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ABA I. CLASSICAL AND SPECIAL PROBLEM STATEMENTS

WITH FREE BORDERS.

I. General characteristics of problems of mass transfer and diffusion with reaction.

I. Initial boundary value problems for level surfaces of the concentration field. Qualitative effects of diffusion processes accompanied by adsorption and chemical reactions.

I. Finite-time stabilization to stationary, spatially localized solutions.

ABA II. STUDY OF NONLINEAR TRANSFER PROBLEMS AND

DIFFUSION OF PASSIVE IMPURITIES IN STRATIFIED ENVIRONMENTS.

A method for separating variables in a quasilinear parabolic diffusion and transport equation.

Exact solutions to problems of diffusion and transfer from concentrated, instantaneous and permanently acting sources in a medium at rest.

ABA III. MATHEMATICAL MODELS OF DIFFUSION PROCESSES

WITH REACTION.

Rothe method and integral equations of the problem.

Problems with free boundaries in the problem of pollution and self-purification by a point source.

THERATURE.

Introduction of the dissertation (part of the abstract) on the topic "Constructive methods for solving boundary value problems with free boundaries for nonlinear equations of parabolic type"

When studying nonlinear boundary value problems that describe the processes of pollution and recreation of the environment, reflecting, along with diffusion, adsorption and chemical reactions, Stefan-type problems with a free boundary and sources that significantly depend on the desired concentration field are of particular interest.

Nonlinear problems with free boundaries in environmental problems allow us to describe the actually observed localization of pollution processes (recreation) environment. The nonlinearity here is due to both the dependence of the turbulent diffusion tensor K and the pollution effluents / on the concentration c. In the first case, spatial localization is achieved due to degeneracy, when at c = O and K = 0. However, it occurs only in at the moment time g and at g yes is absent.

The evolution of diffusion processes with reaction, stabilizing to limiting stationary states with clearly defined spatial localization, can be described by mathematical models with a special dependence of sinks /(c). The latter models the consumption of matter due to chemical reactions of fractional order, when /(c) = . In this case, regardless of the degeneracy of the diffusion coefficient, there is a spatiotemporal localization of the diffusion disturbance of the medium. At any moment of time /, the locally diffusion disturbance occupies a certain region 0(7), limited in advance by the previously unknown free surface Г(7). The concentration field c(p, /) in this case is a diffusion wave with a front Г(/), propagating through an undisturbed medium, where c = O.

It is quite natural that these qualitative effects can only be obtained on the basis of a nonlinear approach to modeling reaction processes.

However, this approach is associated with significant mathematical difficulties when studying the nonlinear problems with free boundaries that arise here, when a pair of functions must be determined - the concentration field c(p,t) and the free boundary Г(/) = ((p,t): c(p ,t) = O). Such problems, as already noted, belong to more complex, little-studied problems of mathematical physics.

Significantly less research has been carried out for boundary value problems with free boundaries due to their complexity, which is associated both with their nonlinearity and with the fact that they require a priori specification of the topological characteristics of the fields being sought. Among the works that consider the solvability of such problems, it is worth noting the works of A.A. Samarsky, O.A. Oleinik, S.A. Kamenomostkoy, etc. With some restrictions on given functions in the works of A.A. Berezovsky, E.S. Sabinina proved existence and uniqueness theorems for the solution of a boundary value problem with a free boundary for the heat equation.

No less important has development effective methods approximate solution of problems of this class, which will allow us to establish functional dependencies of the main parameters of the process on the input data, making it possible to calculate and predict the evolution of the process under consideration.

Due to the rapid improvement of computer technology, effective numerical methods for solving such problems are increasingly being developed. These include the method of straight lines, the projection-grid method, developed in the works of G.I. Marchuk, V.I. Ogoshkov. IN lately The fixed field method is successfully used, the main idea of which is that a moving boundary is fixed and a part of the known boundary conditions is specified on it, the resulting boundary value problem is solved, and then, using the remaining boundary conditions and the resulting solution, a new, more accurate position of the free boundary is found and etc. The problem of finding the free boundary is reduced to the subsequent solution of a number of classical boundary value problems for ordinary differential equations.

Since problems with free boundaries have not been fully studied, and their solution is associated with significant difficulties, their study and solution requires the involvement of new ideas, the use of the entire arsenal of constructive methods of nonlinear analysis, modern achievements mathematical physics, computational mathematics and the capabilities of modern computer technology. In theoretical terms, for such problems there remain topical issues existence, uniqueness, positivity, stabilization and spatiotemporal localization of solutions.

Dissertation work is devoted to the formulation of new problems with free boundaries, modeling the processes of transfer and diffusion with the reaction of polluting substances in environmental problems, their qualitative study and, mainly, the development of constructive methods for constructing approximate solutions to such problems.

The first chapter gives general characteristics problems of diffusion in active media, that is, media in which effluents significantly depend on concentration. Physically based restrictions on flows are indicated, under which the problem is reduced to the following problem with free boundaries for a quasilinear parabolic equation: с, = div(K(p, t, с) grade) - div(cu) - f (с)+ w in Q (/) ,t> 0, c(p,0) = e0(p) in cm c)grade, n)+ac = accp on S(t), c)gradc,n) = 0 on Г if) , where K(p,t,c) is the turbulent diffusion tensor; ü is the velocity vector of the medium, c(p,t) is the concentration of the medium.

Considerable attention in the first chapter is paid to the formulation of initial boundary value problems for surfaces of the concentration level in the case of directed diffusion processes, when there is a one-to-one correspondence between concentration and one of the spatial coordinates. The monotonic dependence of c(x,y,z,t) on z allows us to transform the differential equation, the initial and boundary conditions of the problem for the concentration field into a differential equation and the corresponding additional conditions for the field of its level surfaces - z = z(x,y,c, t). This is achieved by differentiating the inverse functions, resolving the equation of the known surface S: Ф (x,y,z,t)=0->z=zs(x,y,t) and reading the identity back with(x,y,zs, t)=c(x,y,t). Differential equation (1) for c is then transformed into an equation for z- Az=zt-f (c)zc, where

2 ^ Az=vT (K*t*)-[K-b Vz = lzx + jz +k, VT = V-k- . zc dz

When moving from independent variables x,y,z to the independent variables x>y,c, the physical domain Q(i) is transformed into the nonphysical domain Qc(/), limited by the part of the plane c = 0 into which the free surface Γ goes, and by the generally free unknown surface c=c(x, y,t), into which the known surface S(t) goes.

