщального рюняння. Знания функцп Грта дае можливкть розв'язувати цтий клас задач з знаходження власних значень в квантовш теорп поля.

Описана розроблена побудова функцп Грта крайових задач для звичайних лтт-них диференцiальних рiвнянь. Представлен алгоритм i програма в системi Maple для обчислення функцп Грта крайових задач для диференщальних рiвнянь другого та третього порядтв в явному аналтично-му виглядi. Наведем приклади обчислення функцп Грта для конкретних крайових задач. Необхгдна для побудови функцп Грта фундаментальна система розв'язтв звичайних диференщальних рiвнянь з осо-бливими точками обчислюеться в виглядi узагальнених степеневих рядiв за допо-могою розроблених програм в середови-

Maple. Розроблено алгоритм побудови функцп Грта в виглядi степеневих рядiв для диференщального рiвняння другого та третього порядив з заданими крайови-ми умовами. Складено робочi програми в середовищi Maple для обчислення функцп Грта довшьних крайових задач для диференщальних рiвнянь другого та третього порядтв. Наведено розрахунки функцп Грта для конкретних крайових задач третього порядку за допомогою розробле-ног програми. Проведено порiвняння отри-маног наближеног функцп Грта з вИдомими виразами точног функцп Грта i отримана дуже гарна згода

Ключовi слова: функция Грта, звичай-т диференцшльт рiвняння, степеневi ряди,

узагальнеш степеневi ряди, крайовi задачi -□ □-

UDC 621.165

[DOI: 10.15587/1729-4061.2020.19347Q|

CALCULATION OF THE GREEN'S FUNCTION OF BOUNDARY VALUE PROBLEMS FOR LINEAR ORDINARY DIFFERENTIAL

EQUATIONS

I. Belyaeva

PhD, Associate Professor Department of Computer Science, Natural Sciences and Teaching

Methods* N. Chekanov Doctor of Physical and Mathematical Sciences, Professor Department of Applied Mathematics and Computer Modeling*

N. Chekanova PhD, Associate Professor Department of Information Technology Kharkiv Educational and Scientific Institute of SHEI "Banking University" Peremohy ave., 55, Kharkiv, Ukraine, 61174 I. Kirichenko Doctor of Physical and Mathematical Sciences, Professor Department of Physical and Mathematical Sciences National University of Civil Defence of Ukraine Chernyshevska str., 94, Kharkiv, Ukraine, 61023 E-mail: [email protected] O. Ptashny PhD, Associate Professor** T. Yarkho Doctor of Pedagogical Sciences, Associate Professor, PhD, Head of Department** *Belgorod National Research University Pobedy str., 85, Belgorod, Russia, 308015 **Department of Higher Mathematics Kharkiv National Automobile and Highway University Yaroslava Mudroho str., 25, Kharkiv, Ukraine, 61002

Received date 19.12.2019 Copyright © 2020,I. Belyaeva, N. Chekanov,

Accepted date 12.01.2020 N. Chekanova, I. Kirichenko, O. Ptashny, T. Yarkho

Published date 29.02.2020 This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0)

1. Introduction

The Green's function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Let's note that using the Green's function for a homogeneous problem, it is possible to calculate the solution of an inhomogeneous differential equation. Also, using the Green's function, one can solve the problem of finding eigenvalues, which are very relevant in quantum field theory.

Actual and important in mathematical research are the problems of integrating linear ordinary differential equations of the third order, as well as constructing on their basis the Green's function of the boundary value problem for an ordinary differential equation of the third order.

2. Literature review and problem statement

The Green's function in one variable must satisfy the original differential equation of any boundary value problem. Therefore, the Green's function itself can be represented

as a linear combination of linearly independent solutions (fundamental system of solutions) of the original differential equation. The problem of finding linearly independent solutions of the original differential equation is considered in detail in [1]. The problem of solving the initial differential equation when studying the properties of a harmonic oscillator in an electromagnetic field is considered in [2]. The problem of finding solutions to the differential equation in quantum mechanics is considered in [3, 4] when finding the field potential in the Schrodinger equation. Finding solutions of inhomogeneous differential equations with a power law dependence of heterogeneity is considered in detail in [5]. The problem of finding the integrability conditions for second-order differential equations is considered in [6]. However, finding a fundamental system of solutions is a rather complicated, practically unexplored mathematical problem. Thus, researchers in this field have come to the conclusion that it is necessary to use the Green's function to find solutions of differential equations. This problem is considered in detail in [7-9], but it is not solved due to the non-linearity of systems of differential equations. The authors of this article managed to partially solve this problem using the MAPLE computer system in calculating the eigenvalues and eigenfunctions of the Mathieu equation [10]. The authors of this article also used the MAPLE computer system to study the nonlinear Hamiltonian system by the Birkhoff -Gustavson method [11]. The need to obtain the Green's function for solving differential equations in an analytical form is shown in [12-14]. In [15], the problem of the nonlinearity of a system of solutions of differential equations is considered, which makes it impossible to obtain the Green's function in an analytical form. In addition, the search for a fundamental system of solutions is even more complicated if the differential equation has singular points.

