A simple way to solve boundary value problems in technological processes

. The paper proposes a simple approximate method for solving differential equations for boundary value problems. A method is proposed for averaging differential equations over a moving volume, which allows obtaining approximate analytical solutions of differential equations. The control volume is the only one in the considered area of the boundary value problem. In this case, the control volume is considered to be moved in the area under consideration. Based on the averaging of boundary value problems over the volume being moved, an algebraic equation is obtained. When averaging over one of the variables (in the case of a two-dimensional problem), ordinary differential equations are obtained. Examples are given.


Introduction
Most problems in nature, technology and technological processes are described by differential equations. Mathematical modeling of problems in mechanics, physics and other branches of science and technology are reduced to differential equations. In this regard, the solution of differential equations is one of the most important mathematical problems. There are various methods for solving problems of mathematical physics.
Analytical methods (the method of separation of variables, the method of characteristics, etc.) have a relatively low degree of universality. Approximate analytical methods (projection, variational methods, etc.) are more versatile than analytical ones. Numerical methods (the method of finite differences, the method of lines, the control volume method, the finite element method, etc.) are very universal. For engineers investigating the phenomenon described by differential equations, it is more convenient to obtain solutions in the form of simple formulas.

Materials and methods
The proposed method is a method of averaging over the volume being moved and obtaining an approximately analytical solution of boundary value problems. In works [1,2], the moving node method was proposed for the ordinary differential equation of convection-diffusion. In these works, compact schemes were constructed for the problem of convection-diffusion, as well as an approximately analytical solution of these problems based on finite-difference schemes and the control volume method. In [3], an improvement in the construction of discrete analogs based on Richardson's extrapolation was studied. Work [4] is devoted to improving difference schemes based on the choice of a profile on the edge of the control volume. In [5], the moving node method was used to solve the problem of flow in a combined region. Here we propose a method for averaging over the volume being moved for twodimensional boundary value problems. Note that obtaining an approximate-analytical solution of differential equations is based on numerical methods. The nature of numerical methods also makes it possible to obtain an approximately analytical expression for solving differential equations. The control volume method for solving boundary value problems by numerical methods is well known [6][7][8]. If there is only one control volume, in the area under consideration, where the solution is determined, and moved, then it is possible to obtain an approximate analytical form of the solution to the boundary value problem.

One-dimensional problems
For a better understanding of the method, let's start with a simple example.

Flow in a flat pipe
The flow of a viscous fluid in a flat pipe in a one-dimensional formulation is described by the equation [9,10] Δ (1)

Fig. 1. Flat pipe.
where is the U fluid velocity, y is the vertical coordinate perpendicular to the flow, is the pressure drop (const), is the viscosity of flow. Let and motionless walls. We average (1) over the liquid volume. When it changes to segments [0, h], the position of the liquid volume also changes ( Fig. 1 We replace the derivatives in (2) with the difference relation: Here is an approximate value. Thus, averaging (1) over the moved volume has the form: Hence, taking into account the no-slip condition Here is the average solution. For this problem, the averaged solution coincides with the exact solution [9,10].

Heat distribution in the plate
Heat distribution in the plate is described by the equation [11] (5) where k is the thermal conductivity and q is the heat release per unit volume (k and q = const). It is assumed that the source does not depend on temperature. Averaging similarly to the previous problem, we get (6) From equation (6), we obtain (7) Solution (7) coincides with the exact solution.

Two-dimensional boundary value problems: averaging over two variables
We demonstrate a method of averaging over a moving control volume for simple twodimensional problems.

The temperature field in a solid
Consider the problem of temperature propagation on a homogeneous body. The problem is reduced to solving the Laplace equation with boundary conditions [12] (we choose the Cartesian coordinates as shown in Fig. 2): We average the Laplace equation over the control volume (dashed area) [6,7]. Here, the control volume is considered to be moved. (8) since (9) The last expression was obtained under the condition that the derivatives on the faces of the control volume are considered constant and we replace the derivative with the central difference. In (9) the temperature value at the boundary point W, the notation has a similar meaning. Doing similarly with the second expression in (8), we obtain Hence, using the boundary conditions, instead of the Laplace equation, we obtain The maximum absolute difference between the exact and approximate solutions calculated by points ℎ ℎ ℎ is 0.015.

Flow in a rectangular pipe
Consider the problem of the flow of an incompressible viscous fluid through a rectangular pipe [6]. Let us denote the height of the rectangle parallel to the axis as ℎ, and the base parallel to the axis as ℎ, where is any positive constant. We draw the axis through the center of the rectangle and direct it downstream.
The dimensionless equation has the form (for the length scale we take the height, ℎ, and for the velocity scale is the value ℎ , is pressure drop, is flow viscosity) [6,7]: (10) Boundary conditions for (10) (11) If >1 we will average over the section x (Fig.1). Then we have (12) Let's put, Here U is the velocity averaged over the section. Thus, we obtain the averaged equation in the form (13) In equation (13), considering y as a parameter, taking into account the boundary conditions, we obtain On fig. 3 shows a comparison of (14) and (15). The maximum absolute difference between the exact and approximate solutions is 0.003.

Flow at the inlet section of the pipe
With appropriate simplifications, the flow of a viscous incompressible fluid in a dimensionless form is described by the following differential equation [6]: (16) Here, N = -12 is the pressure drop, Re is the Reynolds number. The equations are considered in the area (Fig. 4). Boundary conditions for (16): The convective term is linearizable (the coefficient is replaced by the average speed) Approximating in (16) by the liquid volume we obtain an ordinary equation, solving which we have an approximate solution: (17) here There is no exact solution of problem (16). For comparison, numerical solutions (16) were made using the finite difference method [8]. The resulting system of equations is implemented by the upper relaxation method [13].

Solving the problem of open channel flow by the method of a movable control volume
Consider the currents in a long channel. There is no pressure drop along the length of the flow, the transverse pressure drop is static, the same in all sections.
Let the angle of inclination to the horizon is . Then from the Navier-Stokes equation [9] we get Here .
From the no-slip condition it follows that For the definition we proceed as follows. In (14), let's go to the limit then, we get

Conclusion
The above examples show that it is convenient to apply the method of a moving control volume for engineering calculations. If the dimensions of the two-dimensional region are commensurate with each other, then averaging over both variables is more convenient. If, however, the size of the extent of one of the sides exceeds the other, then it is more reasonable to apply averaging over the narrow side of the area. The advantage of the proposed approach should include its simple form for engineering calculations.