О решениях некоторого класса частичных интегральных уравнений тема автореферата и диссертации по математике, 01.01.01 ВАК РФ

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ACADEMY OF SCIENCES OF THE REPUBLIC OF UZBEKISTAN

INSTITUTE OF MATHEMATICS of the name V. I. ROMANOYSKY

On the rights of the manuscript

¿001- yOH Z UDK.517.968

tro 9Jd

Ekrem CHULFA

ON SOLUTIONS OF SOME CLASSES OF PARTIAL INTEGRAL EQUATIONS

01. 01. 01 - Mathematical Analysis

AUTOREVIEW

A DISSERTATION OF SCIENTIFIC DEGREE OF

DOCTOR OF PHILOSOPHY (Ph.D.) in PHYSICAL AND MATHEMATICAL SCIENCES

TASHKENT - 1997

The work is executed on the Department of theory of functions and functional analysis in the Samarkand Alisher Novai State University

Scientific Leader - Doctor of Physical and Mathematical Sciences, Prof. S.N Lakaev.

OFFICIAL OPPONENTS:

The member of the Academy of Sciences of the Republic of Uzbekistan, Doctor of Physical and Mathematical Sciences, Prof. DJ. Hadjiyev.

Doctor of Philisophy in Physical and Mathematical Sciences, M.A. Berdikulov.

Leading Organization - Urgench State University.

Defence of the dissertation is held /lAv 1997

at at the meeting of Council D. 015.17.01 in the

Institute of Mathematics of the name V. I. ROMANOVSKY.

Adress:

700143 Tashkent, 143, F. Hodjayev str., 29.

The Dissertation is possible to be acquainted at the library of Institute of Mathematics of the name V. I. ROMANOVSKY.

Autoreview is published on QtL^ 1997.

^Scientific Secretary of the Council, Doctor of Physical and Mathematical Scie Sh. A Hashimov

GENERAL CHARACTERISTICS OF THE WORK

1. Actuality of the theme.

In 1952, Abdus Salam considered a "partial" integral equation which arises in the Quantum Theory of Fields. After then, in 1965, In the works of V.A Morozov dedicated to some problems of Building Mechanics, lead to some partial integral equations for two variable functions. After him, in 1975, Whenever L.M. Lichtarnikov had been studying some questions of technique, namely, the problems of stability of rotor also led to a partial integral equation.

Besides these works, Whenever K.O. Friedrichs, S.P. Merkuriev and S.N. Lakaev had been studying on the investigations of spectral properties, particularly, bounded states of many particle operators which appear in Quantum Mechanics, Solid Physics and Quantum Field Theory, their studies led up to the Partial Integral Equations.

2. Goals of the work.

The goals of our work appear as, under some natural condition«; looking for the existence and uniqueness of solution of Partial Integral Equations for three or more non-stop functions with two variables and proving the equivalency of these equations to the Second Type Fredholm's Integral Equation.

3. Methods of the investigations.

During our studies, we applied Modification of the "'Fredholm's Method" for the investigation of "Fredholm's Type" Integral Equations.

4. Scientific Discoveries.

All of our main results are new. During our studies, v/e proved:

i) Partial Integral Equations with Fredholm's Type kernels are, under some conditions based on the kernels, equivalent to the Second type Fredholm's Integral Equation.

ii) Partial Integral Equations with N-dimensional degenerate kernels are equivalent to the Second type Fredholm's Integral Equation with degenerate kernels.

iii) Existence and uniqueness of solution of Partial Integral Equations with 1-dimensional degenerate kernels and for the solution an exact formula obtained,

iv) Decidability problem of Partial Integral Equations with weak singularities are reduced the decidability problem of second type Fredholm's Integral Equations with weak singularities.

5. Theoraticai and practical quality of the work.

Results of the dissertation has theoraticai characters. They can be used in any problems of Integral Equations. Mathematical Physics, Mechanics and Techniques.