This difficulty can be overcome if solutions of differential equations are sought in the form of power series, and if there are singular points, they are searched in the form of generalized power series, for example, according to the Frobe-nius method. However, this entails laborious tasks, such as substituting series in series, their differentiation, comparing coefficients at the same powers. In addition, when constructing the Green's function, problems arise for solving systems of high-order algebraic equations. Therefore, for the successful and accurate solution of such problems, the well-known computer system of symbolic-numerical transformations Maple is used, which allows to perform such necessary transformations in an analytical form with sufficient accuracy and speed.

Therefore, the development of algorithms and compilation of programs in the Maple system for explicitly calculating Green's functions and verifying the operation of these programs for specific boundary value problems is quite reasonable. The main properties and methods of constructing the Green's function, which we denote by G(x, 4), as well as a wide range of different classes of applied problems that can be used to solve the Green's function, are described in many classical textbooks on differential equations, as well as in special manuals [16] and monographs [17] on Green's functions.

equations, linearly independent solutions of which, even if there are singular points, can be effectively calculated using programs using Maple computer-aided symbolic-numerical computing systems.

To achieve the aim, the following objectives are set:

- to develop an algorithm for constructing the Green's function of boundary value problems for ordinary differential equations of the second and third orders;

- to develop an algorithm for finding the fundamental system of solutions of ordinary differential equations of the second and third orders;

- to carry out calculations of the Green's function for specific boundary value problems.

4. Green's function calculation method

Let's introduce the differential operator

dn dn-1 L s p( x) dXn+p( x) ^+

+p2 (x

dn

dxn

-••• + Pn (x ).

(1)

In the interval xe[a, b], let's consider the boundary-value problem for the ordinary differential equation

L { y ( x )} = 0 with homogeneous boundary conditions:

n-1

UAy)^* ■ yik)(a) + k=0

n-1

+lP,k ■ y(k) (b) = 0, k=0

^ = 1,2,3,4.

(2)

(3)

Here y(k)(x) is the k-th derivative of the function y(x), and y(0)(x)=y(x), aik and Pa, are numerical coefficients that are not equal to zero at the same time, i. e. a^ + b\k ^ 0, i, k=0, 1, 2, 3.

For convenience, the boundary conditions (3) are briefly written in the form

uJ y )-U^ ( y ) + Ulb ( y ) = 0.

(4)

Let's give the main properties of the Green's function G(x, 4).

1) the function is continuous and has continuous derivatives with respect to x up to the (n-2) order, inclusive, for all values of x and 4 from the interval [a, b];

2) the derivative of the (n-1) -order for x=4 has a jump equal to 1/p0(4), i. e.

d M) d (n-i) 1

^7-r G (4 + 0,4)--^^ G (4-0,4) = -^;

dx(n-1) lS dx(n-1) lS p0 (4)

3. The aim and objectives of research

The aim of this research is calculation of the Green's function of boundary value problems of ordinary differential

3) in each of the intervals [a, 4) and (4, b], the function G(x, 4) in the variable x satisfies the differential equation and the boundary conditions U^(G)=0, |=1, 2, 3, 4. The function G(x, 4) is called the Green's function or the influence function for a given boundary value problem.

The following theorem is proved in the theory of differential equations: "If the boundary value problem has only the trivial solution y(x)=0, then the operator L, that is, the boundary value problem has one and only one Green's function. It is also equivalent that the number X=0 is an eigenvalue of the operator L".

Using the properties of the Green's function, let's present general formulas for its calculation. For the application of computer calculations, the Green's function is conveniently sought in the form [15]:

G ( x, %) =

Gl (x, Ç), a <x <Ç<b, Gr (x,Ç), a <Ç< x <b,

(5)

where

Gl (x,Ç) = X[^k (Ç) + Bk (Ç)] yk(x),

k=i

Gr (x,Ç) = X[4(Ç)-B (Ç)] yk (x),

k=i

(6)

and y(k)(x) are linearly independent solutions of differential equation (2).

The conditions for the continuity of the Green's function (property 1) are written in the form of two equations

X Bk ( x ) yk ( x ) = 0,

k=1 n-2

X Bk

dyk (x )

(7)

= 0,

and property 3) - the jump of the (n-l)-th derivative at the point x=^ is written in the form of the following equation

X Bk (ç) (ç)=-

1

P0 {%)'

(8)

As a result, let's obtain a linear system of algebraic equations with respect to the functions B^(^):

B • y (4) + B2 • y2 (4) + B3 • y3 (4) + +BA • yA (4)+•••+Bn • yn (4) = 0, B • y,(1) (4)+B2 • yf (4) +B3 • y? (4) + +B4 • yf (4) + •••+Bn • y, (4) = 0, Bi • y12) (4)+B2 • y22) (4)+B3 • y32) (4)+ +B4 • y(2) (4) + •••+Bn • y(2) (4) = 0,

Bi • y1n-1) (|) + B2 • y(n-1) (|) + B3 • y(n-1) (4)-

-1

+BA • y<n-1)(4) +... + Bn • ynn-1)(4) = -

(9)

2 Pc(Ç)'

Since the determinant of this system is equal to the Wron-skian of linearly independent solutions y^(^), k=1, 2, 3, 4, which is not equal to zero, system (9) is defined and has a unique solution A^(^), k=1, 2, 3, 4. To find the functions Ak(^), k=1, 2, 3, 4, let's use the boundary conditions (3):

n-1 n-1

XaJk\a) + XbJk)(b) = 0, (^ = 1, ..., n).