In contrast to the operator divKgrad ■ of the direct problem, operator A of the inverse problem is essentially nonlinear. The thesis proves the positivity of the quadratic form e+rf+yf-latf-lßrt corresponding to operator A, and thereby establishes its ellipticity, which allows us to consider formulations of boundary value problems for it. By integrating by parts, we obtained an analogue of Green's first formula for the operator A c(x,yt) c(t) cbcdy \uAzdc= Jdc d u(KVTz,n)iï- \\viyrv,VTz)dxdy

Vzf x,y,t) 0 c(x,y,t) - í *

We consider a problem with a free boundary for a concentration field c = c(x,y,z,1), when the Dirichlet condition div(Kgradc) - c, = /(c) - Re g c(P,0) = c0 is specified on the surface (P), ReShto), c = (p(p,0, ReB^), ¿>0, (2)

ReG(4 ¿>0. s = 0, K- = 0, dp

In this case, the transition relative to the level surface r = r(x,y,c^) allowed us to get rid of the free surface c=c(x,y,?), since it is completely determined by the Dirichlet condition c(x,y^) = d >(x,y,rx(x,y^),O- As a result, the following initial-boundary value problem for a strongly nonlinear parabolic operator^ - - in a time-varying but already known domain C2c(0:<9/

Az = z(~zc, x,yED(t), 0 0, z(x,y,c,0) = z0(x,y,c), x,y,cePc(O), z(x, y,c,t) = zs (x, y, c, t), c = c(x, y, t), X, y G D(t), t > 0, zc(x,y,0,t )=-co, x,y&D(t), t> 0.

Here we also study the question of the uniqueness of the solution to problem (3). Based on the obtained analogue of Green's first formula for the operator A, taking into account the boundary conditions after elementary but rather cumbersome transformations using Young's inequality, the monotonicity of the operator A on the solutions zx and z2 of the problem is established

Ar2 - Ar1)(r2 -)(bcc1us1c< 0 . (4)

On the other hand, using a differential equation, boundary and initial condition it is shown that

The resulting contradiction proves the uniqueness theorem for the solution of the Dirichlet problem for concentration level surfaces c(x,y,t)

Theorem 1. If the source function w is const, the sink function f(c) increases monotonically and /(0) = 0, then the solution to the Dirichlet problem (2) for level surfaces is positive and unique.

The third paragraph of the first chapter discusses the qualitative effects of diffusion processes accompanied by adsorption and chemical reactions. These effects cannot be described based on linear theory. If in the latter the propagation speed is infinite and thus there is no spatial localization, then the considered nonlinear models diffusion with reaction with the functional dependences of the coefficient of turbulent diffusion K and the density of effluents (kinetics) established in the work chemical reactions) / on concentration c allow us to describe the actually observed effects of the finite speed of propagation, spatial localization and stabilization over a finite time (recreation) of pollutants. The work established that the listed effects can be described using the proposed models if there is an improper integral with w 1

K(w)dzdt = -\Q(t)dt, t>0;

00 dc с(сс^) = 0,К(с)- = 0, z = oo,t>0. dz

The stationary problem in coordinate-free form has the form div(K(c)grade) = f(c) in Q\P (0< с < оо},

K(cgradc,n)) + ac = 0 on 5 = 5Q П Ж, (7) с = 0, (К(с) grade,п) = 0 on Г s (с = 0) = dQ. P D,

JJJ/(c)dv + cds = q. a s

In a semi-neighborhood with eQ of the point Pe Г, the transition to the semi-coordinate form of notation made it possible to obtain the Cauchy problem drj

K(c) dc dt] divT (K(c)gradTc) = f(c) in co rj<0

8) dc c = 0, K(c)~ = 0.77 = 0,

OT] where m] is the coordinate measured along the normal to Γ at point P, and the other two Cartesian coordinates m1, m2 lie in the tangent plane to Γ at point P. Since in co we can assume that c(m1, m2, g/) weakly depends on the tangential coordinates, that is, c(tx, t2,1]) = c(t]), then to determine c(t]) from (8) the Cauchy problem drj drj f(c), TJ follows< О, dc c = 0, K(c) - = 0,77 = 0. drj

An exact solution to the problem has been obtained (9)

77(s)= redo 2 s [ o s1m?< 00 (10) и доказана следующая теорема

Theorem 2. A necessary condition for the existence of a spatially localized solution to the nonlocal problems with free boundaries under consideration is the existence of an improper integral (b).

In addition, it has been proven that condition (6) is necessary and sufficient 1 for the existence of a spatially localized solution to the following one-dimensional stationary problem with a free boundary r(c), 0

00 O tsk = ^- si) o 2 c1c c(oo) = 0, K(c)- = 0, g = oo, c1g that is, it takes place

Theorem 3. If the function /(c) satisfies the conditions f(c) = c ^ , ^< // < 1, при с-» О, а К{с)-непрерывная положительная функция, то при любом д>0 a positive solution to the nonlocal boundary value problem (11) exists and is unique.

Here we also consider issues of environmental recreation in a finite time that are very important for practice. In the works of V.V. Kalashnikov and A.A. Samarsky, using comparison theorems, this problem is reduced to solving the differential inequality -< -/(с), где с - пространственно однородное (т.е. не зависящие от коей1 ординаты) решение.

At the same time, for recreation time the estimate w

T<]. ск х)

In contrast to these approaches, the thesis made an attempt to obtain more accurate estimates that would take into account the initial distribution of concentration co (x) and its carrier “(0). For this purpose, using a priori estimates obtained in the work, a differential inequality was found for the squared norm of the solution Ж

13) from which a more accurate estimate for T t follows<

1+ /?>(())] where c is the root of the equation

Уг^-Р)/ с /1 =(р, = КМГ > = ^-Ш+Р)^1 ■

The second chapter is devoted to the issues of modeling the processes of transfer and diffusion of passive impurities in stratified media. The starting point here is problem (1) with /(c) = 0 and the Dirichlet boundary condition or nonlocal condition c, = (I\(K(p,G,c)%gais)-0 c(p,0) = c0( p) in 0(0),

C(P>*) = φ(р,0 on or = ()((), с(р, Г) = 0, (К(р^, с)%?аес,н) = 0 on Г(Г ).