6. Reports of Seminars and Conferences.

Main results of the dissertation reported in the following seminars and conferences

i) Seminars:

a) Department of Functional Analysis, directed by Prof. S.N Lakaev, Samarkand State University, 1997;

b) Department of Mathematics Analysis, directed by Prof. S.A. Ayupov, institute of Mathematics of Academy of Sciences in Uzbekistan, 1997;

c) Department of Functional Analysis, directed by Prof. V.I Chilin, Tashkent State University, 1997;

d) Department of Functional Analysis, directed by Prof. N.N Ganihod-jaev, Institute of Mathematics of Academy of Sciences in Uzbekistan, 1997.

ii) Conferences:

a) Conference of Young scientists, dedicated to 600 years anniversary of Mirzo Ulughbek's Birthday, Samarkand, 1994.

b) Conference of Young Physicians and Mathematicians, dedicated to 75 years anniversary of Taskent State University, 1995.

c) Conference of Some problems on Complex Analysis, dedicated to 60 • years anniversary of Urgench State University, 1995.

d) International Conference on some topics of Mathematics, October, 199G.

7. Publications.

The main results of the dissertation published in the works of [1-6].

8. Structure and volume of the work

The work consists of an introduction, a short maintenance of dissertation and two chapters. First chapter consists three units. Second chapter has a unit. Volume of the dissertation covers 119 pages, in the list of literature 47 named.

SHORT MAINTENANCE OF THE DISSERTATION

The main purpose of our work was an establishment of Mathematical foundation for the solution of the Nonhomogeneous and Homogeneous Partial Integral Equations and researching the Fredholm's Second Type Integral Equations which are equivalent to these Partial Integral Equations.

In the first chapter, during our investigations on Partial Integral Equations, we considered and examined the following nonhorr.ogeneous and homogeneous partial integral equations

6 h

<p(x,y)-\x J K(x,s)?{s,y)ds - A2 j N(yJ)?(xJ)dt-

a a

b b

-A J J M{x,y,,.%l.)-(6. t)dsdt = f(x,y), (1)

a a

b b V?(x,y)-Ai j K(x,sMs,y)ds- A2 J N(yJ)^(x,t)dt-

i i

— A j J M(x,y\sJ)risJ)dsdt = 0 (!')

a a

where, the known functions A", A' and are continuous functions of their arguments on [a, 6]2 and [a, 6]4 respectively. We are looking for the continuous unknown function tf(x,y). on [a,i]2.

At first, we examined the equation (1) and corresponding homogeneous partial integral equation with the kernels:

i) I\(x,t),N{y,s) and M(x,y,t,s) are any arbitrary nondegenerate continuous functions of arguments on [ct, 6]2 and [a,6]4 respectively.

ii) I\(x,y) = ]T K(x)k,(y), Ar(y. = VAxWAs)

j=i

and

iii) A'(.t, y) -- h\(x)K2(y), -V(y, I) = .\\(y)N2(t)

and M(x, y, t, s) = Mj(x,y)M2(t.s), are degenerate kernels.

Now it's possible to discuss the summaries of each study: In the starting work, for simplicity we assumed the variables x, y,t, and s hold a<x, y, t, s<b.

=i

And then we paid our attensions to search the existence of second type Fredholm's Integral Equations equivalent to the equations (1) and (1). We have denoted Fredholm's determinant and minor by

CO

D,{X,) = 1 + where

t> 0 J "J

K,(tuh) K(tut2) ... A',(i,,<,) A',(t2.t,) K,(t7.t2) ... K,(t2,t,)

dtx...dta. z = l,2.

h\(t.,ti) A',(f„f2) ... A',(t.,i.)

00

Dt(x,y;\t) ~ I\t(x,y) + where dls(x,y) -

6 b

dt\.. .dti% i = 1,2.

A',(.r.y) A',(a:,ii) ... K,(xJs) h\(tuy) K,(U,y) ... AT.(ii,i.)

A'i(is,y) K,(t.,ti) ... A',(<„*,)

If in addition .D;(A,)/0, we have denoted the kernel G;

2/i A.) = A(A[) . , = 1,2

Let's introduce denotations F(x,y) and L(x,y,s.t\Xi,X2,X) for every function f(x,y) and kernels Gi,G2

b

F(x,y) = /(.x,y) + A1 J Gl(x,y,Xlf(t,y)dt +

a 6

+A2 J G2{y,s\X2)f(x,s)ds +

t b

+Ai A2 J J G1{x,t;X1)G2{y,s;X2)f(t,s)dtds,

L(x.y.s.t-.XUX2,X) = XiX2Gi(y.s;Xl)G2(y,s: A2) +

6

+\K(x,y; t,s) + AiA J Gi(xj':Xi,)l\(t\y;t,s)di'A-

a

b

+A2A J G'2(;/..s': A2), I\{x,s';t.s)ds' +

a

b b

4-AA,A2 J J G1(x,t':Xi)G7.(y,s'-,X2)K(t',s'\t,s)dt'ds'

a a

Theorem 1.1. // Fredholm's determinant D,(A,)^0,i = 1,2, then the nonhomogeneous partial integral equations (1) with any nondegenerate kernels A, A* and M is equivalent to the following nonhomogeneous "total" integral equations in two variables

tp(x,y) = F(x,y) +

6 b

Jh

a\y); s,t; A]. A2, X)ip[s,t)dsdt

(2)