(10)

From this system, for known B^(^), let's find the solutions Ak(^), knowing which let's calculate the Green's function according to expressions (5), (6).

If the determinant of system (10) is equal to zero, then the obtained equation for this determinant

det y, (x, X))) = 0

will determine the eigenvalues X when solving the problem on the eigenvalues of operator (1).

As follows from the general scheme of constructing the Green's function described above, it is necessary to calculate the fundamental system of solutions for differential equation (2). Algorithms are developed and programs developed in the MAPLE environment for calculating all linearly independent solutions of differential equations of type (2) in the form of generalized power series [18, 19].

According to the general scheme for computing the Green's function, an algorithm has also been developed, the main steps of which are presented below, and the corresponding programs are compiled using the MAPLE programming system to construct the Green's function of some boundary value problems [20, 21].

5. The algorithm for constructing the Green's function for ordinary differential equations of the second and third order

Algorithm description [20]:

Input:

Pk(x), k=0,1,... are the coefficient-functions of a given differential equation; n is the maximum exponent of the power series used; x0 is the singular point of equation (2), if any; aik and P;k are the coefficients of the boundary conditions (3); a, b are the boundary points of the segment [a, b];

Output:

yk(x) is the fundamental system of solutions of a given differential equation (2); G_left(x, is the Green's function on the interval a<x<t<b; G_right(x, is the Green's function on the interval a<t<x<b.

Description of algorithm steps:

1) calculation of linearly independent solutions yk(x) in the form of power series for differential equation (2);

2) verification of the solutions found by substitution;

3) calculation of the coefficients Bj(^) from the system of equations (9);

4) verification of the found solutions of this system;

5) drawing up a system of equations (10), finding its solutions Ak(^) and checking these solutions;

6) construction of the functions GL(x, GR(x, G(x, according to expressions (5), (6);

7) verification of the main properties of the Green's function G(x,

6. Examples of calculations of Green's functions of boundary value problems for ordinary differential equations of the second order

Let's present the results of calculating the Green's function of boundary value problems for second-order differential equations using the program [20]:

x

Example 1

Let's consider the differential equation x2 y" - xy' + y = x2 ln x + x with boundary conditions

y (1) = 0,

y '(1) + y '(2) = 0.

(11)

Gl ( x, 4) =

2x2 sin (ln x) {3 cos [ln (4 / 2)] - 4 sin [ln (4 /2)]} 43 [5 - 8 sin (ln2)-6cos(ln2)] '

gr ( x, 4) =

x2 {5 sin [ln (4 / 2)] + 6sin [ln 4] cos [ln (x / 2)] - 8 sin [ln (4)] sin [ln (x / 2)]}

43 [ 5 - 8sin[ln2]- 6 cos (ln2)] '

Using the program [20] for the corresponding homogeneous differential equation, the Green's function is obtained in the form:

G ( x, 4) =

Gl ( x, 4), 1 <x <4<2, Gr (x, 4), 1 <4< x <2,

(12)

where

G („ 4)=(ln4-ln2-1)xlnx Gl ( x ' 4)= 42 (2 + ln2) '

Gr (X, 4) =

x (ln x ln 4-2ln 4-ln2ln 4 + ln x ) 42 (2 + ln2)

y ( x ) =

(4 + 2ln2)

G ( x, 4) =

Gl ( x, 4), 1 <x < 4< 2, Gr (x, 4), 1 <4< x <2,

Thus, the Green's function is calculated for the original differential equation. Example 3

Let's consider the differential equation y" - 2y / x2 = 3 x3 sin x + ln x with boundary conditions

y (1) = 0,

y (2) + y '(2) = 0.

The following Green's function is obtained:

G ( x4)jG (x' 4)- 1 < x <4<2 ( X4) G ( X, 4), 1 <4< x <2,

Knowing the Green's function (12) for a homogeneous differential equation, which corresponds to an inhomoge-neous differential equation (11), by the formula

2

y ( x ) = J g ( X, 4)(42 ln 4+4) d 4 i

let's obtain the solution of the inhomogeneous equation

2ln2 x + 2x ln2ln x + +ln2ln2 x + 4 x ln x --10ln2ln x + 10ln x -- ln2 2ln x - 8 x --4x ln2 + 4ln2 + 8

where

gl ( x, 4) =

(1 - x )( x2 + x + 1) 3x4

, 1 < x <4<2,

Thus, a solution to the original differential equation is found in an analytical form. Example 2

Let's consider the differential equation x 2y" - 3 xy' + 5y = 3x2 with boundary conditions

2y (1) + 5y '(1) + y (2)-3y '(2) = 0, y (1)-5y '(1)-y (2) + 3y '(2) = 0.