One-dimensional problems of turbulent diffusion are considered, taking into account the dependence of the diffusion coefficient on scale, time and concentration. They represent local and nonlocal problems for the quasilinear ds equation

1 d dt g"-1 dg p-\

K(r,t,c) ds dg p = 1,2,3,

16) where K(r,t,c) = K0(p(t)rmck; Birkhoff in the form c(r,t) = f(t)B(T1), tj = r7t P>0,

17) where the functions and parameter p are determined in the process of separating variables in (16). As a result, we obtained an ordinary differential equation for B(t]) at] and the representation

Оn+m+p-2)/pBk £® drj

C.B-ij-dtl, oh

For two values of an arbitrary constant C( - C, = and

С1 = ^Ур equation (18) allows exact solutions depending on one arbitrary constant. The latter can be determined by satisfying one or another additional conditions. In the case of the Dirichlet boundary condition c(0,0 = B0[f^)]"p/p (20), an exact spatially localized solution is obtained in the case k > 0, m< 2:

2-t Gf\h;

L/k 0<г <гф(/),

Vd^0(2-m\ p = pk + 2-m, and the exact non-localized solution in the case of k<0, т <2:

1/k 0< г < 00.

22) = [k^2 - t)/?/^1 p = 2-t- p\k\.

Here f(1) = \(p(r)yt; gf (/) = [^(O]^ o

For k -» 0, from the obtained solutions follows the solution of the linear problem c(r,0 = VySht-t) exp[- /(1 - m)2k0f(1)\, which, for f(1) = 1 and m = 0, is transformed into the fundamental solution of the diffusion equation.

Exact solutions were also obtained in the case of instantaneous or permanently acting concentrated sources, when an additional nonlocal boundary condition of the form

23) where o)n is the area of the unit sphere (co1 = 2, a>2 = 2i, a>3 = 4z).

The found exact solutions for k >0 of the form (21) represent a diffusion wave propagating through an undisturbed medium with a finite speed. At k< О такой эффект пространственной локализации возмущения исчезает.

Problems of diffusion from constantly acting point and linear sources in a moving medium are considered, when a quasi-linear equation is used to determine the concentration

Vdivc = -^S(r),

24) where K(g,x,s) = K0k(x)gtsk, 8(g) is the Dirac delta function, O is the power of the source. The interpretation of the coordinate x as time/ also made it possible here to obtain exact partial solutions to a nonlocal problem of the form (21) r 2/(2+2 k) 2 o, 1

2С2 (2 + 2к)К0 к

Solution (25) makes it possible in principle to describe the spatial localization of a diffusion disturbance. In this case, the front of the diffusing wave is determined, separating the regions with zero and non-zero concentrations. For k -» 0, it implies the well-known Roberts solution, which, however, does not allow one to describe spatial localization.

The third chapter of the dissertation is devoted to the study of specific problems of diffusion with reaction in a stratified air environment, which is the following one-dimensional problem with a free boundary uxx-ut = / (u), 0< х < s(t), t>O, u(x,0) = Uq(X), 0< х < 5(0), (26) ux-hu = -h(p, х = 0, t >0, u = 0, their = 0, x = s(t), t > 0.

A numerical-analytical implementation of problem (26) was carried out, based on the Rothe method, which made it possible to obtain the following seven-digit approximation of the problem in the form of a system of boundary value problems for ordinary differential equations with respect to the approximate value u(x) = u(x,1k), and 5 =) V u(x)-u(x^k1): V u"-m~xy = y - m~1 u, 0< х < 5, и"-ки = х = 0, (27) ф) = 0 |ф) = 0.

Solution (27) is reduced to nonlinear integral equations of the Volterra type and a nonlinear equation for x = 0 5 u(x) ~ 4m [i/r-^--* s/r + k^tek -¿r n V l/ g l/g

0 < X < 5, к(р.

For numerical calculations, solving system (28) using finite-dimensional approximation is reduced to finding solutions to a system of nonlinear algebraic equations with respect to the nodal values and. = u(x)) and i-.

Problems with free boundaries in the problem of pollution and self-purification of the atmosphere by point sources are also considered here. In the absence of an adsorbing surface 5(0 (tie&3 = 0) in the case of flat, cylindrical or point sources of pollution, when the concentration depends on one spatial coordinate - the distance to the source and time, the simplest one-dimensional nonlocal problem with a free boundary is obtained

-- = /(s), 00, dt gp~x 8g \ 8g, f,0) = 0, 00; ah

1 I bg + /(c) Г~1£/г=- (30) о о ^ ; ^

The construction of a solution to problem (29), (30) was carried out by the Rothe method in combination with the method of nonlinear integral equations.

By transforming the dependent and independent variables, the nonlocal problem with a free boundary about a point source is reduced to the canonical form

5l:2 8t u(x,0) = 0, 0< л; < 5(0), (5(0) = 0), (31) м(5(г),т) = мх(5(т),т) = 0,

Pmg + = d(r), m > 0, containing only one function defining the function d(r).

In particular cases, exact solutions of the corresponding nonlocal stationary problems with a free boundary for the Emden-Fowler equation with 12 and 1 in l are obtained

2=х иН, 0<Х<5, с!х ф) = м,(5) = 0, \х1~/*и1*сЬс = 4. (32) о

In particular, when /? = 0 m(l:) = (1/6)(25 + x)(5-x)2, where* = (Зз)1/3.

Along with the Rothe method, in combination with the method of nonlinear integral equations, the solution to the nonstationary problem (32) is constructed by the method of equivalent linearization. This method essentially uses the construction of a solution to a stationary problem. As a result, the problem is reduced to the Cauchy problem for an ordinary differential equation, the solution of which can be obtained by one of the approximate methods, for example, the Runge-Kutta method.

The following results are submitted for defense:

Study of qualitative effects of spatiotemporal localization;

Establishing the necessary conditions for spatial localization to limiting stationary states;

Theorem on the uniqueness of the solution to a problem with a free boundary in the case of Dirichlet conditions on a known surface;

Obtaining by separation of variables exact spatially localized families of partial solutions of degenerate quasilinear parabolic equations;

Development of effective methods for the approximate solution of one-dimensional non-stationary local and non-local problems with free boundaries based on the application of the Rothe method in combination with the method of integral equations;

Obtaining accurate spatially localized solutions to stationary diffusion problems with reaction.

Conclusion of the dissertation on the topic "Mathematical Physics", Doguchaeva, Svetlana Magomedovna

The main results of the dissertation work can be formulated as follows.

