THE MEAN VALUE THEOREM FOR LAPLACIAN ON THE GRAPH Текст научной статьи по специальности «Математика»

ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ

Abdulkhayeva Z. T.

Master's Degree student Kazakh National Research Technical University after K.I. Satbayev

Penkin O.M.

Doctor of Math and Physics Sciences, professor Kazakh National Research Technical University after K.I. Satbayev

THE MEAN VALUE THEOREM FOR LAPLACIAN ON THE GRAPH

Summary: This article deals with the problem of the mean value theorem for harmonic functions, defined on some given domain. Analysis of recent studies has been conducted. That allowed us to replace the classical metric sphere by so-called probabilistic sphere and the harmonic functions were the limits of the discrete functions, which describe the Brownian motion on the graph. The advantage of the described theory is that it can be devoted to the graph of arbitrary structure, and further it will be developed to the arbitrary stratified set. Several numerical experiments were made to demonstrate feasibility of our method.

Keywords: the mean value theorem, harmonic function, geometric graph, random walk, probabilistic sphere.

Introduction

The theory of the ordinary differential equations on graphs (on stratified sets in more general context) has begun its development in 80-th years of last century in the works of Покорный Ю.В., Павлов Б.С., Фадеев М.Д., G. Lumer and others. Some significant results had been achieved to the present moment: the questions of solvability of the boundary value problems on graphs, the elements of spectral theory had been constructed.

Graph G is defined to be a connected subset of Rd consisting of a finite collection of points vi,v2, ...,vk called vertices and some open intervals et connecting this vertices called edges. This graph is supposed to be divided into two parts: G0 and dG0. dG0 consists of the

vertices of multiplicity one (multiplicity means the number of edges adjoining to this vertices).

Sub-graph G' is defined to be an open and connected subset of G. Sub-graph G' c G has the interior vertices only from G0, i.e. every interior vertex of subgraph is also interior for graph G. There is another situation for boundary vertices of G'. Let et is an edge, which contain X0 £ dG', then it belongs to the subgraph G' not all, but with one-piece cutting-of X0. It is the main difference between our definition of subgraph and algebraic interpretation. [1]

If the edges admit a sufficiently smooth parametri-zation and do not have self-intersections, we can assume that they are rectilinear intervals. Thus, it is convenient for us to assume that G0 consists of some set of disjoint intervals

e = [X ERd:X = Vi + tF--^0 < t < 1}

where I equals the length of the edge e. The value of the parameter t, corresponding to the point X £ G, may be interpreted as a coordinate of the point X. Using this parametrization we can interpret a restriction Uie(X) of the function u: G ^ R as a function ue (t) of numerical argument defined by

ue(t) = uw(n(t))

where n:t ^ X is a coordinate mapping. Using this numerical representation, we can define the derivative u'(X).

The differentiation of the function u(X): G ^ R inside the some edge e is done by the natural parameter, besides it is assumed that the orientation is chosen for this edge (one of two possible directions). For example, on the edge e = (a, b) with orientation 'from b to a' the derivative ^ (X) is defined like

d ,, d ( (a-b) u'(X) = —ue(t) = —u (b + t

dt ew dt \ Wa-b\\

If we change the orientation, the sign of u' will change to the opposite. However, the sign of the second derivative u'' is not depend from the orientation of the edge.

The harmonic function on the graph is a solution of equation

! !■

36 Wschodnioeuropejskie Czasopismo Naukowe (East European Scientific Journal) #4(44), 2019 SMI

¡Au = 0 on each edge

^ U' = 0 on inernal vertices

The first equation means, that the function u is a linear function. The second one- the sum of the directional derivatives (orientation is from the given vertex to the opposite one) is zero. In addition, we can take some values of function at boundary vertices. If, for example, take uaGo = 0, then we will get a zero solution. Let us take nonzero values of u on boundary, and then we will get a Dirichlet problem