For the corresponding homogeneous equation with the same boundary conditions, the Green's function is obtained from the program [20]

(1 -4)(42+4+1) (x, 4)= 1 4X44 1, 1< x <4<2.

Similarly to the previous examples, from the given values of the Green's function let's find

y (x) = 60x sin x +120 cos x --3 x3 sin x + 1/6ln2 x - 1/9ln x.

Thus, a solution to this inhomogeneous differential equation is obtained: Example 4

Let's consider the differential equation // 1 / m

y +—y--2 y = 0, m = 1,2,...

x x with boundary conditions y (0) = y (1) = 0.

The following Green's function is obtained:

Gl (x, 4), 0 < x <4<1, Gr(x,4), 0<4<x<1,

X^m (4-m 4m )

where

G ( x, 4) = where

Gl (x,4) =

2m

Gr ( x,4) =

4™ (x-m - xm )

2m

which coincides with the known result [22]. Example 5

Let's consider the following boundary value problem [23] d

d f(i + ar )4 ^ j + X2 y = 0,

dx [ dx J

y (0) = y (L) = 0.

y1 ( x ) = acos X

X

(1+

a(1+ax ) X

a(1 + ax )

y1 ( x ) = -a sin

/ \ X

X

+

(1 + ax )

a(1 + ax ) X

a(l + ax)

Eigenvalues are found from the equation det (uj y{ (x, X))) = 0, which leads to the following transcendental equation XL

X -a Lctg

1+aL

X + a2 + a3L = 0.

with boundary conditions

y (0) = 0, y (1) = 0, y'(0)-y'(l) = 0.

For this homogeneous differential equation, the Green's function is obtained in the form:

G ( x, 4) =

Gl ( x ,4), 1< x < 4 < 2, Gr ( x,4), 1<4<x <2,

(13) where

This problem is an eigenvalue problem that arises when studying the stability of a cone-shaped rod under the action of an external longitudinal force. The parameter a determines the geometric configuration of the truncated cone. In this problem, the critical force at which the rod loses stability is equal to the product of Young's modulus and the smallest eigenvalue X.

Equation (13) has the following linearly independent solutions:

Gl (x,4) = 1 x(4-l)(x-4),

gr (X, 4) = -14( x - i)(4-x).

Next, a solution to the inhomogeneous equation is obtained:

y (x) = — x6 +197 x2 - 99 x - 2 x2 sin (x)' 40 40 20 w

7

-—x2 sin (l) - 4 x2 cos (l) -11 x cos (x) + 31

+21sin(x) +15xcos(l)—— xsin(l).

Thus, a solution to the original differential equation is found in an analytical form. Example 2

Let's consider the differential equation y"' = 0

with boundary conditions

y (0) = 0, y (1) = 0, y (0) = y' (1).

Using the developed program, the Green's function is obtained:

(14)

G ( x, 4) =

The value X obtained by formula (14) differs by less than 2 % from the same value obtained in [23] in another way.

The examples of solving boundary value problems in this section are used by the authors in solving the equations of heat conduction and oscillations with partial derivatives.

7. Examples of calculations of the Green's functions of some boundary value problems for ordinary differential equations of the third order

We present the results of calculating the Green's function of boundary value problems for third-order differential equations using the program [20].

Example 1

Let's consider the boundary value problem for the differential equation

y'" = x sin x + 2x2 cos x + 3 x3

Gl (x, 4), 0 <x <4<1, Gr(x,4), 0<4<x<1,

where

GL = 2 (x 24-*42 -*2 +*4), Gr = 2 (*24-*42 -*4+42).

The resulting expression coincides with the exact expression and is anti-self-adjoint. Example 3

For a differential equation y"' (x) + y' (x) = x cos2 (x) with uniform boundary conditions y (0) = 0, y ^ ] = 0

+

and

/ (0)=✓ (f )

the Green's function is found in the form \gl(x,4), 0<X<4<1,

G ( x, Ç) =

Gr(x,4), 0<4<x<1,

where

Gl (x, 4) = 2

Gr (X, 4) = -2

cos x -sin x + cos X cos 4- cos x sin 4+ sin X cos 4+ + sin X sin 4-cos 4+sin 4-1

cos X - sin X - cos X cos 4 --cosX sin4 + sinX cos4--sin x sin4 - cos 4 + sin4 + 1

To construct the Green's function of the boundary value problem (15), (16), let's first solve the Cauchy problem at the point x0, find linearly independent solutions for equation (15) in the form of series:

y ( * ) = ck} ■(* - *o )k -

k=3

y2 ( X )= ( * - X0 )+ Z ck2 > ■ ( x - x0 )k -

k=3

y3 (X) = (x-x0)2/2 + Zck3) ■(x-x0)k -

k=3

where cf, c(k2\ are the numerical coefficients. Let's find the Green's function in the form

(17)

(18) (19)

G ( x, 4) =

Gl (x, 4), a <x <4<b, Gr (x, 4), a <4<x <b,

where let's find:

/ \ n 2 . 11 . n .

y ( x ) =--sin x--sin x + — sin x -

v ; 32 72 6

x n2 n n2 11

--cos x sin x +--cos x-----1---+

6 32 6 32 72

x2 11 2 n 11

+----cos x + —cos x +--cos x.