1. Qualitatively new effects of spatio-temporal localization have been studied.

2. The necessary conditions for spatial localization and stabilization to limiting stationary states have been established.

3. A theorem on the uniqueness of the solution to the problem with a free boundary in the case of Dirichlet conditions on a known surface is proven.

4. Using the method of separation of variables, exact spatially localized families of partial solutions of degenerate quasilinear parabolic equations were obtained.

5. Effective methods have been developed for the approximate solution of one-dimensional stationary problems with free boundaries based on the application of the Rothe method in combination with the method of nonlinear integral equations.

6. Exact spatially localized solutions to stationary problems of diffusion with reaction were obtained.

Based on the variational method in combination with the Rothe method, the method of nonlinear integral equations, effective solution methods have been developed with the development of algorithms and programs for numerical calculations on a computer, and approximate solutions of one-dimensional non-stationary local and non-local problems with free boundaries have been obtained, allowing one to describe spatial localization in pollution problems and self-purification of stratified water and air environments.

The results of the dissertation work can be used in formulating and solving various problems of modern natural science, in particular metallurgy and cryomedicine.

CONCLUSION

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Please note that the scientific texts presented above are posted for informational purposes only and were obtained through original dissertation text recognition (OCR). Therefore, they may contain errors associated with imperfect recognition algorithms. There are no such errors in the PDF files of dissertations and abstracts that we deliver.

Doguchaeva, Svetlana Magomedovna AUTHOR

candidate of physical and mathematical sciences ACADEMIC DEGREE

Nalchik PLACE OF PROTECTION

2000 YEAR OF PROTECTION

01.01.03 RF Higher Attestation Commission CODE

RGB Lach

rights of hands

Doguchaeva Svetlana Magomedovna

Constructive methods for solving boundary value problems with free boundaries for nonlinear equations of parabolic type

Specialty 01.01.03 - Mathematical physics

dissertation for the degree of candidate of physical and mathematical sciences

Nalchik -

The work was carried out at the Kabardino-Balkarian State University named after. HM. Berbekov and the Institute of Mathematics HAH of Ukraine.

Scientific supervisor: Doctor of Physics and Mathematics

Sciences, Professor Berezovsky A.A.

Official opponents: Doctor of Physics and Mathematics

Sciences, Professor Shogenov V.Kh. Candidate of Physical and Mathematical Sciences, Associate Professor Bechelova A.R.

Leading organization: Research Institute

Applied Mathematics and Automation KBSC RAS

The defense will take place on December 28, 2000. at 1022 hours at a meeting of the specialized Council K063.88.06 at the Kabardino-Balkarian State University at the address:

360004, Nalchik, st. Chernyshevsky, 173.

The dissertation can be found in the KBSU library.

Scientific secretary DS K063.88.06 Ph.D. Kaygermazov A.A.

General characteristics of work

Relevance of the topic. When studying nonlinear boundary value problems that describe the processes of pollution and recreation of the environment, reflecting, along with diffusion, adsorption and chemical reactions, Stefan-type problems with a free boundary and sources that significantly depend on the desired concentration field are of particular interest. In theoretical terms, the issues of existence, uniqueness, stabilization and spatial localization of solutions remain relevant for such problems. In practical terms, the development of effective numerical and analytical methods for solving them seems especially important.

The development of effective methods for approximate solution of problems of this class makes it possible to establish functional dependencies of the main parameters of the process on the input data, making it possible to calculate and predict the evolution of the process under consideration.

Among the works that consider the solvability of Stefan-type problems with a free boundary, noteworthy are the works of A.A. Samarsky, O.A. Oleinik, S.A. Kamenomostkoy, L.I. Rubenstein and others.

Purpose of the work. The purpose of this dissertation is to study problems with free boundaries in a new formulation that models the processes of transfer and diffusion, taking into account the reaction of pollutants in environmental problems; their qualitative research and, mainly, the development of constructive methods for constructing approximate solutions to the problems posed.

General research methods. The results of the work were obtained using the Birkhoff method of separation of variables, the method of nonlinear integral equations, the Rothe method, as well as the equivalent linearization method

Scientific novelty and practical value. Statements of problems such as the Stefan problem studied in the dissertation are considered for the first time. For this class of problems, the following main results were obtained for defense:

1. Qualitatively new effects of spatio-temporal localization have been studied

2. The necessary conditions for spatial localization and stabilization to limiting stationary states have been established,

The results of the dissertation work can be applied in formulating and solving various problems of modern natural science, in particular metallurgy and cryomedicine, and seem to be very effective methods for forecasting, for example, the air environment.

Approbation of work. The main results of the dissertation were reported and discussed at the seminar of the Department of Mathematical Physics and Theory of Nonlinear Oscillations of the Institute of Mathematics of the HAH of Ukraine and the Department of Mathematical Physics of Taras Shevchenko University of Kyiv, at the International Conference "Nonlinear Problems of Differential Equations and Mathematical Physics" (August 1997, Nalchik), at seminar of the Faculty of Mathematics of Kabardino-Balkarian State University on mathematical physics and computational mathematics.

Structure and scope of work. The dissertation consists of an introduction, three chapters, a conclusion and a list of cited literature containing 82 titles. Scope of work:

It is 96 pages typed in Microsoft Office 97 environment (Times Roman style).

The introduction substantiates the relevance of the topic, formulates the purpose of the research, provides a brief overview and analysis of the current state of the problems that are studied in the dissertation, and provides an annotation of the results obtained.

с, = div(K(p,t,c)gradc)~ div(cu)- f(c) + w in Q(i), t > 0, сИ = с0ИвП(0)

(K(p,t,c)-grad(c,n))+ac - accp on S(t), (1)

c(p,t) = 0, (K(p,t,c) grad(c,n)) = 0 on T(i),

where K(p,t,c) is the turbulent diffusion tensor; and is the velocity vector of the medium, c(p,t) is the concentration of the medium.

