Birgebayev

THE ROLE OF PROOF OF DIFFERENTIAL OPERATORS’ SEPARABILITY IN UNDERSTANDING THE ENVIRONMENT

(Almaty, Kazakh national pedagogical university named after Abai)

Abstract. The present scientific paper examines differential equation as a model of certain physical phenomena. It determines the reasons for existence, uniqueness and correctness of formulated problems for the studied model. It demonstrates the manner in which the given problem was formulated for corresponding differential operator in functional space. The present paper discusses the role of the considered problems in understanding of the environment.

Keywords: Differential operator, functional, functional space, hyperbolic and parabolic equations.

The majority of physical laws of nature can be formulated in terms of equations with partial differential coefficients and ordinary derivatives.

In this connection it is worth mentioning Sturm Liouville equation, Maxwell’s equation, Newton’s heat transfer law, Navier–Stokes equation and Schrodinger equation in quantum mechanics. All these equations describe physical phenomena using derivatives of space and time. The specified equations contain derivatives, which characterize very important physical values, such as velocity, acceleration, friction, flow, current, etc. Thus, there is a differential equation that must be correspondingly determined. Among problems included into the

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differential equations there is a class of problems having the only one available solution and depending on certain initial conditions. These problems are the most important ones.

It may appear natural, but nevertheless, assumed model needs their proof. Proof of correctness is the first approval of a mathematical model. Its essence consists in the following:

if there is no contradiction in the model (solution available), then the model unambiguously characterizes physical processes (unique solution), and the model weakly depends on the physical values measurement errors (the solution continuously depends on the task’s data).

The definition of the correctness of the statement of problems was first given J. Hadamard in the beginning of the last century. According to this definition the statement of problem shall be considered correct, if the following conditions are met: the solution exists, the solution of the problem is unique and the solution of the problem continuously depends on the initial conditions. First condition requires repeated uncertainty of the problem. The second conditions requires determining sufficiency of data essential for solution of the problem, while the third condition is connected with the following terms. If the problem is connected with the description of physical phenomena, but the problem data cannot be treated as absolutely accurate, then we may only guess that we know the only some approximants. Thus, in the absence of a continuous dependence on data, the solution of this problem does not comply with the relevant physical arrangement.

The aforementioned correctness conditions need to be defined. Particularly, in the theory of boundary value problems and solutions, that data are considered as elements of certain functional spaces, while the correctness of conditions is formed in the following manner:

1) Solutions exists for any data and normalized vector spaces (C® , Lp, Wpl ...) determined in the internal enclosed spaces;

2) The solution of the problem is unique in one of the specified spaces;

3) In the space considered for infinitesimal variation of data (provided that the data shall exist in such space) there is an infinitesimal variation of solutions in such space. Existence of solutions and continuous data dependence on the initial conditions proves the problem was stated correctly.

With respect to elliptical equations the boundary conditions used for determining the solutions of stated problems within the boundary regions are correct. In solving hyperbolic and parabolic equations the prescribed boundary regions of Cauchy problems and mixed problems are also correct. When considering levels of physical processes complying with the mathematical problems the Cauchy data means the condition of field was determined in the initial immediate time.

Let us consider the requirements to the problems connected with differential equations at the level of the functional analysis theory.

The linear operator in functional spaces is identified as the general state of the concept of linear transformations in finite-dimensional spaces (function, matrix, etc.). In this connection it is worth mentioning that complete definition of the operator À requires determination of its definition range first, i.e., the value Àõshall be defined for the set of elements x in the space H. Operators of unequal spaces are identified as different operators. The definition range represents the linear set belonging to the plane H. If the operator is linear and Àõ is defined for each

õ

 S, then we can define single-valued Àó for each combination, which shall make ó elements in the set S considering from its linearity. However, all operators cannot be linear in the same direction.

Formal definition of the linear operator: the internal set D (А), which is called the definition range of the linear operator А or in the transformation plane H is determined with linear mapping with respect to internal set R (A) which is called the range space. The linear variety shall be the definition range and the range of values. The operator A is considered

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bounded, if we find a constant Ê , then the following condition is met Au K u for any u

A D( )

 the considered norm of operator in plane shall be the smallest among constant numbers Ê .