4 36 6 72

Thus, the solution of the initial inhomogeneous differential equation in the analytical form is obtained.

8. Construction of the Green's function for third-order differential equations in the form of power series

Let's consider a third order differential equation

Po (*) y'" + Pi (*) y" + Pi (*) y' + p3 (*) y = 0 (15)

with boundary conditions

«1,0 y (a) + «1,1 y (a) + «1,2 y" (a) +

+Plj0 y (¿) + Pu y' (6) + P1, y''(b) = 0, a20 y (a) + a21y' (a) + a22 y''(a) +

+P2,0 y (b) + P2,iy' (b) + P2,2y"(b) = 0,

a3j0y (a) + a3jly' (a) + a32y"(a) +

+P3,o y (b) + P31/(b) + P3,2y"(b) = 0, (16)

where p0(x), pi(x), p2(x), p3(x) are continuous functions together with continuous derivatives of the first and second orders on the interval [a, b], aj,0, a^, a^2, a20, a2,i, a2,2, a30, a3,1, a3,2, Pi,0, Pi,1, Pi,2, P2,0, P2,1, P2,2, P3,0, Ps,1, P3,2 are the coefficients in the boundary conditions (16) for a particular boundary-value problem,

XX«2 *0, XXP2 * 0, ¿=1, 2, 3, k=0, 1, 2.

where

Gl ( 4) = t [ A (4) + Bk (4)]-y ( x ),

k=i

3

gr (X, 4)=X[ A (4)-Bk (4)]-yk (X).

k=i

Using the properties of the Green's function given above, let's find the coefficient functions Ak(4), Bk(4) and with their help let's construct the Green's function according to formulas (17)-(19). Since linearly independent solutions are represented by power series, the Green's function is also in the form of power series.

In accordance with the above formulas, an algorithm is developed for constructing the Green's function in the form of power series for the boundary value problem (15) in the Maple environment.

9. An algorithm for constructing the Green's function for third-order equations in the form of power series

Using three linearly independent solutions found in the form of power series, the Green's function is constructed on the basis of the developed algorithm and its calculation program.

Input:

n is the desired maximum order of the power series;

P0(x)^0, P1(x), P2(x), P3(x) in general, the coefficient functions in a given differential equation of the third order (15);

a, b are the boundary points of the segment [a, b] on which the Green's function is sought;

a1,0, a1,1, a1,2, a2,0, a2,1, a2,2, a3,0, a3,1, a3,2, P1,0, P1,1, P1,2, P2,o, P24, P2,2, P3,0, P3,1, P3,2 are the coefficients in the boundary conditions (16) for a particular boundary-value problem.

Output:

y1(x), y2(x), y3(x) are the fundamental system of solutions for a given differential equation of the third order (15);

G_L(x, 4) is the Green's function on the interval a<x<4<b;

G_R(x, 4) is the Green's function on the interval a<4<x<b.

Description of algorithm steps:

1) procedure for calculating linearly independent solutions of a third-order differential equation in the form of power series;

2) verification of the found solutions;

3) calculation of coefficient-functions B1(i;), B2(^), B3(^) for the Green's function G(x, Ç);

4) verification of the calculated coefficient-functions

B1(fj), B2(fj), B3(fj);

5) specification of particular boundary conditions in the interval [a, 6];

6) solving a system of algebraic equations for determining coefficient-functions A1(Ç), A2(Ç), A3(Ç);

7) verification of the calculated coefficient-functions

A1(fj), A2(fj), A3(fj);

8) construction of the Green's function G_L(x, Ç), (a<x< <£<b) and G_R(x, Ç), (a<£<x<b);

9) verification of all properties of the Green's function G(x, Ç).

10. Examples of constructing the Green's function of ordinary differential equations of the third order in the form of power series

Example 1

Let's consider the differential equation y '"- 6 y" + lly '- 6 y = 0 with boundary conditions

y (0) = 0, y (1) = 0, y''(0) = 0.

For differential equation (20), the solutions

y = e" - % = e2" - y3 = e3"

(20)

(21)

(22)

are a fundamental system of decisions. According to the above, the Green's function G(x, Ç) can be constructed from the general solutions of (22) using the formulas:

G ( x, Ç) =

Gl (x, Ç), a < x <Ç<b, Gr (x,Ç), a <Ç<x <b,

where

Gl (x,Ç) = X[¿(Ç) + B(Ç)]yk (x),

k=i

G, ( x,Ç) = X [ A(Ç)-B (Ç)] yk ( x ).