Au = 0 on each edge ^ U' = 0 on internal vertices ^u(vi) = on boundary vertices

Consider the solution of Dirichlet problem , which equals

(" U = 1 on \-u = 0 otherwise

One easily can be see that Uj is a solution of the initial problem. If we multiply it to a;, the new combination will be the solution of described above Dirichlet problem. On the other side, the solution Uj has a probabilistic point of view. Let particle begins it motion at the point X0 £ G0. What is probability of particle to achieve some fixed boundary vertex, assuming that other boundary vertices are absorbing? The function may be interpreted as a solution to this kind of problem: particle starts at the point X0 = v, i.e. the probability to achieve vertex v is one and particle is absorbed instantly. Particle starts at another vertex ^ v, i.e. it has never achieved vertex v, because it will be absorbed, and that is why the probability is zero. And so on.

Analysis of recent research and publications

Let H be a domain in R" and u is a C2 (H) function. The Laplacian of u, denoted Au, is defined by

Au = > = div

The function u is called harmonic (subharmonic, super harmonic) in H if it satisfies there

Au = 0(>0,<0) [3]

The first result, related to our theme, belongs to C.F.Gauss. It claims that if u is a harmonic function in the domain G c R", i.e. the function satisfies the equation Au = 0, then for each £in the domain and the ball Br c G centered at ^ the following formulas are hold:

= t1- f u(x)dZ = T~2 f u(x)dS "(0 = J^T f

In general case we can comprises the well-known mean value properties of harmonic, subharmonic and super harmonic functions.

Theorem Let u £ C2(H) satisfy Au = 0(> 0,

< 0) in H. Then for any ball B = (y) cc H, we have 1 r

w(y) = (<, >)-r I uds,

udx

u(v) = (<, >) 1 f

where is volume of unit ball.

For harmonic functions theorem thus asserts that the function value at the center of the ball B is equal to the integral mean values over both surface 3B and B itself. These results, known as the mean value theorems, in fact also characterize harmonic functions. [3]

Because of importance of Gauss theorem, the several repeated attempts were made to generalize it. An analogue of the Gauss theorem for the harmonic functions on stratified sets has been obtained in work of EXXo^ O.M.neHKHH in 90-th of the last century. A stratified set H (fig. 1) is defined to be a connected subset of R", consisting of a finite number of smooth manifolds (strata) afcj- adjacent to each other according the following rules:

1) For all couples afcj-, either afcj- n = 0 or afcj- n ami consists of strata.

2) The boundary d5fcj = ofcAofc,- consists of strata.

3) If ofc- > and Y 6 ^fc-ij, then T^Ot/ (tangent space) has a limitposition (as X ^ Y) containing

Tyak-lj-

Fig. 1. Example of the stratified set

Many Laplacians Ap can be determined on stratified set. Harmonic functions in mean of this Laplacians have some properties similar to the properties of simple harmonic functions in the domain of Euclidean space, but there is also significant differences. Roughly speaking, there are two extreme cases of Laplacians - soft and hard. The soft Laplacian is closer in its properties to the classical one, than the hard one. In the last decade, interest in equations on stratified sets has steadily increased. Their systematic study began with a one-dimensional case by the works of G. Lumer and Yu. V. Pokorny. Multidimensional problems began to be studied by S. Nicaise,W. Zhikov. [4]

Let X 6 ofc- c H0 and r0 (X) is the minimum of distances from point X to (fc — 1) -dimensional stratum, which do not contain X in their closure, and let 5r(X) be a ball in R". Let the set = 5r(X) n H will be

called a stratified ball for r < r0. Define 3sr(x) (u) as the weight average

3sr(x)(u) =

of the function u for the ball and 3aBr(x)(u) as similar average for the corresponding stratified

sphere. Then there is the following statement:

Theorem Let u 6 C2(H0) is a p-harmonic function. Then

3Br®(u) = 3sBr(*)(w) = u(X)

However, in the work of Е.Г.Гоц, O.M.neHKHH the mean value theorem has proved only for the stratified balls of so-called 'allowable radius'. If we are dealing with the graph, the definition 'allowable radius' means that it cannot exceed the distance from the center to the nearest vertex. In this case, sphere is defined metrically as the set of points equidistant from a given point (for example, center of a ball) to a distance, which is not exceeding a fixed positive number.