Considerable attention in the first chapter is paid to the formulation of initial boundary value problems for surfaces of the concentration level in the case of directed diffusion processes, when there is a one-to-one correspondence between concentration and one of the spatial coordinates. The monotonic dependence c = c(x,y, z,t) on z allows us to transform the differential equation, the initial and boundary conditions of the problem for the concentration field into a differential equation and the corresponding additional conditions for the field of its level surfaces z = z(x,y,c ,t) .This is achieved by differentiating inverse functions, resolving the equation of a known surface S:<$>(x,y,z,t) = 0 functions, resolution of the equation of the known surface S: y, z, t) = 0 -» z = zs (x, y, t) and inverse pro-

reading the identity c(x,y,r5^)=c(x,y^). Differential equation (1) for C is then transformed into an equation for r - Ar - r, - /(c)rc,

where Ar = Ym(K-Ugg)-

Yr = rx1 + r y] + k,

When moving from independent variables x, y, z to independent variables x, y, c, the physical domain is transformed into a non-physical domain limited by part

the plane c=O, into which the free surface Г goes, and the generally free unknown surface c=c(x,y,1), into which the known surface 5(1) goes.

In contrast to the operator cYu^ac1c of the direct problem, the operator A of the inverse problem is essentially nonlinear. The thesis proves the positivity of the quadratic equation corresponding to operator A

form +m]2 +y£2 -2a^ - 2/3m]^ and thus its ellipticity is established, which allows us to consider problems for it in this formulation. By integrating by parts, we obtained an analogue of Green's first formula for the operator A

c(x,y,1) c(0

jjdxdy |and Azdc-

We consider a problem with a free boundary for a concentration field c = c(x, y, 1,1), when the Dirichlet condition is specified on the surface £(£)

diviK.grayc) - c, = /(c) - c>, Re * > O c(P,0) = co(P), ReI(0),

c =

с = 0, K- = 0, PeY(t), t> О ôn

In this case, the transition relative to the level surface z = z(x,y,c,о) allowed us to get rid of the free surface c = c(x, y,t), since it is completely determined by the Dirichlet condition c(x,y,0 =

known area: Qc(i) :

Az = z, - (/(с) -w(z)]zc x,yeD(t), 0<с O, z(x,y,c,0) = Zq (x,y,c), x,ye D(t), (3)

z(x,y,c,t) = zs(x,y,c,t), c = c(x,y,t), x,y e D(t), t> 0, zc(x,y ,0,0 = -°°, x,yeD(t), t> 0,

Here we also examine the question of the uniqueness of the solution to problem (3).

The following theorem holds

Theorem 1. If the source function W = COïlSt, the sink function f(c) increases monotonically and /(o) = 0, then the solution to the Dirichlet problem (2) for level surfaces is positive and unique.

The third paragraph of the first chapter discusses the qualitative effects of diffusion processes accompanied by adsorption and chemical reactions. These effects cannot be described based on linear theory. If in the latter the propagation speed is infinite and thus there is no spatial localization, then the nonlinear models of diffusion with reaction under consideration, with the functional dependences of the turbulent diffusion coefficient K and the effluent density (kinetics of a chemical reaction) f on the concentration c established in the work, make it possible to describe the actually observed effects of co-reaction.

finite speed of spread, spatial localization and stabilization over a finite time (recreation) of pollutants. The work established that the listed effects can be described using the proposed models if there is an improper integral

¡K(w)~2dw< оо (4)

The corresponding (1) nonlocal initial-boundary value problem with d - O is considered

ffed^ 1 Ac), o o,

oz\ oz) at c(z,0) = 0, 0< z < то, /00 / \\\ct+f{c)\lzdt = -\Q{t)dt, t>0; 00 0 dc

c( ,t) = 0, K(c)- = 0, z =°o>0. dz

The stationary problem in coordinate-free form has the form: div(K(c) grade) = f(c) in Q \ P (0< с < да},

(.K(c)grad(c,n))+ac = 0 on S = dQf)dD, (5) c = 0, (K(c)grad(c,n)) = 0 on Г=(с = 0) = aoP£>, jff/(c)dv + afj cds = Q.

In a semi-neighborhood of the point P e G, the transition to the semi-coordinate form of notation made it possible to obtain the Cauchy problem

Divx(K(c)gradTc) = /(c) in (O (^<0),(6)

c = 0, K(c)- = 0.7 = 0.07

where 17 is the coordinate measured along the normal R to Γ at point P, and the other two Cartesian coordinates r, r2 lie in the tangent plane to Γ at point P. Since in o we can assume that c(r, r2 μ) weakly depends from tangential coordinates, that is

c(r,m2 Г]) = c(t]), then to determine c(//) from (6) the Cauchy problem follows

Ad- =/(c), g|<0,

c = o, ad-=0.7 = 0.

An exact solution to problem (7) is obtained.

77(s) = |l:(i>) 21 K(y)/(y)<ь (8)

o |_ 0 and the following theorem is proven

Theorem 2. A necessary condition for the existence of a spatially localized solution to the considered nonlocal problems with free boundaries is the existence of an improper integral (4).

In addition, it has been proven that condition (4) is necessary and sufficient for the existence of a spatially localized solution to the following nonlocal stationary problem with a free boundary:

0 < г < оо,

c(oo) = 0, DG(c)-= 0, g

that is, it takes place

Theorem 3. If the function f(c) satisfies the conditions f(c) = c2/M, V2 0, and K(c) is a continuous positive function, then for any Q> O a positive solution to the nonlocal boundary value problem (9) exists and is unique.

Here we also consider issues of environmental recreation in a finite time that are very important for practice. In the works of V.V. Kalashnikov (1974) and A.A. Samarsky (1982) with the help of comparison theorems, this problem is reduced to solving the differential inequality

- < -/(с), где с - пространственно однородное (т.е. не завися-dt

depending on the coordinate) solution. At the same time, the estimate for recreation time was obtained

In contrast to these approaches, the thesis made an attempt to obtain more accurate estimates that would take into account the initial distribution of the concentration of CD (x) and its carrier 5(0).