B (Ç) = -4B2 (Ç) =1 e-2Ç, (Ç) = -1

4

To find the coefficient functions Aj(^), (¿=1,2,3), let's use the boundary conditions (21) and as a result let's obtain the system

A (Ç) + A2 (Ç) + A3 (Ç) = - B (Ç) - B2 (Ç) - B3 (Ç), A (Ç) + 4 A2 (Ç) + 9 A3 (Ç) = - B (Ç) - 4B2 (Ç) - 9B3 (Ç), A, (Ç) + ^2 (Ç) + e2A3 (Ç) = B, (Ç) + eB2 (Ç) + e2B3 (Ç).

From this system let's find solutions: 1

A (Ç) = - 4(3e2 -8e + 5) X x(-20e1+Ç + 5e2Ç + 10e2 + 8e1-2Ç -3e2+2Ç)e~3Ç,

A (Ç) = : '

2(3e2 -8e + 5) x(-8e1+Ç + 8e2Ç + 8e2 -5eÇ -3e2+Ç)e~3Ç,

a (ç)=--

4 (3e2 - 8e + 5) x(-12e1+4 - 5 + 6e24 + 3e2 + 8e) e-34.

Using the found expressions for Aj(^), Bj(^), (k=1,2,3), let's find the exact Green's function for the boundary value problem (20), (21) in the form:

G ( x, Ç) =

where

Gl (x,Ç) =

Gl (x, Ç), 0 <x <Ç<1, Gr (x, Ç), 0 <Ç<x <1,

ex (-5 / 2e~Ç + 5e1"2 Ç- 5/2e2-3 Ç)

3e2 - 8e + 5

e2 x (4e-Ç- 8e1-2 Ç + 4e2- Ç) ----2- +

3e - 8e + 5 e3 x (-3/2e-Ç+ 3e1-2 Ç- 3/2e2- Ç) 3e2 - 8e + 5 '

ex (3/2 e2-Ç - 4 e1^ + 5 e1"2Ç - 5 / 2 e

(24)

Gr ( x, Ç) =

2-3 Ç\

From the continuity conditions for the Green's function, its first derivative, and also the jump of the second derivative, let's obtain a system for determining the coefficients of the functions Bfâ), B2(Ç), B3(Ç):

B (Ç)y, (Ç) + B2 (Ç)y2 (Ç) + B3 (Ç)y (Ç) = 0, B (Ç) y,(Ç)+B2 (Ç) y2 (Ç)+B3 (Ç) y' (Ç)=0, (23)

B (Ç)yi'(Ç) + B2 (Ç)y2'(Ç) + B3 (Ç)y3'(Ç) = -1/(2P0 (Ç)).

System (23) is always solvable and has a unique solution, because P0(^)^0, and therefore, the main determinant of this system is the Wronskian W [yi, y2, y3], which is not equal to zero.

From system (23) let's find solutions:

3 e2 - 8 e + 5

e2 x (4 e-Ç- 3 e2-2 Ç- 5 e~2Ç + 4 e2-3Ç)

3 e2 - 8 e + 5 e3 x (-3/2e-Ç+ 3 e1-2 Ç + 5/2e~3 Ç- 4 e1"3 Ç) 3 e2 - 8 e + 5

(25)

Using the developed program for the boundary value problem (20), (21), the approximate Green's function is obtained in the form of power series, the first terms of which are given below

G ( x, Ç) =

gl(x,Ç), 0<x<Ç< 1, gr (x, Ç), 0 <Ç<x <1,

where

19663 82489 „ 308167

g(7) =-x--x4+-x42 -

L 35489 35489 70978

216293 3 1059601 ,.3 907379 3(.

-x —______ x43 +-x 4-

212934 212934 212934 98315 4 3462703 3389837

x 44-:

70978 851736 425868

412445 ^ 4699457 +-x 44-

-x 342 +

70978 4258680

19663

117978

GR) = ^^ x +1/ 242 ^ x4 +

35489

35489

+1/ 2x2 -43 + 521101x42 - 3x24 -70978

3359 3 25„4 324471

--x +—44--x43 +

212934 24 35489

25 2^2 3359 q^ 292555 <

+— x 42+-x 4--x -

4 35489 851736

-3 /445 + 6656713 x44 - 15/2x243 -851736

195827 3tt2 292555 4e 1505447 -x 42 +-x44-

425868 141956 4258680

Comparisons are calculated for different values 4 and for the values of the power series n=13 and n=16. From the calculations it follows that the obtained approximate Green's function differs from the exact one by 1.3 % and 6-10-3 % at n=13 and n=16, respectively. Example 2

Let's consider the differential equation

x3 y"' + xy' - y = 0

with boundary conditions

y (1) + y' (1) + y''(1) + y (2) + y' (2) + y"(2) = 0,

2y (1) + y' (1) - 2y"(l) + y (2) + y' (2)- 2y"(2) = 0,

2 y (1) + y' (1) + 3 y"(l) - 2 y (2)- y' (2) + y"(2) = 0.