If the set of values of operator A is the set real numbers R (A) then it shall be defined as a functional. The functional is the special case of operators. Study of the functions dependence on the real variables is based on the concept of limit (lim) in the set of real numbers. Therefore, introduction of the limit concept in the considered set is absolutely natural. Thus, introduction of this concept of limit in certain set allows us to define the concept of space. In general, the concept of space in science has different values. In terms of philosophy it is considered as a form of matter, in geometry it is understood as three-dimensional space –R3, determined using relevant axiomatic statements. In the majority of cases we consider the ï- dimensional space, which is located higher then used three-dimensional spaces. Thus, time, velocity, acceleration and several other variables shall be added to the coordinates of points. This results in abstract description of three-dimensional space R3 on ï - dimensional and infinite-dimensional space and study of their properties. These spaces include metric and measurable spaces, as well as Hilbert, Banach, Sobolev spaces and etc.

Functional analysis was formed as the mathematical discipline in the beginning of the XX century. Owing to rampant development it became one of the most important directions in mathematical science, which is used in many branches of mathematics, including several parts of applied mathematics. It has appeared as a result of description of mathematical methods and concepts known and used in mathematical disciplines. This description was created from amongst crossover to the highest levels of the mathematical abstraction peculiar to modern mathematical methods. Consideration of a number of mathematical, physical and technical problems from the perspective of abstract view in many cases affords an opportunity for better determination of its basic patterns, the most correct revelation of general in methods of its solution, even if the problems have different contents.

At present it is impossible to assume solution of equations of complex physical processes, differential equations of mathematical physics, approximate computations of summarized spaces and other important issues without functional-analytic approaches.

Issues considered in the given problem may have relevance to different mathematical problems with the stated content. However, if we do not take into account their real content, then using the mathematical abstraction the considered issues are united into one or two problems.

If we inherently assume thatX is the set consisting of the

õ

elements (elements, number, function, vector, matrix and etc.), the T shall be the operator reflecting the set X of itself, i.e., Тх X.Then in the equality

Тх = (1), Zero element  ( X) remains in the setX , while the operator Т searches for the element where х0Xtransforming the zero element - . This problem can be explained with the following system of linear equations:

, 1 . 2 ..., .

1

n j

r x a

_{k j}

n

k

jk

 



 (2)

Inherently, using the following symbols Тх= À

õ

-r , where

53































 

nn n n

n n

a a a

a a a

a a a A

...

...

...

2 1

2 22 21

1 12 11

, ,

...

2 1























xn

x x

x























rn

r r

r ...

2 1

(2) Solution of the system Т

õ

₀=, results in search of elements –

õ

₀ (where =(0,0…0)) fulfilling conditions.

A kind of that,

n j

y dx a

dy ⁿ

j , 1.2...,

1







 

 

The problem that requires determination of solutions to the system of linear differential equations or determination of the eigenvalue of the Sturm Liouville problem with boundary conditions results in the following solution of equation:

X

TX  (3), Where  is the real number, i.e., using the operator Т we determine the element – Х changeable in collinear element.

With respect to the problems (1) - (3) several problems are provided, while the main of which are the following

I the problem with existing solution II the problem with the unique solution

III the choice of methods for determining direct and approximate solutions IV the problem of stability of solutions

V the problem for identification of errors in determination of direct and approximate solutions Consideration of differential equations with supplementary conditions allows determination of the operator A. It correspondingly arranges set of functions located in supplementary conditions to each solution in spaceuE .

Having considered the function f in this total as an element of the functional space F we find the solution of differential equation

Au=f (4) as an equivalent of solution of fulfilled conditions uEof operator’s equation

Containment of an image of element of any Efunction, i.e., operator A in the set of values is both necessary and sufficient for the availability of the solutions of the equation (4).

Thus, the set of data (values) of the problem is fully characterized with determination of the space E, being the space of solutions.