For this purpose, using a priori estimates obtained in the work, a differential inequality was found for the squared norm of the solution

from which follows a more accurate estimate for T

T< ,(1+/?жо)

where c is the root of the equation

"(1 -ru2lUg

2_0-/у с /2 =<р,

y(t) HkMI2 , s(0) = ~-p(l + /))c

The second chapter is devoted to the issues of modeling the processes of transfer and diffusion of passive impurities in stratified media. The starting point here is problem (1) with /(c) 3 O and the Dirichlet boundary condition or the nonlocal condition ct = div(K(p,t,c)gradc) - div(cü) + с in Q(t), t> ABOUT

с(р,0) = со(р) in OD,

c(p,t) = q>(p,t) on S(t) or jc(p,t)dv = Q(t), (13)

c(p,t) = O, (K(p,t,c)grad(c,n)) = 0 on Г(0) One-dimensional problems of turbulent diffusion are considered taking into account the dependence of the diffusion coefficient on scale, time and concentration. They represent local and nonlocal problems for the quasilinear equation

where K(g,(,c) =K0<р(()гтс1!; <р^) - произвольная функция;

K0, m and k are some constants. Particular solutions of this equation are sought by the method of separation of variables in the form

c(r,t) = f(t)B(rj), р>О,

where the functions /(/),5(r]),φ(/) and the parameter p are determined in the process of separating variables in (14). As a result, an ordinary differential equation for B(t]) was obtained

and presentations

c(r,t)^(t)f B(rj), =

meaning

arbitrary

constant

C, - Cx and Cx = (t ^/equation (16) allows for exact

ny solutions depending on one arbitrary constant. The latter can be determined by satisfying certain additional conditions. In the case of the Dirichlet boundary condition

с(0.0 = В0[ф(0]У* (18)

an exact spatially localized solution was obtained in the case k>0,m<2:

t)0 = [v*K0(2 - t)p / k]P"(2~t\ p = pk + 2-t.

and the exact non-localized solution in the case of<0, т<2:

0<г<гф(0 , гД0<г<со

s(r,1)=В«Ш-п

ABOUT< Г < 00. (20)

u = [k0(2-t)r/vU1|4"(2_t)5 R = 2-t-p\k[

Here= |f(t)s1t; gf (/) = . When k 0 from received-

ny solutions follows the solution of the linear problem

cM = vM) G/(1"t) exp[- g2- /(1 - t)gK^)\

which, for φ(() = 1 and m - 0, is transformed into the fundamental solution of the diffusion equation.

Exact solutions were also obtained in the case of instantaneous or permanently acting concentrated sources, when an additional nonlocal boundary condition of the form

Q =

where son is the area of a unit sphere (i>1 = 2, eog = 27u, o)b = 4l").

The exact solutions found for k > O of the form (19) represent a diffusion wave propagating through an undisturbed medium with a finite speed. At k< 0 такой эффект пространственной локализации возмущения исчезает.

where K(r,x,c) = KcK(x)gtsk, ô(r)~ Dirac delta function; Q-source power. The interpretation of the coordinate X as time / also made it possible to obtain exact partial solutions for (22)

0<г <гф(х), Гф(х)<Г< 00,

" 2Скг(2 + 2к)Кь ko

lky(2 + 2ku

Solution (23) makes it possible in principle to describe the spatial localization of a diffusion disturbance. In this case, the front of the diffusing wave is determined, separating the regions with zero and non-zero concentrations. For k -> 0, it implies the well-known Roberts solution, which, however, does not allow one to describe spatial localization.

theirx~u1=/(u)> 0< лт < £(/), />0,

u(x,0) = u0(x), 0<х< 5(0), (24)

their -II = ~)r<р, х = 0, ¿>0,

u- 0, their= 0, x = ¿>0.

A numerical and analytical implementation of problem (24) was carried out, based on the Rothe method, which made it possible to obtain the following approximation of the problem in the form of a system of boundary value problems for ordinary differential equations with respect to the approximate value u(x) = u(x^k), and

u(x) = u(x,1k_)):

u"-t~1u = ir - r"1u, 0< дг <

u"-Ui = -bср, x = 0, (25)

n(l) = 0 n"O) = 0.

The solution to problem (25) is reduced to the nonlinear Volterra integral equations

u(x) - l/t ¡зИ-^

For numerical calculations, solving (26), (27) using finite-dimensional approximation is reduced to finding solutions to a system of nonlinear algebraic equations with respect to the nodal values u] = u(x]) a sj.

Problems with free boundaries in the problem of pollution and self-purification of the atmosphere by point sources are also considered here.

by precisionists. In the absence of an adsorbing surface S(t) (mesS = 0) in the case of flat, cylindrical or point sources of pollution, when the concentration depends on one spatial coordinate - distance to the source and time, the simplest one-dimensional nonlocal problem with a free boundary is obtained

-^=/(s),0<г<гф(0,">0,

1 d f „_, 8 s

g""1 dg( dgu

c(r,0) = 0, 0< г < (0) (28)

с(r,0 = 0, - = 0, r = gf(0, t> 0;

2--- = xx~rir, 0<л 0,

I 1 T + - \QiDdt (29)

The solution to problem (28), (29) was constructed using the Rothe method in combination with the method of nonlinear integral equations.

By transforming the dependent and independent variables, the nonlocal problem with a free boundary about a point source is reduced to canonical form

u(x,0) = 0, 0<л; <5(0), (5(0) = 0), (30)

m(5(g),g) = m;s(5(g),g) = 0, g>0

In particular cases, exact solutions of the corresponding nonlocal stationary problems with a free boundary for the Emden-Fowler equation are obtained

■ xx~ßuß, 0

u(s) = ux($) = 0, Jjf2 pußdx = q

] = (1 / 6)(2 s + x)(s -x)r, where

Along with the Rothe method in combination with the method of integral equations, the solution to the non-stationary problem (31) is constructed by the method of equivalent linearization. This method essentially uses the construction of a solution to a stationary problem. As a result, the problem is reduced to the Cauchy problem for an ordinary differential equation, the solution of which can be obtained by one of the approximate methods, for example, the Runge-Kutta method.

1. Berezovsky A.A., Doguchaeva S.M. Spatial localization and stabilization in diffusion processes with reaction //Dopovda HAH Decoration. -1998. -No. 2. -WITH. 1-5.

2. Berezovsky N.A., Doguchaeva S.M. Stefan's problems in the problem of pollution and self-purification of the environment by point sources // Nonlinear boundary value problems of mathematical physics and their applications. - Kyiv: Institute of Mathematics HAH of Ukraine, 1995. -

3. Berezovska JI.M., Doguchaeva S.M. D1r1hle problem for the top r1vrya of the concentration field // Mathematical methods in scientific and technical advances - Kshv: Institute of Mathematics HAH Ukrashi, 1996.-P.9-14.

4. Berezovsky A.A., Doguchaeva S.M. Mathematical model of obstruction and self-purification of the otuchuny middle point by point dzherel //Problems with free boundaries and nonlocal problems for nonlinear parabolic equations. - Kyiv: Institute of Mathematics HAH of Ukraine, 1996. P.13-16.

