CC BY
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Вестник КРАУНЦ. Физ.-мат. науки. 2016. № 3(14). C. 14-22. ISSN 2079-6641
DOI: 10.18454/2079-6641-2016-14-3-14-22
MSC 35M10
BOUNDARY-VALUE PROBLEMS WITH GENERALIZED GLUING CONDITIONS FOR A LOADED DIFFERENTIAL EQUATION WITH A PARABOLIC-HYPERBOLIC OPERATOR
B. Islomov, U. I. Baltaeva
National University of Uzbekistan named after Mirzo Ulugbek, 100174, Uzbekistan, Tashkent c., University st. 4 E-mail: [email protected]
In the present work we study the unique solvability of local boundary value problems with generalized gluing conditions for the third order differential equation with a loaded parabolic-hyperbolic operator.
Key words: loaded equation; equations of mixed type; parabolic-hyperbolic operator; integral equations; gluing condition.
© Islomov B., Baltaeva U.I., 2016
УДК 517.956
КРАЕВЫЕ ЗАДАЧИ С ОБОБЩЕННЫМИ УСЛОВИЯМИ СКЛЕИВАНИЯ ДЛЯ НАГРУЖЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ С ПАРАБОЛИЧЕСКО-ГИПЕРБОЛИЧЕСКИМ ОПЕРАТОРОМ
Б. Исломов, У. И. Балтаева
Национальный Университет Узбекистана им. Мирзо Улугбека, 100174, Узбекистан, г. Ташкент, ул. Университетская, 4 E-mail: [email protected]
В настоящей работе исследуем однозначная разрешимость локальных краевых задач с обобщенными условиями склеиваниями для дифференциального уравнения третьего порядка с нагруженным параболо-гиперболическим оператором.
Ключевые слова: нагруженное уравнения; уравнения смешанного типа; параболо-гиперболический оператор; интегральное уравнение; условия склеивание.
(с) Исломов Б., Балтаева У. И., 2016
Introduction
The theory of mixed type equations is one of the principal parts of the general theory of partial differential equations. The interest for these kinds of equations arises intensively due to both theoretical and practical uses of their applications. Currently, the concept of mixed-type equations has expanded to include all possible combinations of two or three classic types of equations. In this the necessity of consideration the theory of parabolic-hyperbolic type equations for the first time was specified in 1956 by I.M.Gelfand [1].
On the other hand, recently, in connection with intensive research on problems of mathematical biology [2], optimal control of the agro-economical system [3, 4], long-term forecasting and regulating the level of ground waters and soil moisture [3], and also in the study of inverse problems [4], the numerical solution of integro-differential equations [4], the linearization of nonlinear equations [5] it has become necessary to investigate a new class of equations called "loaded equations".
Basic questions of the theory of boundary value problems for partial differential equations are the same for the boundary value problems for the loaded differential equations. However, existence of the loaded operator does not always make it possible to apply directly the known theory of boundary value problems for non-loaded partial differential equations.
Boundary value problems for mixed type equations are considered in mixed domains. To determine the solution of boundary value problem in the whole domain, the desired function and its derivatives should be glued continuously or by a special gluing condition on the lines, which separates various parts of the mixed domain. Special gluing conditions sometimes generalize continuous gluing conditions and they are used because of their physical meaning [6], [7]. Here we consider the boundary value problems (such as Tricomi) for the linear loaded differential equation of third order, with the parabolic-hyperbolic operators with the following gluing conditions
u(x, +0) = a1(x)u (x, -0) + Yi (x), 0 < x < 1, (1)
d+Q) = ft (x)du(x: 0) + a2(x)u(x,-0) + %(x), 0 < x < 1, (2)
dy dy
a1(x)ft1(x) = 0. This special gluing condition, is usual in the theory represents the equality of temperatures and streams on the boundary of oscillation bodies with different tenses [6], [7].