Existence of backward operator A^1 will be the guarantor of the solutions availability for the equation (4) and determination of the uniqueness.

u=A^1

f

(5) In the given case, please bear in mind that the basic purpose of the mathematical problem’s solution is the determination of the mathematical description. In addition, the problem data have certainly been obtained from the aforementioned experiment, and thus, they cannot be measured precisely, on the other words, the problem data usually contain erroneous measurements. Additional requirements, determining physical data, in which minor changes of the solution corresponds to minor changes in the problem data, should be applied to problems, describing real physical processes using the mathematical models. Generally, when considering equation with operators, minor deviations from the initial values can be considered as the

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availability of the solution stability. Let us summarize the mentioned above using the relevant mathematical term.

Let the u serve as the solution of equation (4). Let the operator influences from the normalized space E on the normalized space F. Solution of equation (4) fF is called stable and minor deviation of the right-hand side, if the number  0 will be determined from any number  0, and the following conditions\ 

F

f f

  and condition

 ;



E

u u



 _u^ _E , are also met for the following equation Au=f,



  f u

A .

Availability of the solution (4), uniqueness and stability of the solution to minor deviations of data is called the stability of mathematical problem.

In this connection let us give more precise formulation of the present notion Let the operator influences from the normalized space E on the normalized space F.

F E

A: ^A

If the operator meets the requirements mentioned below, the problem of equation (4) is called correctly constructed problem in the paired space E, F.

1) Any solution for fFexists, and it is located in the space Е;

2) Solution of the problem in the space E will be unique;

3) For any element fFthe solution of the problem is stable.

Notes: the problem may be correct in one paired space and incorrect in another paired space. For example, in case of extension of the space F the solution of any fF may possibly result in complementary decision.

Let us give an example of incorrectly constructed problem (this example belongs to Hadamard, who had introduced the concept of correctness).

Let us consider the potential equation (Laplace’s equation) in plane:

2 0

2 2

2  

dy u d dx

u

d (6)

Let us consider a Cauchy problem along the x-axis. For this purpose we shall use the band 0у as the zone of the search for solutions, where  means any positive number.

Let us mark the band =y,0y. Let us consider y as the non-osculating direction.

Let assume that the conditions of Cauchy problem are the following:

) (

0 ),

( _0,

0  _   

  x

dy x du

u _y  _{y (7).}

With respect to the unique function  x of the determined data, let us consider this function as the confined function on a continuous axis and on the axis of all numbers. Then we may take continuous confined function in the band C   as the area for determination of operators of the boundary problem (6), (7), then take the set of functions in the space C()

with continuous second derivative and satisfying the condition _y0 0 dy

du . Let us prove the correctness of the problem statement (6), (7) in the paired space E, F. We may easily prove the uniqueness of the problem solution. This implies accordance with the solution u0 of function

x0

 . Let us give slight deviation of function  x in the sense of measurement of the space F, and thus, for the equation (6) according to Cauchy data:

cos , 0

0

0  _ 

 y

y dy

du n

u nx (8)

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Let us consider the Cauchy problem: where n – is sufficiently great natural number.

Solution of the new problem

 

cos ,

, n

nxchny y

x

u  may be checked by putting it into equation (6), (8). It will be understood that

, 1 0

max cos

cos    

 _





 x n

F

F n n

nx n

nx



  .

Among other things,



 

 

  

 n

y

F x n

chn n

nxchny

u 



max cos

0





In the other words, slight deviation of data in the sense of measurement of the space F corresponds to the same large deviation in the sense of measurement of the space E. Thus, considered Cauchy problem is incorrect in the paired space for the potential equation (Laplace’s equation).

Thus, we can prove the existence of the solution for the boundary problems formulated using the differential equations considered in the normalized space. In connection with application of functional analysis methods (Riesz theorem, Titchmarsh’s methods, and etc.) classical statement of problems has been replaced with the universal laws. Computed solution is universal and specifies functional spaces for determination of this solution.

