On the solvability of the Cauchy problem for a singularly perturbed integro-differential equations in partial derivatives of the first order with a turning point Текст научной статьи по специальности «Математика»

On the solvability of the Cauchy problem for a singularly perturbed integro-differential equations in partial derivatives of the first order with a turning point Kydyraliev T.

О разрешимости задачи Коши сингулярно-возмущенных интегро-дифференциальных уравнений в частных производных первого порядка с точкой поворота Кыдыралиев Т. Р.

Кыдыралиев Торогелди Раимжанович /Kydyraliev Torogeldi Raimjanovich - старший

преподаватель,

кафедра информатики и вычислительной техники,

Кыргызский национальный университет им. Ж. Баласагына, г. Бишкек, Кыргызская Республика

Abstract: in this study we investigated the solvability of the Cauchy problem solution and its structure for a singularly perturbed integro-differential equations with a turning point derivatives. In solutions found an integral representation.

Аннотация: в работе изучена разрешимость решений задачи Коши и ее структура для сингулярно-возмущенных интегро-дифференциальных уравнений в частных производных с точкой поворота. В решении найдено интегральное представление.

Keywords: integral equation, partial differential equation of first order, the principle of contraction mappings, Lipschitz condition, nonlinearity.

Ключевые слова: интегральное уравнение, дифференциальное уравнение в частных производных первого порядка, принцип сжатых отображений, условие Липшица, нелинейность.

The essence of the proposed method is converting solutions, finding the solutions of the initial transformation of the Cauchy problem for a singularly perturbed integro-differential equations in partial derivatives with a turning point, and bringing to the Voltaire equivalent integral equation of II kind. Here are algebraically-functional bases of conversion method making the theory of differential, integral equations.

In many problems of analytical and asymptotic theory of differential and integral equations applied the method of converting solutions. For example, in Paper [1] Chapter VIII is devoted the method of converting solutions, allowing to integrate a predetermined differential equation or explore the properties of its solutions.

In this regard, we introduce a definition. Let's О - a set, the operators A and K represent it in himself. Consider the equation

Ax = b (1)

where b - a fixed element of О and conversion

From (1), (2) directly have

X = Ky AKy = b

(2)

(3)

Hence, if there is a "semi-inverse" operator {AK) to the operator of AK, we obtain

y = {AK Г b (4)

and from (3), (2) we have solution of equation (1) in the form

X = K {AK )-1 b

Definition!. The operator K will be called the operator of converting solutions of A.

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NOTE1. If (2) has the form x = Kx

then (1) can be written as

x = (AK )-1 b

in particular, if it appears А К = E, E -the unit operator, K = A 1.

NOTE 2. It is necessary to choose the operator K, so as to obtain a simplified new operator equation (3), to which it would be possible to apply one of the following methods:

- topological methods for proving the existence of solutions, for example, the principle of compressed mappings;

- methods expansions of solutions, for example, methods of making the expansion in power series (the first Lyapunov method);

- directly produce various, including asymptotic evaluation using assumptions with respect to A and K and the element b, the independent variable tends to a limiting value.

NOTE 3. We note that the definition 1 includes the methods of integral transforms

y = Fx

J , Fourier transforms (Laplace), substituting (2) in the form y = Fx where F = K 1,

l.e pre-suppose the existence of the inverse operator K 1, whereas in (4) assumes the

existence of semi-inverse operator (AK) .

Now consider the singularity perturbed integro-differential equation in partial derivatives with a turning point

r du du

s--------1--

'ydt dx ^

with the initial condition

u(0,x) = q>(x). (6)

Here are mathematical notation used in this paper:

R-number line

Cafi'''Л)- space of functions bounded and continuous together with its derivatives to the corresponding order;

Lip(L I u) - class of functions satisfying a Lipschitz condition and with coefficient L. Assumption (T). Let n e N - fixed number,

f (t,x,u) e C([0,T]xRxR)nLip(Ц\u), (p(x) e C(R),

K (t ,r, u) e C (( 0 <r < t < T )x R )n Lip (L2|a).