5. Doguchaeva S.M. Free boundary problems in environmental problems // Nonlinear boundary value problems Math. physics and their applications - Kyiv: Inst. Mathematics HAH of Ukraine, 1995.-

6. Doguchaeva Svetlana M., Berezovsky Arnold A. Mathematical models of scattering, decomposition and sorption of gas, smoke and other kinds of pollution in a turbulent atmosphere // International Conference Nonlinear Differential Eguations, Kiev, August 21-27, 1995, p. 187.

7. Doguchaeva S.M. Spatial localization of solutions to boundary value problems for a degenerate parabolic equation in an environmental problem // Nonlinear boundary value problems Math. Physicists and their applications.-Kyiv: Institute of Mathematics HAH of Ukraine,

1996.-S. 100-104.

8. Doguchaeva S.M. One-dimensional Cauchy problem for level surfaces of the concentration field //Problems with free boundaries and nonlocal problems for nonlinear parabolic equations. -Kiev: Institute of Mathematics HAH of Ukraine, 1996 - P. 27-30.

9. Doguchaeva S.M. Qualitative effects of diffusion and mass transfer processes, accompanied by adsorption and chemical reactions // Nonlinear problems of differential equations and mathematical physics. -Kiev: Institute of Mathematics,

1997,-S. 103-106.

10. Doguchaeva S.M. Problems with free boundaries for a degenerate parabolic equation in the environmental problem //Dopovts HAH Decorations. - 1999. - No. 12 - P.28-29.

ABA I. CLASSICAL AND SPECIAL PROBLEM STATEMENTS

WITH FREE BORDERS.

I. General characteristics of problems of mass transfer and diffusion with reaction.

I. Initial boundary value problems for level surfaces of the concentration field. Qualitative effects of diffusion processes accompanied by adsorption and chemical reactions.

I. Finite-time stabilization to stationary, spatially localized solutions.

ABA II. STUDY OF NONLINEAR TRANSFER PROBLEMS AND

DIFFUSION OF PASSIVE IMPURITIES IN STRATIFIED ENVIRONMENTS.

A method for separating variables in a quasilinear parabolic diffusion and transport equation.

Exact solutions to problems of diffusion and transfer from concentrated, instantaneous and permanently acting sources in a medium at rest.

ABA III. MATHEMATICAL MODELS OF DIFFUSION PROCESSES

WITH REACTION.

Rothe method and integral equations of the problem.

Problems with free boundaries in the problem of pollution and self-purification by a point source.

THERATURE.

Introduction dissertation in mathematics, on the topic "Constructive methods for solving boundary value problems with free boundaries for nonlinear equations of parabolic type"

Nonlinear problems with free boundaries in environmental problems make it possible to describe the actually observed localization of environmental pollution (recreation) processes. The nonlinearity here is due to both the dependence of the turbulent diffusion tensor K and the pollution effluents / on the concentration c. In the first case, spatial localization is achieved due to degeneracy, when at c = O and K = 0. However, it occurs only at a given moment of time r and is absent at z.

It is quite natural that these qualitative effects can only be obtained on the basis of a nonlinear approach to modeling reaction processes.

Equally important is the development of effective methods for approximate solution of problems of this class, which will make it possible to establish functional dependencies of the main parameters of the process on the input data, making it possible to calculate and predict the evolution of the process under consideration.

Due to the rapid improvement of computer technology, effective numerical methods for solving such problems are increasingly being developed. These include the method of straight lines, the projection-grid method, developed in the works of G.I. Marchuk, V.I. Ogoshkov. Recently, the fixed field method has been successfully used, the main idea of which is that a moving boundary is fixed and a part of the known boundary conditions is set on it, the resulting boundary value problem is solved, and then, using the remaining boundary conditions and the resulting solution, a new, more accurate position is found free boundary, etc. The problem of finding the free boundary is reduced to the subsequent solution of a number of classical boundary value problems for ordinary differential equations.

Since problems with free boundaries have not been fully studied, and their solution is associated with significant difficulties, their research and solution requires the involvement of new ideas, the use of the entire arsenal of constructive methods of nonlinear analysis, modern achievements of mathematical physics, computational mathematics and the capabilities of modern computing. technology. In theoretical terms, the questions of existence, uniqueness, positivity, stabilization, and spatiotemporal localization of solutions remain relevant for such problems.

The dissertation work is devoted to the formulation of new problems with free boundaries that model the processes of transport and diffusion with the reaction of polluting substances in environmental problems, their qualitative study and, mainly, the development of constructive methods for constructing approximate solutions to such problems.

The first chapter provides a general description of diffusion problems in active media, that is, media in which effluents significantly depend on concentration. Physically based restrictions on flows are indicated, under which the problem is reduced to the following problem with free boundaries for a quasilinear parabolic equation: с, = div(K(p, t, с) grade) - div(cu) - f (с)+ w in Q (/) ,t> 0, c(p,0) = e0(p) in cm c)grade, n)+ac = accp on S(t), c)gradc,n) = 0 on Г if) , where K(p,t,c) is the turbulent diffusion tensor; ü is the velocity vector of the medium, c(p,t) is the concentration of the medium.

2 ^ Az=vT (K*t*)-[K-b Vz = lzx + jz +k, VT = V-k- . zc dz

When passing from independent variables x, y, z to independent variables x>y, c, the physical region Q(i) is transformed into the non-physical region Qc(/), limited by the part of the plane c = 0, into which the free surface Г passes, and free in in the general case, an unknown surface c=c(x,y,t), into which the known surface S(t) goes.

Vzf x,y,t) 0 c(x,y,t) - í *

ReG(4 ¿>0. s = 0, K- = 0, dp

Az = z(~zc, x,yED(t), 0 0, z(x,y,c,0) = z0(x,y,c), x,y,sePc(O), z(x, y,c,t) = zs (x, y, c, t), c = c(x, y, t), X, y G D(t), t > 0, zc(x,y,0,t)=-co, x,y&D(t), t> 0 .

Ar2 - Ar1)(r2 -)(bcc1us1c< 0 . (4)

On the other hand, using the differential equation, boundary and initial conditions it is shown that

The resulting contradiction proves the uniqueness theorem for the solution of the Dirichlet problem for concentration level surfaces c(x,y,t)

Theorem 1. If the source function w is const, the sink function f(c) increases monotonically and /(0) = 0, then the solution to the Dirichlet problem (2) for level surfaces is positive and unique.

