Ибратжон Алиев – Все науки. №12, 2024. Международный научный журнал (страница 2)

Based on certain data, it is possible to state the change of function (2) to (14) and the problem of the following boundary conditions (15—18), given that the boundary conditions and the dynamic phenomenon are known, the Fourier method of variable separation can be adopted as a solution for it.

Now that the initial and boundary conditions, as well as the corresponding equation, have been determined at the specified moment, it is necessary to pay attention to the effect of the Laplace equation in static form, and it, as a partial equation from the Helmholtz equation, can be interpreted as follows. Namely, for the reason that in this case the phenomenon of energy transfer is observed, and in this case the Laplace equation is used to display in a global sense a model of an emitter or an energy-emitting «charge» in the face of the Sun. Thus, on a more local scale, the harmonic function taken will satisfy, based on these conditions, the homogeneous equation of thermal conductivity or energy conductivity (19), including based on the transformation model for the connection of the Helmholtz equation and the wave equation.

Where, the coefficient of energy conductivity is determined in (20), along with all the determined parameters, including the coefficients of energy conductivity of the vacuum between the Sun and the Earth (21), the specific energy capacity (22) and the available energy density under the circumstances in the specified area (23).

Based on the calculated parameters according to (21—23), expression (20) obtains a numerical indicator (24).

Based on the conditions obtained, it is possible to determine that the problem can be solved by taking the form of an equation of the form (25), where, after substitution, a transformation can be obtained according to (26), with an equated coefficient (27).

From expression (27), 2 partial differential equations are formed – 1 ordinary with respect to time in the first degree and the second in the square of partial derivatives. The first equation is solved by adopting a general solution with an exponential form, where, after substitution, a characteristic form is presented, from which a general form of the function is formed – the solution of the resulting ordinary differential equation in time (28).

With respect to time, initial conditions (3—4) have already been obtained, which can be substituted to form initially the coefficient value from (27) to (29), the independent variable in (30) and the resulting form of a function with known constants in (31).

The resulting function is the solution of only one differential equation, the second (32) is formed relative to the Laplacian in a spherical coordinate system with a known constant.

The solution of this equation is presented initially after the disclosure of the Laplacian for the Ψ-function in a spherical coordinate system, where the Fourier variable separation method is applied, which was originally applied in (26). Then subsequently, after substitution, the resulting separation expression is revealed, forming separate groups of derivatives in the specified system (33).

Taking into account the resulting transformation and taking into account the transformation of the original ratio, the Laplacian ratio of the function and the function itself can be substituted into the transformed form after separating the variables, from which a separate additional ratio is created for each function – radius, first and second angles, as well as for the second derivatives of these expressions (34).

From the first ratio in (34), a second-order differential equation with respect to radius is formed, which can be solved after revealing the ratio and using the integral of the second degree with respect to the radius variable in the second degree. When integrating in both directions, in the right case, a known ratio is obtained, in the left case, the function itself is used as a variable, which allows us to arrive at the resulting equation between the value of the function and the variable of the radius of this function.

Transformations with respect to double logarithm, followed by further exponentiation after transformations and repeated logarithm in kind, allow us to arrive at a function relation that becomes complete after being reduced in an algebraic transformation (35)

Taking into account the obtained type of function, as well as the known ratio, it can be noted that in (34) an additional second coefficient was introduced, which took part in (35) and the resulting formula of the radius function. The value of this coefficient can be calculated based on the appropriate type of function, taking into account the fact that the radius is a constant equal to a single astronomical unit, calculations become the simplest and most definite (36).

Thus, the function from (35), taking into account the value of the coefficient (36), takes the form (37) with a single value of the function at a given radius in (38).

Since the type and value of the radius function has been determined, the ratio in (35—36) can be used later to operate with the function of the first angle from a given spherical coordinate system. After converting the ratio, a third additional coefficient is introduced, from which, consequently, a new ordinary differential equation of the second degree is created using trigonometric functions. Subsequently, after the transformation, the operation of integration, exponentiation, logarithmization and transformations with logarithms, which were carried out within the framework of calculations in (35), are applied to the function of the first angle using the general form of this function (39).

With respect to the obtained function of the first angle in the spherical coordinate system, which also depends on the independent constant and the introduced constant, there are also boundary conditions derived from the available empirical data (18). The application of each of them creates 3 forms of the function with the specified values of the angle variable and the value of the function as a whole, while the third form causes the variable to be replaced in the first and further transition from a system with 3 equations to 2 equations, and then, after deducing the function for the independent constant into a single equation. The expression formed in this way, after elementary algebraic transformations, leads to the value of the introduced third coefficient (40), its substitution into the formula of the independent constant (41), which can be substituted into the form of a function (42).

As a result, a uniform form with constants for the first function is obtained, on the basis of which it is possible to continue the given ratio with transformation into the form of an ordinary differential equation of the second degree relative to the second angle. The solution is carried out after the conversion of the function, where all 3 specified constants are enclosed, which are used during the conversion. During the double integration on the left side, due to the fact that the first angle is used as a variable in the square of the sine, double integration relative to the second angle cannot be performed in principle, which is why the first and second independent coefficients appear.

Thus, a relation is created with respect to which natural logarithm is performed, which, after appropriate algebraic operations, leads to a single form of the function with respect to the second angle (43).

Based on the experimental data in (15—17) [1—5; 13—17; 19], boundary conditions can be used [17—19] and, consequently, 3 equations for the second angle, each of which is solvable after converting the third equation and reducing to 2 equations. After applying the substitution method, a single equation is output for the system of equations, after which the value for the first coefficient is calculated, as well as the square of the sine of the first angle relative to the specified boundary conditions, which can then be used in the substitution method, creating a single form of the equation (44).

As a result of the calculations, the function has already been determined in time, the first and second corners, taking into account that the value of the function in radius is equal to one, the general appearance of the function looks according to (45).

The resulting function can describe the energy value taking into account the empirical coefficient, so that a graph of the function (45) can be presented. It is important to note that the function depends on 3 variables – the first and second angles, as well as time, which can be represented as an animation, as well as by a single time.

Results

In this case, the graph is plotted relative to each angle and shape relative to a given time of 4.5 billion. years after the formation of the Sun, which is also used in a given function (Fig. 1—2).

Fig. 1. The first perspective of the constructed three-dimensional graph at the time of 4.5 billion years after the formation of the Sun

Fig. 2. The second perspective of the constructed three-dimensional graph at the time of 4.5 billion years after the formation of the Sun