Application of financial instruments in organizations of the agro-industrial complex. Shishkin V., Kudryavtseva G.

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 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.

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.

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.

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 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.

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 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,

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, r/) weakly depends on the tangential coordinates, that is, c (mx, m2,1]) = c(t]), then to determine c(m]) 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 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(/),

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 exact solutions found 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;

Establishment of 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.

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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