The third paragraph of the first chapter discusses the qualitative effects of diffusion processes accompanied by adsorption and chemical reactions. These effects cannot be described based on linear theory. If in the latter the speed of propagation is infinite and thus there is no spatial localization, then the nonlinear models of diffusion with reaction under consideration, with the functional dependences of the coefficient of turbulent diffusion K and the density of effluents (kinetics of chemical reactions) / on concentration c established in the work, make it possible to describe the actually observed effects of a finite speed of propagation , spatial localization and stabilization over a finite time (recreation) of pollutants. The work established that the listed effects can be described using the proposed models if there is an improper integral with w 1

K(w)dzdt = -\Q(t)dt, t>0;

00 dc с(сс^) = 0,К(с)- = 0, z = oo,t>0. dz

The stationary problem in coordinate-free form has the form div(K(c)grade) = f(c) in Q\P (0< с < оо},

K(cgradc,n)) + ac = 0 on 5 = 5Q П Ж, (7) с = 0, (К(с) grade,п) = 0 on Г s (с = 0) = dQ. P D,

JJJ/(c)dv + cds = q. a s

In a semi-neighborhood with eQ of the point Pe Г, the transition to the semi-coordinate form of notation made it possible to obtain the Cauchy problem drj

K(c) dc dt] divT (K(c)gradTc) = f(c) in co rj<0

8) dc c = 0, K(c)~ = 0.77 = 0,

An exact solution to the problem has been obtained (9)

77(s)= redo 2 s [ o s1m?< 00 (10) и доказана следующая теорема

Theorem 2. A necessary condition for the existence of a spatially localized solution to the nonlocal problems with free boundaries under consideration is the existence of an improper integral (b).

00 O tsk = ^- si) o 2 c1c c(oo) = 0, K(c)- = 0, g = oo, c1g that is, it takes place

At the same time, for recreation time the estimate w

T<]. ск х)

13) from which a more accurate estimate for T t follows<

1+ /?>(())] where c is the root of the equation

Уг^-Р)/ с /1 =(р, = КМГ > = ^-Ш+Р)^1 ■

C(P>*) = φ(р,0 on or = ()((), с(р, Г) = 0, (К(р^, с)%?аес,н) = 0 on Г(Г ).

1 d dt g"-1 dg p-\

K(r,t,c) ds dg p = 1,2,3,

16) where K(r,t,c) = K0(p(t)rmck;

17) where the functions and parameter p are determined in the process of separating variables in (16). As a result, we obtained an ordinary differential equation for B(t]) at] and the representation

Оn+m+p-2)/pBk £® drj

C.B-ij-dtl, oh

For two values of an arbitrary constant C( - C, = and

С1 = ^Ур equation (18) allows exact solutions depending on one arbitrary constant. The latter can be determined by satisfying certain additional conditions. In the case of the Dirichlet boundary condition c(0,0 = B0[f^)]"p/p (20), an exact spatially localized solution is obtained in the case k > 0, m< 2:

2-t Gf\h;

L/k 0<г <гф(/),

Oh, gf(/)<г< оо,

Vd^0(2-m\ p = pk + 2-m, and the exact non-localized solution in the case of k<0, т <2:

1/k 0< г < 00.

22) = [k^2 - t)/?/^1 p = 2-t- p\k\.

Here f(1) = \(p(r)yt; gf (/) = [^(O]^ o

Exact solutions were also obtained in the case of instantaneous or permanently acting concentrated sources, when an additional nonlocal boundary condition of the form

23) where o)n is the area of the unit sphere (co1 = 2, a>2 = 2i, a>3 = 4z).

Problems of diffusion from constantly acting point and linear sources in a moving medium are considered, when a quasi-linear equation is used to determine the concentration

Vdivc = -^S(r),

Gf(x)<Г<СС,

Mk 0<г<гф (х), Ф

2С2 (2 + 2к)К0 к

Solution (27) is reduced to nonlinear integral equations of the Volterra type and a nonlinear equation for x = 0 5 u(x) ~ 4m [i/r-^--* s/r + k^tek -¿r n V l/ g l/g

0 < X < 5, к(р.

-- = /(s), 0<г<гф(О,/>0, dt gp~x 8g \ 8g, f,0) = 0, 0<г<гф (0) (29) 5с с(г,0 = 0, - = 0, г = гф(0, ^>0; ah

1 I bg + /(c) Г~1£/г=- (30) о о ^ ; ^

The construction of a solution to problem (29), (30) was carried out by the Rothe method in combination with the method of nonlinear integral equations.

By transforming the dependent and independent variables, the nonlocal problem with a free boundary about a point source is reduced to the canonical form<х<^(г), г>0,

5l:2 8t u(x,0) = 0, 0< л; < 5(0), (5(0) = 0), (31) м(5(г),т) = мх(5(т),т) = 0,

Pmg + = d(r), m > 0, containing only one function defining the function d(r).

In particular cases, exact solutions of the corresponding nonlocal stationary problems with a free boundary for the Emden-Fowler equation with 12 and 1 in l are obtained

2=х иН, 0<Х<5, с!х ф) = м,(5) = 0, \х1~/*и1*сЬс = 4. (32) о

In particular, when /? = 0 m(l:) = (1/6)(25 + x)(5-x)2, where* = (Зз)1/3.

The following results are submitted for defense:

Study of qualitative effects of spatiotemporal localization;

Establishing the necessary conditions for spatial localization to limiting stationary states;

Theorem on the uniqueness of the solution to a problem with a free boundary in the case of Dirichlet conditions on a known surface;

Obtaining by separation of variables exact spatially localized families of partial solutions of degenerate quasilinear parabolic equations;

Obtaining accurate spatially localized solutions to stationary diffusion problems with reaction.

Conclusion of the dissertation on the topic "Mathematical physics"

The main results of the dissertation work can be formulated as follows.

1. Qualitatively new effects of spatio-temporal localization have been studied.

2. The necessary conditions for spatial localization and stabilization to limiting stationary states have been established.

3. A theorem on the uniqueness of the solution to the problem with a free boundary in the case of Dirichlet conditions on a known surface is proven.

4. Using the method of separation of variables, exact spatially localized families of partial solutions of degenerate quasilinear parabolic equations were obtained.

6. Exact spatially localized solutions to stationary problems of diffusion with reaction were obtained.

The results of the dissertation work can be used in formulating and solving various problems of modern natural science, in particular metallurgy and cryomedicine.

CONCLUSION

List of sources dissertation and abstract in mathematics, candidate of physical and mathematical sciences, Doguchaeva, Svetlana Magomedovna, Nalchik