Calculation of the green’s function of boundary value problems for linear ordinary differential equations

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. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found


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.

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
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 Frobenius 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, ξ), 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.

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 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.

Green's function calculation method
Let's introduce the differential operator In the interval xϵ[a, b], let's consider the boundary-value problem for the ordinary differential equation with homogeneous boundary conditions: Here y (k) (x) is the k-th derivative of the function y(x), and y (0) (x)≡y(x), α ik and β ik are numerical coefficients that are not equal to zero at the same time, i. e. 2 2 0, For convenience, the boundary conditions (3) are briefly written in the form Let's give the main properties of the Green's function G(x, ξ).
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 ξ from the interval [a, b]; 2) the derivative of the (n-1) -order for x=ξ has a jump equal to 1/p 0 (ξ), i. e.
3) in each of the intervals [a, ξ) and (ξ, b], the function G(x, ξ) in the variable x satisfies the differential equation and the boundary conditions U μ (G)≡0, μ=1, 2, 3, 4. The function G(x, ξ) 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 ˆ, L that is, the boundary value problem has one and only one Green's function. It is also equivalent that the number λ≡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]: 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 and property 3) -the jump of the (n-1)-th derivative at the point x≡ξ is written in the form of the following equation As a result, let's obtain a linear system of algebraic equations with respect to the functions B k (ξ): Since the determinant of this system is equal to the Wronskian of linearly independent solutions y k (ξ), k=1, 2, 3, 4, which is not equal to zero, system (9) is defined and has a unique solution A k (ξ), k=1, 2, 3, 4. To find the functions A k (ξ), k=1, 2, 3, 4, let's use the boundary conditions (3): From this system, for known B k (ξ), let's find the solutions A k (ξ), 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 will determine the eigenvalues λ 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].

The algorithm for constructing the Green's function for ordinary differential equations of the second and third order
Algorithm description [20]: Input: P k (x), k=0,1,… are the coefficient-functions of a given differential equation; n is the maximum exponent of the power series used; x 0 is the singular point of equation (2) Output: y k (x) is the fundamental system of solutions of a given differential equation (2); G_left(x, ξ) is the Green's function on the interval a≤x≤ξ≤b; G_right(x, ξ) is the Green's function on the interval a≤ξ≤x≤b.
Description of algorithm steps: 1) calculation of linearly independent solutions y k (x) in the form of power series for differential equation (2); 2) verification of the solutions found by substitution; 3) calculation of the coefficients B k (ξ) 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 A k (ξ) and checking these solutions; 6) construction of the functions G L (x, ξ), G R (x, ξ), G(x, ξ) according to expressions (5), (6); 7) verification of the main properties of the Green's function G(x, ξ).

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]:

Example 1
Let's consider the differential equation 2 2 ln x y xy y x Using the program [20] for the corresponding homogeneous differential equation, the Green's function is obtained in the form: Knowing the Green's function (12) for a homogeneous differential equation, which corresponds to an inhomogeneous differential equation (11), by the formula 10ln .
ln 2ln 8 4 ln 2 4ln 2 8 Thus, a solution to the original differential equation is found in an analytical form.
Example 2 Let's consider the differential equation For the corresponding homogeneous equation with the same boundary conditions, the Green's function is obtained from the program [20] ( ) ( ) ( ) , 2 sin ln 3cos ln / 2 4sin ln / 2 , 5 8sin ln 2 6cos ln 2 Thus, the Green's function is calculated for the original differential equation.

Example 3
Let's consider the differential equation The following Green's function is obtained: Similarly to the previous examples, from the given values of the Green's function let's find ( ) 3 2 60 sin 120cos Thus, a solution to this inhomogeneous differential equation is obtained: Example 4 Let's consider the differential equation The following Green's function is obtained: , 5sin ln / 2 6sin ln cos ln / 2 8sin ln sin ln / 2 . 5 8sin ln 2 6cos ln 2 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 α 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 λ.
Equation (13) has the following linearly independent solutions: Eigenvalues are found from the equation The value λ 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.

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

Example 2
Let's consider the differential equation Using the developed program, the Green's function is obtained: The resulting expression coincides with the exact expression and is anti-self-adjoint.
where let's find: ( ) Thus, the solution of the initial inhomogeneous differential equation in the analytical form is obtained.
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.

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.
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, ξ) is the Green's function on the interval a≤x≤ξ≤b; G_R(x, ξ) is the Green's function on the interval a≤ξ≤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;

Example 1
Let's consider the differential equation 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 B 1 (ξ), B 2 (ξ), B 3 (ξ): System (23) is always solvable and has a unique solution, because P 0 (ξ)≠0, and therefore, the main determinant of this system is the Wronskian W[y 1 , y 2 , y 3 ], which is not equal to zero.
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 Comparisons are calculated for different values ξ 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 сonsider the differential equation  Using the developed program, the Green's function is obtained: where the first terms of the Green's function are: 120ln .
Thus, the Green's function of the initial homogeneous differential equation is obtained.

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.

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.

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