The main results
Statement of the problem and main functional relations
Let consider the loaded equation
d
(a— + c)Lu = 0 (3)
d x
_ 1 - sgny 1 + sgny . t Lu = uxx--2-Uyy--2-Uy - ^u - ^0)
for (x,y) g Q, where Q is a simple connected domain bounded by y > 0 with segments AA0,BB0,A0B0, (A(0,0), B(1,0),A0(0,h),B0(1,h)) and by y < 0 with characteristics AC : x + y = 0, BC : x — y = 1 of equation (3). We let
Q1 = Qn{y > 0}, Q2 = Qn{y < 0}, I = {(x,y) : 0 < x < 1, y = 0}.
Then a,c,X,/ are given real parameters in Q;, i.e. X = (— 1)i+1Ài,/ = / for i = 1,2 respectively and a = 0.
We investigate the following problem:
Problem 1. To find a regular solution of (3) from the class of functions
W2 = {u(x, y) : ux g C(AA0), u g C(Qn C1 (Q; UIU AC), i = 1, 2}; satisfying boundary conditions
y)|AA0 = y)|BB0 = y)|AA0 = <P3(y), 0 < y < h
(4)
y)|AC =
d u(x, y)
d n
AC
= V2(x), 0 < x < 2,
(5)
and the gluing conditions (1) and (2), where n is the interior normal, ^1(y), ^2(y), ^(y), 1i(x) and 12 (x) are given functions, such that ^1(0) = 11(0). Theorem 2.1. If the following conditions
A > 0, Çj(y) G C1 [0;h], Çs(y) G C[0;h] nC1(0;h), j = 1,2,
(6)
H(x) G C1
0;2
nC^0;1 ), ^2(x) G C
0;2
n C2(0-
ai(x), Yi(x) G C1 [0; 1] n C3(0; 1), fr(x), a2(x), 72 (x) G C[0,1] n C2(0; 1),
are fulfilled, then Problem 1 has a unique solution.
Proof. We start by introducing the following notations:
u (x, +0) = t1 (x), u (x, -0) = T2(x),
(7)
(8)
(9)
uy (x, +0) = v1 (x), uy (x, -0) = v2(x),
"yy (x, +0) = jU1(x), "yy (x, -0) = ^2(x).
Then instead of (1), (2) we have
T1(x) = «1(x)T2 (x) + Y1 (x),
Supposing
V1 (x) = ß1 (x) V2 (x) + «2 (x) T2 (x) + 72 (x).
u(x, y) =
"1 (x,y), (x,y) G ^1, "2(x,y), (x,y) G ^2,
(10) (11)
(12) (13)
equation (3) can be represented in the form of two systems:
LiUi = Uixx - Uly - À1U1 - ßiUi = Ui(x,y),
au1x + cu1 = 0,
(x, y) G ßi,
L2U2 = U2xx - U2yy + A2U2 - M2U2 = U2(x,y), \ , x ^ n
aU2x + CU2 = 0, py) G ^
(14)
(15)
where u1(x,y), u2(x,y) arbitrary continuous functions.
By virtue of the second representation, system (15) are reduced to the following form in the domain Q.2
L2U = w2(y) exp ,
and to change of variables £ = x + y, n = x - y, we get
¿2 M2 ( £ + n ^ 1 (£ - n .
U2£n + -4 U2 - "JM —, 0 ) = 4W2Î I X
(16)
X exp
c(£ + n )
2a
boundary-value conditions (5) are reduced to the form
U2=0 = Vi(y) , 0 < n < 1,
and
d U2
d£
£ =0 = * (2) , 0 < n < 1.
(17)
(18)
(19)
It is know that every regular solution of equation (17) in with the boundary condition (18), and
(U2| - U2n)|n =£ = V2(£), 0 < £ < 1, (problem Darboux) is represented as
(20)
x+y
U(x,y)= y V2(£)J^A2 (£ -x -y) (£ -x + y)J d£ +
0
+W(0)Jb
x-y
^2 (x2 - y2) + - J vi B (0, n ; x+y, x - y) dn+
(21)
x+y x—y
+1 /d£ / B(£,n; x+y,x—y) i^) + i--exp(—^^ dn,
were B (£, n ; x + y,x — y) is the Riemann-Hadamard function [9], J0 [z] is the Bessel function[10].
