The mathematical model of the energy process of earthquakes and the construction of an end-to-elemental regulatory solution to a one-dimensional reverse decision of seismic

This article has developed a mathematical model of energy processes of Earthquakes of South Kyrgyzstan. Here, an end-to-its-characterized regulatory solution to a one-dimensional reverse problem of seismic was built here.

The author revealed, from the point of view of the practical task of seismic, the best method is the end-and-playing method, which needs to establish the stability of the solution.

Keywords: the natural phenomenon, earthquake, seismic equations, energy process, of course, regulatory solution, Aikonal method, task, method of highlighting features,inverse task.

Introduction. Earthquake. It is known that an earthquake is tectonic deformations of the earth's crust, due to accumulating stress, which come to the surface of the earth in the form of shocks in different power [2]. With each earthquake in the bowels of the Earth, a certain amount of energy is released, which accumulated constantly and continuously. To identify, recognize natural phenomena preceding earthquakes, monitoring (constant tracking) is carried out behind the time (or invariable) physical conditions in the area under consideration, the characteristics of the waves and deformation of the rocks.

To achieve such goals, it is necessary to develop new effective methods of mathematical modeling [8].

1. Formulation of the task. For the seismic equation, consider the reverse task:

(1)

(2)

Let the solution of the direct task be given, to solve the inverse task,

(3)

Let the conditions be met with respect to the coefficients of the equation (4)

where (5)

Equation (1) is an equation of hyperbolic type, so the task can be considered in the field of ∆(Т) [1]:

. (6)

The inverse task is to determine the Lame coefficient at known values: - — the density of the medium, as well as additional information about solving the direct decision (3).

Let's denote by

The singular and regular part of the solution of the direct decision (1) -(2) will be allocated according to the method of V. G. Romanov (the method of allocating features) for this purpose we present the solution of the problem in the form of [1]:

where - — smooth continuous function, - — Heaviside function,

The last calculations are substituted into equation (1), and we get (8)

Collect the members at the same coefficients and equate them to zero:

Then we will get tasks for

(9)

(10)

(11)

To get:

(12)

If we consider that as well as in the above calculations, we get the following inverse problem with a rectilinear characteristic:

(14)

(15)

The inverse problem (13) — (15) is to define afunction, , with a known function , with a known function - additional information about solving a direct decision.

If we define functions

(16)

we can also define an unknown function .

Using the D'Alembert formula for a direct task (13), (14), we get a solution to a direct problem [7]:

(17)

With (17) get

2. Let's build a finite difference solution. Let's introduce a grid area for solving problem (13)-(15):

where is the grid step on

The difference analogue of the differential equation (13) will be:

(19)

From (19) we get

From the last expressions we can get the recurrence formula [3]:

(21)

The last expressions are substituted sequentially in the right part (21), and also again writing the same recurrent formula and supplying it to (21) and continuing this process we get a difference analogue of the integral formula of Alembert(17): (22)

Suppose that in the last formula (22) and also taking into account formulas (14), we get a difference analogue of the integral formula (18):

The difference formulas (22) and (23) constitute a system of difference nonlinear equations of the second kind.

In the difference analogue (22) we wrote without small quantities Thus, it is possible to obtain for the formula (22) with a small magnitude Denote a solution with a small quantity through and

Then for и

(24)

(25)

Let's introduce notation

(26)

Given these designations from (24) and (25) we will get estimates

(27)

Let then

(29)

From the last formula, using a discrete analogue of Gronulla-Bellman, we get

(30)

The following theorem on the convergence of the finite-difference solution of the difference task (22), (23) to the solution of the differential decision (13) — (15) is proved.

Theorem 1. Let the solution of the differential task(14) (15) exist

Build a finite-difference regularized solution. Let the additional information about solving the direct decision, for solving the inverse task, be given in the form and executed

- a small number. (31)

Then for and

(32)

(33) subtracting from formulas (22)-(23) formulas (32)-(33) we get

Taking into account the introduced norms, we will evaluate the latest equations (34)

(35)

Let than now from the last expressions we get

Then again using the Gronole–Bellman formulas, we get an estimate

(37)

And if we consider that the ratings (30), we have

(38)

The last estimate is an estimate of the regularized solution of the inverse task.

Theorem 2. Let the solution of the direct task(13)-(15) exist and let then the constructed finite-difference solution of the inverse task converge to the exact solution of the inverse task (13)-(15) with the velocity of order

Of course, the difference regularized solution of the original inverse problem (1) — (3) is obtained from formula (16):

. Let's integrate the latter From here

Thus, the finite-difference regularized solution of the inverse problem (1) — (3) has the form:

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