Presentation of the main material

Here we give a probabilistic interpretation of the following Dirichlet problem for Laplacian A on graph G:

f Au = 0 on G

Mac0 = 9 (function) where A acts on functions in C2(G) as follows:

u''on edges

Au = '

E

= 0 on internal vertices v

here a notation et > v means that the et is adjacent (attached) to the vertex v and stands for the directional derivative of u at the vertex v, correspond to internal direction 'form v to the interior or e; '. In other words, Vi is a unit vector at the vertex v directed to another vertex of the edge e^

V

C2(G0) denotes the set of all scalar valued functions, which are continuous on G0 and have continuous derivatives up to order two on each edge.

Let denote by uv the solution of the problem: what is the probability to attain form point X to some fixed vertex v, assuming that other vertices are absorbing. Then the well-known Dirichlet problem has the form

Auv = 0 u(v0) = 1 [u(v) = 0,ve G0\{v0}

Theorem The solution to the described Dirichlet problem on the graph has the form u = ^v£gGo 9 (v)uv.

Proof It is easily known that each harmonic function is a linear combination. In our case uv is a harmonic function, cp (v) is a fixed number, summing these we again get the harmonic function because of linear operator A. Our solution must satisfy the boundary conditions: at the vertex v0 we get u(v) = cp(v) and on the others- zero.

For arbitrary point X0 fix the vertex v. Let u„(X0) be a probability of point X0 to achieve vertex v, assuming that all other boundary vertices are absorbing. In the limit, there is a continuous random walk (S ^ 0,n ^ x>,u„ ^ uv). The function uv equals

(1 at the vertex v

I 0 otherwise

Besides, this function is harmonic. Assume that particle begins its motion on the some edge at the point X0. Then

II

usv(X0) =-usv(X0 + S)+-usv(X0 - 5)

11 1

0 =~usv(X0 + 5) -~usv(X0) - (usv(X0) -~usv(X0 - 5))

2 1 2 1 2

0=-u'vs(Xo)8--u'vs(Xo-S)S

212

-u"vs(X0)S2 = 0

i

Dividing last equation by - and letting 8 tends to zero we get the well-known harmonic function on the graph. Such assumptions might be applied to the case when particle begins from the some interior vertex (here sum of the directional derivatives will be zero).

We know that each harmonic function might be represented as a linear combination of

=Y

CiUv vedGn

Therefore, it is sufficient to prove the mean value theorem for the function uv, and then it will be true for arbitrary harmonic function.

The analogue of the mean value theorem for harmonic function on the graph

Theorem Let G' is a sub-graph in G and X0 £ G'. uv(X) is a probability for Brownian particle starting its motion from the point X to achieve the vertex v (before being absorbed by another boundary vertex). Then

uv(X0) = p1uv(v'L) + V2Uv(V2) + P3uv (v3) + - + PkUv(vk')

where vt' are boundary vertices of G', pt denotes a probability for the particle starting its motion form X0 of the first passage through the vertex vt.

Corollary Let pt = p2 = — = pk and its sum is one, then v[,..., vk' lies in probabilistic sphere and

k

k/ u"

i=1

Remark Probabilistic sphere does not depend of choosing v.

Main theorem Let u be a discrete harmonic function and G' is a probabilistic ball centered at X0. Then

u(Xo) =

k

Proof We know that

k k u.Jv/) = — w

(Xo)=^y u(v)uv(Xo) ="y u^YuM^yy u(v)uv(vi')

¿—>vedG0 ¿—>vedG0 kj-j k^Z—ivegG0

k =

(Xo)=lYu(vi')

ky u(

i=1

Conclusion and universal methods, but we were interested in the

The article is devoted to develop the analogue of methods, which later may be applied to stratified sets

the mean value theorem for the harmonic functions on also. The mean value theorem for harmonic functions

the graph. The results can be achieved using the simple on stratified sets without 'allowable radius' of the

u

sphere is one of the interesting serious and perspective problems nowadays. It had be seen that instead of metric sphere at the Gauss theorem should be used so-called probabilistic sphere, and harmonic functions in this case are limits of the discrete functions which descried the Brownian motions on the graph.