Using the developed program, the Green's function is obtained:

G (x, 4) =

gl(x,4), o<X<4<1,

G(X,4), 0<4<X<1, where the first terms of the Green's function are:

GL (x, 4) =

242 (44 + 90 ln2 2 - 248 ln 2)'

x[l80ln2 4-120ln 4 + . +],

Gr (x' 4)= 242 (44 + 89 ln2 2 - 248 ln2)X x[l36ln2 4-120ln 4 + . +].

Thus, the Green's function of the initial homogeneous differential equation is obtained.

11. Discussion of the results obtained on the construction of the Green's function of ordinary differential equations

Methods for constructing the Green's function for linear ordinary differential equations of the second and third orders are developed. Knowing the Green's function [8] allows to calculate the solution of a linear inhomogeneous differential equation with given boundary conditions, as well as to find the eigenvalues and functions of the boundary value problem.

An algorithm is developed for constructing the Green's function in the case when in the Maple system it is possible to obtain in explicit form three linearly independent solutions of a given third-order differential equation with boundary conditions. The description of the algorithm for constructing the Green's function for ordinary differential equations of the third order in an explicit analytical form is given. The Green's function calculations for specific boundary value problems using the developed program are presented.

An algorithm has been developed for constructing the Green's function in the form of power series for a third-order differential equation with given boundary conditions. The description of the algorithm for constructing the Green's function for third-order equations in the form of power series is given. The Green's function is calculated for specific third-order boundary value problems using the developed program, and the obtained approximate Green's function is compared with the exact, if known, and the accuracy of their agreement is shown.

Based on the known properties of the Green's function, in this research, an algorithm is developed and a program of symbol-numerical calculations of the Green's function is developed using computer systems of analytical calculations; any boundary value problems for ordinary differential equations of the second and third orders can be stated. An essential and important node in the calculation of the Green's function is the search for a fundamental system of solutions for a given differential equation. In this paper, this problem is solved by calculation using working programs.

Calculations are carried out for a number of boundary value problems, and the corresponding Green's functions are obtained. For the answers of problems known from the literature [22], a very good agreement is found (generally exact coincidence) with the calculations of the Green's functions presented in this paper. This proves the efficiency of constructing Green's functions in the proposed approach. Similar calculations in the known literature have not been identified. It can and is important to note that the accuracy of the calculation of the Green's function is determined by the accuracy of the calculation of the fundamental system of solutions, which in the general case is automatically controlled by the number of terms in power series and the number of decimal places in decimal numbers.

In conclusion, it is possible to say that thanks to the methods and algorithms proposed in this article, any boundary-value problem for ordinary differential equations of the second and third orders can be solved. The only drawback of this work is that the calculation of the Green's function and finding all the linearly independent solutions of the fundamental system of solutions necessary for this is a difficult operation, manual calculations are practically impossible for homogeneous equations, and even more so for heterogeneous ones.

12. Conclusions

1. A method for constructing the Green's function for linear ordinary differential equations of the second and third orders having singular points is described in the form of generalized power series using computer systems of algebraic transformations.

2. The construction of a fundamental system of solutions in the form of convergent series allows, in subsequent numerical calculations, to obtain the desired accuracy by increasing the number of terms in the series and by increasing the number of digits after the decimal point, it means with such accuracy to calculate the Green's function itself.

3. Examples of calculations of the Green's functions of boundary value problems for ordinary differential equations of the second and third order in the form of power series in the Maple system are presented, which allows one to efficiently and accurately perform all the necessary transformations when constructing the Green's function. The calculated Green's functions are compared with those available in the literature and the accuracy of their agreement is shown. The exact coincidence of the calculated Green's functions with the known from other sources is obtained, which proves the effectiveness of the calculation method used and the developed program.

References

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12. Lewis, H. R. (1968). Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators. Journal of Mathematical Physics, 9 (11), 1976-1986. doi: https://doi.org/10.1063/U664532

13. Khurshudyan, A. Z. (2018). New Green's functions for some nonlinear oscillating systems and related PDEs. International Journal of Modern Physics C, 29 (04), 1850032. doi: https://doi.org/10.1142/s0129183118500328

14. Khurshudyan, As. Zh., Frasca, M. (2018). Green's functions for higher order nonlinear equations. Available at: https://www.re-searchgate.net/publication/326110263_Green's_functions_for_higher_order_nonlinear_equations

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18. Belyaeva, I. N., Ukolov, Yu. A., Chekanov, N. A. (2005). Postroenie obshchego resheniya differentsial'nyh uravneniy fuksovskogo tipa v vide stepennyh ryadov. Svidetel'stvo ob otraslevoy registratsii razrabotki No. 50200500089.