Corollary 1.1.if the Fredholm's determinant D,( A,)^0, » = 1,2, then the hornogeneous partial integral equation (1') with any nondegenerate kernels K, .V and M is equivalent to the following homogeneous "total" integral equation

& b

f y/>(.T,y);M;A1,A2,A)ss(.M

)dsdt

(2')

Here we proved that if the kernels of partial integral equations are degenerate and satisfy naturally some conditions, then the obtained kernels of equivalent second type Fredholm's Integral Equations are also degenerate. And explicit forms of the degenerate kernels of the required Integral Equations are obtained. Before the theorem, we may introduce the Fredholm's determinants Ak and A(v as follows

AK =

1-AiA'n -AjA'u ••• -AiA'in —AiA'2i -A[A'22 ... — AiA'2„

-AiA'm — Ai A'„2

An;

A v =

l-A2An -A2A'i2 ••• --^jYin -A2/V2. — A2jV22 ... ->2;V2n

—A2JV„I — A2A„2

, where,

t b KtJ= j k,(x)Kj(x)dx,i,j = T^,Nst = J N,(z)N,(z)dz,

□ a

. / = 1: 772 and A1*1 the first minor of the element in the sth row and ilh column of Ar- and can be defined similarly. Let

Drj = {—l)r+'&[,j\r,j = T7m and the functions $ and "I* are expressed in terms of Fredholm's determinants A,y and A,v. and the degenerate kernels K, and A',.

*.(<) = ¿rA''W4» = (-U^ii0. = iT^. iZ Ak

m n

= E = t-1)^^' = ^

and also the degenerate kernel I\ expressed by K„N,, Mk, Mk, and

tyj. i = 1; n, j = 1; ?72,

K(x, y\ t,s\ A) = XM{x,y:t,s)+

0

+a£/v.(.v) J <S>t(t')M(t',y,s,t)di' +

J(z)M[x.z;t,»)d: +

f

+ A $>'(!/) / 4M

■> = 1 a

b t

n m . .

h'MNAy) / ^,(t')<fr{z)M(t'.z-t,s)dt'dz

.=i j=i

+

Theorem 1.2. Let Aa-^0 and AjV^0. Then the equations (I) and (V) with degenerate kernels K. N and M are equwahnt to the fallowing nonhomogeneous and homogeneous second type Fredhotm's integral equations with degenerate kernel K respectively

b i

- J J K(x,y,t,s)?(t<s)dUls = F(x,y), (3)

a a

b b

V{x,y)~J J k{x,y,t,s)v(t,s)dtds (3'")

a a

Corollary 1.2. Let the arbitrary numbers Ai,A2€C and the degenerate kernel M(x,y; i,s)sO. Then the nonhomogeneous and homogeneous partial integral equations (1) and (1') with degenerate kernels A' and N are equivalent to the following nonhomogeneous and homogeneous second type Fredholm's integral equations with degenerate kernels respectively

n m " 6

^x,y) = F(x,y) + \1\2J2y£ j f h\(x)N}(y)x

;=i j=i J J

x$,{t)qj(z)p(t,z)dtdz, b b

n m .

'=' J=1 a {

In the work, for simplicity we assumed the variables x,y,t and s hold a<i\y,t,s<b and the functions A'i, A'2, Ari, N2, Mi and M2 are continuous functions of their arguments, and we look for the unknown function ^(x,y) which is continuous on [o, 6]2.

Here, we paid our attention to search the existence of the solutions of equations (1) and (1') and then we arrive at'the following important results;

Under some conditions:

i) Nonhomogeneous and homogeneous partial integral equations (1) and (T) are equivalent to the second type Fredholm's Integral Equations;

ii) Nonhomogeneous partial integral equation (1) has a unique solution;

iii) Nomogeneous partial integral equation (!') has nontrvial solution.