The solution of the Cauchy problem (1) - (2) in the form

t a, „ Bs л

C —(t-s )+B- 1

u(t, x) = ф(x -1) +1 e s s — Q(s, x -1 + s)ds,

J s

0 s

-sin ntl

l

u(t, x) = f (t, x, u(t, x)) + J K (t, s, u(s, x) ) ds (5)

(7)

Where Q (t, x) - a new unknown function to be determined; a, fie R+ and their values will be determined later.

Successively differentiating with respect to and relation (7), we have

t-s) +Bs 1

1 fit t

1 — a г

ut (t, x) = —ф(x -1) +—e s Q(t, x)----(u -ф) - I e

—Qfis, x -1 + sxds; (8) s

0

46

ux(t, x) = ф'( x -t) + J From whence

t a „ Bs i ■ —(t-S )+B- l

0

— QX, x -1 + s)ds. (9)

s

ut (t,x) + ux(t,x) = — e s Q(t,x) -—u + —p(x -1).

s s s

Multiplying both sides of this equation by S , we have

B

s(ut + ux) + au(t,x) = e s Q(t, x) + ap(x -1). (10)

In view of (6)

pt

s(

(ut + ux) + sintu = e s Q(t,x)-{a-sint)u(t,x) + ap(x-1)

Where, taking into account (7), we obtain

pt

Q(t, x) = e

t

f (t, x, u) + J K (t, t, u(t, z))d т

pt

+e s {a-sinnt)

t aч p- I

p(x -1) + J e s s — Q(s, x -1 + s)ds

J c

pt

- e sacp(x -1),

or

-p p -BtsyPi i

Q(t, x) = e s f (t, x,p(x -1) +1 e s s — Q(s, x -1 + s)ds +

J c*

pt t

+e s J K(t,s,p(x-t) + Je s ' s — Q(s,x-t + s)ds

т a, ч ps -i

-----( t-s)+— —

dT -

(11)

+e s {a- sin nt )J e

‘ _at_Ap

-(‘

pt

— Q(s, x -1 + s)ds + e s sin ntp(x -1) = P [Q1. s

Then, to prove the existence and uniqueness of solution of the Cauchy problem (5) - (6) to the Voltaire nonlinear integral equation of II kind apply the contraction mapping principle, l. e to the operator equation

u = Pu,

where the operator Pu - right side of equation (11).

Let the set

П = {Q(t, x) :Q(t, x) e C<U) ([0,T] x «)u|Q < h}

Values T and h will be determined later.

From (11), the assumption (T), we have

-Ж P

||PQ|| < e s [M\ + M2T] + (a + 1)Je

0

Л a +1 ,, ,,

< e s (M + M2T + M ) +--------- Q .

a + p

Hence, by the definition of Q, we have

pT

< e s

t (a+p)

(t-—) 1

p s ,

s ' — \\Q(s, x -1 + s)\\ds +e s M <

s

— a +1

PQ\\ < e s (M + M2T + M ) + — h.

a + p

0

0

47

a +I

If we choose T, а, ft, h so that

pr

e ^ (M + M2T + M ) + -ai1 h < h, (12)

a + ft

the operator P[Q] puts the set of P\Q\: Q ^ Q

Now we show that the operator P is a contraction operator. From (11) using the assumption (T), we obtain

p -a(t-s)+— —

? s [f (t, x,p(x -1) + J e s s — Q (s, x -1 + s)ds) -

P [Q ]-p Q ]<

0

. -a(t-s )+ft -f (t, x, p(x - t) +1 e s s — Q2 (s, x - t + s)ds)

j c*

+

x a. , fts 1

- —xs I

f I f —(r-!)^ I I (• I /• —(r-s)+— I

JK\ t,s,p(x-x) + J e s s -Q/s,x-x + s)ds I-JK\ t,s,p(x-x)+Je s s -Q2(s,x-x + s)ds

x a. , fts 1

- —(r-Jls I

ftt t a, , fts 1

--/ \ Г (t-s)+ I г л

e e (a-sinnt)J e s s—[Qft,x-1 + s)-Q ft,x-1 + s)]ds

0 s

where it is considered that ||sin nt|| < I, V n G N. Now we impose on a, ft the following restrictions:

L + L +(a+i)

L + L +(a +1)

a + ft

||Q i(t, x) - Q 2(t, x)||,

a + ft

-< I. (13)

Then (11) implies that operator P [Q] is a compression operator on the set Q. On the principle of contraction mapping equation (11) has a unique solution Q(t, x) gQ .