Thus, in the solution of boundary problems accordingly formulated for the differential equations, let us consider Lu f of the differential operator, determined in the appropriate normalized space H. In this connection the solution of the specified classical boundary problems will be determined as follows:

1) Determination of the operator L^-1;

2) Boundedness in the space Н of the operator L^-1; 3) Fulfillment of conditions _h

h

c f

f

L

^1



_;

4) Of the L operator’s separability;

If the differential operator 1), 2) will be linear, then the proof of availability of the uniqueness of the equation’s solution becomes obvious;

3) In addition, it is worth mentioning a coercitive valuation, which express the continuous dependence on the initial conditions, i.e. determines stability; and 4) allows determining smoothness of differential equations’ solutions.

Pioneer researches in this sphere have been conducted by Everett V.M. and Geertz M.[1]

for singular differential operators. Thereafter, these studies have been continued in scientific researches performed by Otelbaev M. [3], Boymotov K.H. [2], Muratbekov M.B. [4], Birgebaev A. [5] and many others.

Thus, the purpose of formulated problems and their solution is to validate correctness and feasibility of the studied model for the surrounding phenomena using relevant mathematical fictions.

A possibility of proving existence, uniqueness and stability of solutions of formulated problems gives us an objective confidence that the scientific researches have been conducted in the right direction. It is very difficult to appraise its significance, as examination of such issues first requires complete understanding and correct formulation of the relevant problem.

The study of separability of differential operators will provide future teachers of mathematics with information on the environment and will definitely shape their worldview and lead to understanding of the specified problems, allowing them solving relevant problems formulated in the normalized space of the mathematical model.

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1. Everitt W.N., Yiertz M. On some properties of the prower-ties of a formally self-adjoins differential expessions. Proc. London Math. Soc., 24(3), 1972, 756-768.

2. Бойматов К.Х. Теоремы разделимости.-Докл. АН СССР, 1973, т. 213, № 5, с. I009-I0II.

3. Отелбаев М. О разделимости эллиптических операторов.-Докл;. АН СССР, 1977, т.

234, № 3, с. 540-543.

4. Муратбеков М.Б. Теоремы разделимости и спектральные свойства одного класса дифференциальных операторов с нерегулярными коэффициентами. //Автореферат док.дис. физ.-мат. наук Алматы, 1994-30с.

5. Биргебаев А. Элементы теорем вложения и теории разделимости. - Уч. пос. - КазНПУ им.Абая, Алматы-2008, 88 стр.

Аңдатпа. Жұмыста дифференциалдық теңдеу белгілі бір құбылыстың моделі ретінде қарастырылған. Сол модель үшін қойылған есептің жалғыздығы және корректілігінің мағнасы айқындалады. Сонымен қатар проблеманың функционалдық кеңістіктегі сәйкес дифференциалдық оператор үшін қалай қойылғаны қарастырылады. Зерттеліп отырған есептің қоршаған ортаны танудағы ролі қарастырылған.

Түйін сөздер: Дифференциалдық оператор, гиперболалық және параболалық теңдеулер, функционал, функционалдық кеңістік.

Аннотация. В работе рассматривается дифференциальное уравнение как модель некоторых физических явлений. Определяется смысл существования единственности и корректности поставленной задачи для изучаемой модели. Показывается, как поставлена данная проблема для соответствующего дифференциального оператора в функциональных пространствах. Обсуждается роль исследуемых задач в познании окружающего мира.

Ключевые слова: Дифференциальный оператор, функционал, функциональное пространство, гиперболическое и параболическое уравнения.