After entering the necessary denotations which are abbreviations of conditions of the theorem, we stated tiie theorems of our main results:

¿(A;) - 1 - A. J A\(s)K3(.i)ds, t

6(A2) = 1 - A J ^ Nt(t)N2(t)di and t s

=1 " ï

a a

6 t

x M2{t,s) J J KiMNiWMiit'^'Wds'^Gis^dsdt

where.

t>

AA /"

G(s,<) = AWi(i,i) + ^A',(i) J K2(*')M1(s',t)ds'+ b

6 b

x J J Kils^Xiit^M^s'jys'dt'

And we abbreviated ,f(x,y) by

b

f(x,y) = f(x,y)-f^y I I<2(s)f(s,y)ds +

¿gfV*«*'.')**

b i

+ J J KMWMdsdt.

a a

Theorem 1.3. Let i/(A,)^0. £(A2)/0.1 - ¿(X,) - S{A,)*0 a) If in addition. Ji(A], A2, A)^0, Men the nonhomogencous partial integral equation (lj with simplest degenerate kernels K, N and M has a unique solution for any arbitrary function f(x,y) in the form of

•p(-r-y) = f(x,y) +

b

jg^//«*.*«»*-

+

x

/> i>

AiWM

x

J J M2{s,t)f(s,t)dsdt

G(r. !/) + j _ ¿^j _ ¿(A2") X

: j J^ K2{s)N2{t)G(s,t)dsdt

where,

b b

Pw/J Kli'bWiMdsdi.

" 1 -d(A) -<5(A2)

a a

¿J // hi addition. A(Ai,A2,A) = 0. iAfii iAe homogeneous partial integral equation (1') icith simplest degenerate kerne's K. A' a"(/ M has a nontrivial solution.

Suppose M(i,y\i,s)=0. Then automatically the nonhomogeneous and homogeneous equations (1) and (1') become

b

y-(r,y)-A] j I\'[x,s)ifi{s,y)ds-

a

b

-A, j N(y,t)^x.t)dt = f{x,y), (4)

tfi{x,y)~Xi J K{x,s)f[s,ij)d6-

a

b

-hj N(y,tMx,t)dt = 0 (4')

Corollary 1.3. Lei d(Ai <5( A2)^0 and if

i) a'(Ai) + i5(A2)/1. Then the nonhomogeneous partial integral equation (4) with simplest degenerate kernels K and M has n unigue solution as follows

6

*(*,!/) = /(*,!/)+ - J I<2(t)f(t,y)dt+

a

6

+slt)Nl[y) J

a

b i

+7(Ai,A2)A',(i)ATl(y) J J K2(s)K2(s)f(s,t)dsdt,

a a

where,

(X ).)- +

71 " 2> ¿A1)5(A2)(ci(A1) + 5(A2) - 1)

ii) ii(Ai) + r5jA2) = 1, then the homogeneous partial integral eqwuation (4 ') with simplest degenerate kernels 1\ and M has a nontrivial solution.

In the second chapter, we have considered the nonhomogeneous and homogeneous partial integral equations in two variables with weak singularities and examined that whether we can reduce the decidablity problem of these Partial integral equations to the decidablity problem of "Second Type Fredholm Integral Equations" with weak singularities, or not. ?

We have examined the following nonhomogeneous and homogeneous partial integral equations with weak singularities in the forms of

l

= /o(a'.y) + j t}'p[t,y)ilt+ o

+ / --(a)

J If -

, n iHAxj)

= J j^zjf-At

n

+ / ^ w

J IP - si

o

where, all considered kernels //j, //2 and the function /0 are known and continuous functions of their arguments on the rectangular region Q = {0<.r.t/<l} and a,0 hold the condition 0 < a,0 < j.