References

1. Ю.В. Покорный, О.М. Пенкин, «Дифференциальные уравнения на геометрических графах»

2. R. Courant, K. Friedrichs, H. Lewy, 'On the Partial Difference Equations of Mathematical Physics '

3. D. Gilbarg, N. Trudinger, 'Elliptic Partial Differential Equations of Second Order'

4. Е.Г. Гоц, О.М. Пенкин, «Теоремы о среднем для лапласианов на стратифицированном множестве»

5. E. Zauderer, 'Partial Differential Equations of Applied Mathematics'

Bovin A.A.

physics teacher,secondary school № 63, Krasnodar

Ivankova P.N. student,secondary school № 63, Krasnodar

APPLICATIONS OF THE FRACTAL METHOD FOR DETERMINING THE LENGTH OF THE KUBAN COASTLINE IN THE KRASNODAR CITY RIGION

Бовин Александр Александрович

учитель физики,средняя общеобразовательная школа № 63, г. Краснодар

Иванкова Полина Николаевна

учащаяся,средняя общеобразовательная школа № 63, г. Краснодар

ПРИМЕНЕНИЕ ФРАКТАЛЬНОГО МЕТОДА ДЛЯ ОПРЕДЕЛЕНИЯ ДЛИНЫ БЕРЕГОВОЙ ЛИНИИ КУБАНИ В ЧЕРТЕ ГОРОДА КРАСНОДАРА

Summary: Using fractal theory, the formula for the length of the coastline of the Kuban River within the city of Krasnodar is obtained and its fractal dimension is determined. It was revealed that for the river bank coastline there is a limiting minimum size of the square grid cell size used in the processing of a satellite image, less than which the coastline fractality is not observed.

Аннотация: Пользуясь фрактальной теорией, получена формула длины береговой линии реки Кубань в черте города Краснодара и определена её фрактальная размерность. Выявлено, что для речного побережья существует предельное минимальное значение размера ячейки квадратной сетки, используемой в процессе обработки космического снимка, меньше которого фрактальность береговой линии не наблюдается.

Key words: fractal method, fractal length, coastline.

Ключевые слова: фрактальный метод, фрактальная длина, береговая линия.

Введение.

В настоящее время теория фракталов имеет широкое распространение и применяется во многих областях человеческой деятельности. В частности, для определения длины береговой линии. Полагая, что форма береговой линии представляет собой фрактал можно определить её фрактальную размерность и фрактальную длину. Решение подобных задач имеет большое практическое значение, поскольку позволяет оценить экономические затраты на строительство прибрежных коммуникаций, дорог и других сооружений. Фрактальный метод был использован, например, для определения длины береговой линии Норвегии [1], Великобритании [2,3], южного побережья Крыма [4]. В этих работах в качестве объекта использовалась морская береговая линия. Для речных береговых линий фрактальный метод не применялся. В частности, для береговой линии реки Кубань. Исходя из практических целей, в данном случае рассматривался только участок реки, входящий в черту города Краснодара.

Описание фрактального метода Е. Федера для измерения длины береговой линии.

Бенуа Мандельброт, работая в американской исследовательском центре IBM, проявлял свой интерес к приложениям математики в различных областях. В частности, его заинтересовала проблема измерения длины береговой линии Великобритании. Оказалось, что ответить на этот вопрос весьма затруднительно. Свои исследования Б. Мандельброт опубликовал в журнале «Science» в 1967 году [2]. В этой публикации рассматривается парадокс береговой линии, заключающийся в том, что её длина зависит от способа её измерения. Если оценка длины береговой линии осуществляется путём наложения на карту N равных отрезков длиной 5, то окажется, что чем меньше длина отрезка измерений, тем больше становится конечная измеряемая длина. При этом в случае стремления длины отрезка измерений к нулю значение длины береговой линии возрастает до бесконечности. Таким образом, чтобы говорить о длине береговой линии, нужны какие-то другие средства количественной оценки.

Для определения длины береговой линии Ман-дельброт воспользовался эмпирическим законом,