УДК 517.958:004

Б.С. Дарибаев¹, Д.В. Лебедев¹, В.А. Перепелкин² РЕАЛИЗАЦИЯ ЧИСЛЕННОЙ МОДЕЛИ ДВУМЕРНОГО

ЭЛЛИПТИЧЕСКОГО УРАВНЕНИЯ В СИСТЕМЕ ФРАГМЕНТИРОВАННОГО ПРОГРАММИРОВАНИЯ LUNA

(¹г. Алматы, Казахский национальный университет им. аль-Фараби, ²г. Новосибирск, Россия, Институт вычислительной математики и математической геофизики СО РАН)

Аннотация. При численном моделировании явлений методами конечных разностей на современном суперкомпьютерном оборудовании требует построения сложных программ, адаптирующихся к вычислителю и ходу моделирования для обеспечения эффективности проводимых вычислений. В работе проводится сравнительный анализ двух параллельных программ, реализующих попеременно-треугольный метод для решения задачи Дирихле для уравнения Пуассона в единичном квадрате. Первая реализация построена «традиционным»

способом — с использованием интерфейса передачи сообщений MPI, а вторая — с помощью системы LuNA, автоматизирующей конструирование численных программ для мультикомпьютеров. Сравниваются масштабируемость и эффективность распараллеливания программ.

Ключевые слова: Фрагментированное программирование, LuNA, численное решение, MPI, параллельная программа, попеременно-треугольный метод.

Введение. Численное моделирование методами конечных разностей с использованием суперкомпьютеров всё чаще наталкивается на вопросы эффективности

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алгоритмов и реализующих их параллельных программ. Для реализации моделей с большим количеством точек необходимо использовать алгоритмы, допускающие эффективную (по ресурсам, времени и т.п.) и масштабируемую параллельную реализацию. В частности, плохой масштабируемостью обладают программы, использующие централизованное управление и/или хранение данных, передачу данных на далёкое расстояние (в смысле сетевой топологии).

Примером алгоритма, допускающего эффективную и масштабируемостью реализацию, является попеременно-треугольный метод с чебышевским набором параметров. Он является быстро сходящимся двухслойным итерационным методом, применяемым для решения разностных эллиптических уравнений, возникающих при аппроксимации эллиптических уравнений второго порядка.

Тем не менее, параллельная реализация такого рода алгоритмов является сложной задачей системного параллельного программирования, т.к. в программе требуется обеспечить синхронизацию отдельных вычислительных процессов, передачу данных по сети, и т.п.

Для снижения трудоёмкости разработки подобных параллельных программ применяют инструменты, частично автоматизирующие их конструирование. К таким системам относятся LuNA [1], PaRSEC [2,3], Charm++ [4].

Целью настоящей работы является сравнительное исследование свойств параллельных программ, реализующих попеременно треугольный метод, применяемый для решения модельного двумерного эллиптического уравнения. Сравниваются две параллельные программы, одна из которых получена с помощью традиционного подхода (на базе интерфейса передачи сообщений MPI), а другая — с использованием системы фрагментированного программирования LuNA.

В первом разделе статьи приводятся необходимые определения и описание системы LuNA. Во втором разделе описывается постановка задачи, прикладной алгоритм и схема его параллельной реализации. В третьем разделе приводятся результаты численных экспериментов и сравнительный анализ их производительности

Система фрагментированного программирования LuNA. В системах автоматизации конструирования параллельных программ используются модели вычислений, отличные от распространённой модели взаимодействующих последовательных процессах, используемой, в частности, в стандарте MPI (Message Passing Interface). Это связано с тем, что различные модели вычислений обладают разными свойствами. Например, динамическая миграция MPI-процесса с одного узла на другой как способ выравнивания нагрузки практически невозможна из-за необходимости переносить всю память процесса по сети, возможных открытых дескрипторов файлов и тому подобных проблем, которые возникают из модели вычислений, используемой в MPI.

В системе LuNA используется модель вычислений, называемая фрагментированной программой (ФП). В этой модели данные задачи представляются как множество отдельных единиц, называемых фрагментами данных (ФД). ФД иммутабельны и являются переменными единственного присваивания. Значения ФД могут иметь как базовый тип (целочисленный, вещественный, и т.п.), так и составной (фрагмент сетки, вектор, и т.п.).

Вычисления задачи задаются множеством процессов, каждый из которых связывается с набором входных и выходных ФД и вычисляет значения выходных ФД из значений входных. ФВ не имеет побочных эффектов.

Вычислительный процесс состоит в том, что ФВ, для которых известны значения всех их входных ФД и неизвестны значения выходных, исполняются, что приводит к

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