At first, we introduced the necassary denotations given below

1 i

i I*-'I y J

0

1

0

■>(»,«') = /

0 (|»-«r)(|i-i'n

We have introduced the Fredholm's determinant and minors for the kernels A',, 1 = 1,2 and resolvent function G;. (See p.6)

We now introduce the function F interms of the function / and the kernels G'i and G2 as

l

F(.T,y) = f(x,y) + J G,(i,a)/(a,y)di +

o

i l I

+ J GAy,s)f(x<s)ds + J J G1(xJ)G2(y,s)f(i,s)dtds

0 0 0

and

S(x, y, t,s) = G\{x,s)

K2(y,s) + J G2(y.t)k'i(t,s)dt

x\x ~ s\°\y - sf + K{x,t)

H2{y,s)+\y - sfx

ft(«'.«)

(|t'-*n

dt'\ + H,{xJ)

1

+1 v-*t j

G2(y,

\t'-sf

Here the function A' denoted by

Main result of the chapter 2 as follows

Theorem 2.2. If Fredholm's determinant D,^ 0, i = 1,2, then the non-homogeneous and homogeneous partial integral equation (6) and (6') with weak singularities are equivalent to the following nonhomogeneous and homogeneous "Second type Fredholm's Integral Equation" with iceak singularities respectively

<p{x, y) = F(x,y) +

S{X'y;Us) Mt,s)dtds (6)

and

i I

vU\y)= i [ <j(t,s)dids.

J J ix-in«-(r

o

0

10. Main results of the dissertation announced in the following theses and articles:

1. S.N Lakaev, E. Chulfa, "On solutions of Partial integral Equations," Conference of the works of young scientists dedicated to the 600 years anniversary of Mirzo Ulughbek's Birthday, p.38, Samarkand,199 )

2. J.I Abdullaev, E. Chulfa, "Solutions of partial integral equations with degenerate kernels, " Conference of Some problems on Complex Analysis dedicated to 60 years anniversary of I'rgench State University, p.57, 1995

3. S.N Lakaev, E. Chulfa, "On partial integral equations of two variable functions," Conference of young Physicians and Mathematicians Which is dedicated to 75 years anniversary of Toshkent State University, p.161, 1995

4. S.N Lakaev, E. Chulfa, "Equivalent partial integral equations to the Fredholm's integral equation", International Conference on some topics of Mathematics, p.63, Samarkand , 1996

5. E. Chulfa, "Equivalent partial integral equations to the Fredholm's Integral Equation," The journal of Uzbek Mathematics, V.2, p.109 113, Tashkent. 1997.

6. E. Chulfa, "Fredholm Solutions of partial integral equations," The journal of Academy of Sciences of Uzbekistan, V.7, p.9-13, Tashkent, 1997.

Хусусий интеграл тенгламалар баъзи синфлари ечимлари да^ида.

Диссертация Фредгольм типидаги хусусий интеграл тенгламаларнинг баъзи синфларининг ечимлари мавжудлиги ва ягоналигини урганишга багишланган.

Диссертацияда олинган асосий натижалар ^уйидагилардан иборат:

1. Фредгольм типидаги умумий ядроси хусусий интеграл тенгламанинг кккинчя тур Фредгольм интеграл тенгламасига эквивалентлиги исботланган.

2. Ы-улчамли ажралган ядроси хусусий интеграл тенгламанинг ажралган ядроси иккинчи тур Фредгольм интеграл тенгламасига эквивалентлиги курсатилган ва интеграл тенглама ядроси берилган хусусий интеграл тенглама ядролари ор^али ифодаланган.

3. Бир улчамли ажралган ядроси хусусий интеграл тенглама ягона ечимининг ани^ ифодаси топилган.

4.Кучсиз махсусликка эга булган Фредгольм типидаги хусусий интеграл тенглама ечими ягоналиги масаласи махсусликка эга булган Фредгольм интеграл тенгламаси ечими ягоналиги масаласига келтирилган.

О решениях некоторого класса частичных интегральных уравнений.

Диссертация посвящена исследованию вопросов существования и единственности решений некоторого класса частичных интегральных уравнений типа Фредгольма.

В диссертации получены следующие основные результаты:

1. Доказано, что частичное интегральное уравнение типа Фредгольма с общим ядром эквивалентно интегральным уравнениям Фредгольма второго рода.

2. Показано, что частичное интегральное уравнение с Галерным вырожденным ядром эквивалентно к интегральным уравнениям с вырожденным ядром. Найдено выражение для ядра полученного интегрального уравнения через ядра данного частичного интегрального уравнения.

3. Найдено явное выражение решения частичного интегрального уравнения с одномерным вырожденным ядром.

4. Вопрос о единственности решений частично интегрального уравнения типа Фредгольма со слабой особенностью сводится к проблеме единственности решений интегрального уравнения со слабой особенностью.