Substituting the function found in (7), we obtain the solution of the Cauchy problem (5)-(6).

Now we investigate the differential properties of solutions of the Cauchy problem (5)-(6) in the region Q. For all Q(t,x) gQ from the equation (7) implies the inequality

\u(t, x)|| <|p( x -1 ^ +

From (8) we have

\\ul(t, x)\\ =

t a . fts 1

I e s s — Q(s, x -1 + s)ds l s

< K,„ + e

-aT h

a+ft

-- Kn = const.

I — a г -a(t-s)+—s I

-p'(x -1) + — e s Q(t, x)-----(u-p) -J e s s — Qx (s, x -1 + s)ds

< N - const.

< II P(x -1)|| +

I a t a. . ft 1 i* —(t-s)+^s -

- e s Q(t, x) + (u-p) + -I e s s — Q (s,x-1 + s)ds

s s J s 0 s

From (9) we have \Ux(t, x^ = ||p'(x - *^1 ■

< N2 - const.

t a. . fts л

* (t-s)+— I . 1

e s s — Qx (s, x -1 + s)ds

0 s

Thus we have proved that all the derivatives included in the equation (5) are uniformly bounded. Thus, we have

0

ftt

+

e

<

+

+

48

THEOREM. Let the assumptions (T), (12), (13). Then 3 T0 > 0 such that the Cauchy

problem (5) - (6) has a solution u(t,x) e C ^u)([0, T ]xR) , which has a representation in the form of the integral (7).

References

1. Erugin N.P. The book to read on the general course of differential equations: Edition-3, reworked and enlarged. - Minsk: Science and Technology, 1979. - p. 743.

2. Imanaliev M.I., Imanaliev T.M., Kakishov K. The Cauchy problem for nonlinear differential equations with partial derivatives of sixth order // Study on integral-differential equations. Bishkek: Ilim, 2007. - Issue 36. - p. 19-28.

3. Baizakov A.B., Aitbaev K.A. On a solution of Volterra equations with irregular Singularities // Abstracts of the IV Congress of the Turkic World Mathematical Society, Baku, 1-3 July, 2011. - Baku, 2011. - P. 145.

4. Imanaliev M.I., Baizakov A.B., Aitbaev K.A. The solvability of the Cauchy problem for integro-differential equations in partial derivatives // Report. International Scientific Conference "Functional analysis and its applications". - Astana, 2012. - p.135

5. Imanaliev M., Baizakov A., Kydyraliev T. Sufficient conditions for the existence of solutions of the Cauchy problem of partial differential equations of third order // Abstracts of the V Congress of the Turkic World Mathematicians, Kyrgyzstan, “Issyk -Kul Aurora”, 5-7 June, 2014. - P. 179.

Свойства времени Романенко В. А.

Романенко Владимир Алексеевич /Romanenko Vladimir Alekseevich - ведущий инженер-

конструктор,

Нижнесергинский метизно-металлургический завод, г. Ревда

Аннотация: в статье с критических позиций рассматривается вопрос о времени, которое входит в формулы специальной теории относительности. Анализируются свойства времени на основе бицилиндрической системы координат.

Abstract: this article critically examines the question of the time, which is included in the formula of special relativity. Analyzes the properties of time-based bitsilindricheskoy coordinate system.

Ключевые слова: синхронное время, 3-интервал, гиперболические функции,

преобразования Лоренца, постулат Бора, бицилиндрическая система координат. Keywords: synchronous time, 3-interval, hyperbolic functions, Lorentz transformations, Bohr's postulate, bitsilindricheskaya coordinate system.

1. Введение.

В статье автор предлагает осознать вопрос о времени Мира, в котором мы живём. Проблема далеко не проста. Начиная с древних времён, человечество пытается ответить на вопрос о том, что такое длительность. Древними цивилизациями были разработаны календари, по которым определялись промежутки времени, связанные с вращением Земли вокруг своей оси и вокруг Солнца. В более поздние времена были изобретены часы - приборы для измерения промежутков времени, связанных с сутками. Принцип действия часов самый различный, но суть не меняется - все они измеряют время в единицах времени: секундах, минутах, часах.

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