19. Belyaeva, I. N., Chekanov, N. A. (2010). Simvol'no-chislennoe integrirovanie lineynogo differentsial'nogo uravneniya tret'ego poryadka v vide obobshchennyh ryadov. Svidetel'stvo o gosudarstvennoy registratsii programmy dlya EVM No. 2010614257. Zayavka No. 2010612592 ot 11 maya 2010 g.

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23. Mihlin, S. G. (1947). Prilozheniya integral'nyh uravneniy k nekotorym problemam mehaniki, matematicheskoy fiziki i tehniki. Moscow-Leningrad: OGIZ izdatel'stvo tehniko-teoreticheskoy literatury, 304.

Запропоновано генетичний метод для прогнозування показнитв здоров'я населения на основi нейромережевих моде -лей. Принципова вiдмiннiсть запропонованого генетичного методу вiд кнуючих аналогiв полягае у використант диплогд-ного набору хромосом в особин популяцп, яка еволющонуе. Така модифжацш робить залежнкть фенотипу особини вiд генотипу менш детермтованою i, врештi, сприяе збереженню рiзноманiтностi генофонду популяцп i варiабельностi ознак фенотипу впродовж виконання алгоритму. Крiм цього, запропоновано модифгкащю генетичного оператору мутацш. На в^мшу вiд класичного методу, особини, як пИддаються дп оператору мутацп, обираються не випадковим чином, а у вiд-повiдностi до гх мутацшног стiйкостi, що вгдповгдае значен-ню функцп пристосованостi особини. Таким чином, мутують особини, що характеризуются гiршими значеннями цтьовог функцп, а геном сильних особин залишаеться незмтним. У цьому випадку зменшуеться вiрогiднiсть втрати досягнуто-го впродовж еволюци екстремуму функцп внаслИдок дп оператора мутацш, а перехгд до нового екстремуму здшснюеться у випадку накопичення достатньог питомог ваги кращих ознак впопуляцп.

Порiвияльний аналЬ роботи моделей, синтезованих за допо-могою розробленого генетичного методу, показав, що найкращi результати досягнутi у моделi на основi нейронног мереж дов-гог короткочасног пам'ятi. Шд час створення i навчання моделi на основi мереж довгог короткочасног пам'ятi було дослгджено можливкть використання методу рою часток для оптимЬацп параметрiв мережi. Результати експериментальних дослiд-жень показали, що розроблена модель дае найменшу помилку передбачення кiлькостi нових випадтв туберкульозу - серед-ня абсолютна помилка складае 6,139, що менше у порiвняннi з моделями, побудованими за допомогою тших методiв.

Шрактичне використання розроблених методiв дасть мож-ливкть своечасно коригувати плановаш лжувально^агно-стичт, профiлактичш заходи, завчасно визначати необхШж ресурси для локалЬацп та лгквгдацп захворювань з метою збе-реження здоров'я населення

Ключовi слова: нейронн мережi, генетичний алгоритм, фенотип, модифгкований генетичний оператор мутацп, прогнозування показнитв здоров'я населення

■а □-

UDC 004.93

IDOI: 10.15587/1729-4061.2020.197319]

CONSTRUCTION OF A GENETIC METHOD TO FORECAST THE POPULATION HEALTH INDICATORS BASED ON NEURAL NETWORK

MODELS

I. Fedorchenko

Senior Lecturer* E-mail: [email protected] А. Ol i i nyk PhD, Associate Professor* А. S tepanen ko PhD, Associate Professor* T. Zai ko PhD, Associate Professor* S. Korniienko PhD, Associate Professor Software Developer A. Kharchenko Software Developer *Department of Software Tools Zaporizhzhia Polytechnic National University Zhukovskoho str., 64, Zaporizhzhia, Ukraine, 69063

Received date 21.10.2019 Accepted date 17.02.2020 Published date 29.02.2020

Copyright © 2020, I. Fedorchenko, A. OlUnyk, A. Stepanenko, T. Zaiko, S. Korniienko, A. Kharchenko This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0)

1. Introduction

Quality of life of population is determined by different indicators, in particular health indicators, whose condition is predetermined by environmental factors. According to medical research conducted in recent years [1], there is a close relationship between the anthropogenic air pollution in certain areas and the increased population morbidity. As estimated by the World Health Organization (WHO), air pollution is the biggest factor of environmental health risks at present [2]. Based on this assessment, about 3.7 million of additional deaths are related to ambient air pollution, 4.3 million - to

air pollution indoors. Since many people are exposed to both indoor and outdoor polluted air, causes and deaths from various diseases caused by different sources cannot be determined through the usual generalization of data. The biggest health problems caused by direct influence of air pollution are related to diseases of blood circulation, respiratory diseases, cancer, neuro-mental disorders, as well as some others [3, 4].

Consequently, the health condition and population morbidity in a region can be considered as derivatives from the environment.

The use of known statistics methods for forecasting the dependence of health indicators, as well as mathematical