Using boundary condition (19), we take into consideration the property of B(£,n; nO [8] we get
n n
Mz/ T2 (0 dt + /w2 (-0 exp (-= ^ (2) +
0 0
n
11 (£) dt - 21 (0) - 4V2(0) - A2n 11 (0),
(22)
Differentiating (22) with respect to n, taking account of v2(0) = u2y(0,0) = u1y(0,0) = (0) and (0) = 2 v/212(0) - (0) , we find the function w2(-2):
W2 (-2) = {-M2T2 (2) + (2) + A2I1 (2) - A2H(0)|exp n, (23)
where w2 (-2) in -2 < -^ < 0. As in characteristic triangle be realized inequality 0 < £ < n, then - 2 < < 0. Therefore in place of w2 (-72) we can take w2 ^.
Substituting into (21) expression of w2 () with regards (23) and
u, n )|n=£ = T2),
and after some transformation we find main functional relation between the function T2(x) and v2(x) on AB in the domain Q2:
T2(x)^ v2(t)/0 VÄ2(x-t) dt + M(x)+ 0
(24)
x y
+tJ d'SJ"
0 t
\/A2(t - x) (s - x)
T2( - T2( V) exp(- 2a(t+s)
ds,
where
M(x) = - 11 (0) J0 [ v^x
- A22 J x^1
0
V7Ä2x(x - t)
11 dt + (25)
+2 J dtJ J0
0t
\JÄ2 (t - x) (s - x)
V21W V) + A211 ( ^ ) - A211 (0)
e a ds.
Hence, after some transformations we have:
T2(x) - ß2j n(x,t)T2(t)dt = J V2(t)J0 VÄ2(x-1) dt + M(x), (26)
00
x
x
xx
x
x
where
x—2/
f J0yjA2(s - x) (2t - s - x)dt + / J0 a/A2(s - x)(2t + s - x)
n(x, t) = < * exp (-a (t + s)) dt, 0 < t < 2,
j J0 V¿2(s - x)(2t - s - x)
2t x
dt, 2 < t < x.
Representation (26) is the main functional relation in ß2. Present we need to get second relation of betweenness these functions. To this end equation (3) for y > 0, bearing in mind (14) rewrite in view of
L1U1 = w1(y) exp (- -xj ,
(27)
Passing to the limit in (27) at y ^ +0 taking into consideration necessary conditions problem, (9), (10), [8] we have:
<(x) - Vi (x) - (¿i + ji ) Ti (x) = wi (0) exp (-£x) ,
(28)
where w1(0) is an unknown constant to be defined. Equality (28) is the second functional relation between t(x) and v(x), transferred from the domain to AB.
Uniqueness and existence of the solution
From (26) and (28) bearing in mind (12), (13), taking account of
A-
T2(0) = Vi(0), T2(x) = yi(0) + / T2(t)dt, T2(0) =
*3(0) «1(0) w(0) - m, (29)
a1 (0) a1 (0)
ai(0)'
using integration by parts and after some transformations we have
x
t2 (x) -J n1 (x, t) t2 (t )dt = w1 (0) M2(x) + M1 (x),
in which
n ( ßi(x) 1 («i(t)ßi(t)) r
Hi^ t ) = -— I1--¡r^-J0
«i(x) ß2(t )
J0 (x - s)]
\/Ä2(x - t) +\fhl
-«1 (t ) !ßi(t )
J1
(x - t )
ßi(s)
(a1 (s) - (Ai + Ui) ai (s) - 02(s)) + j2n(x,s)
Mi(x) = OH{( ^3(0) - Yi(0) - a(0M(0)) -M(x)+
(30)
ds}, (31)
+ yi(0) i 1 - J
J0 [VA2(x -1 )] ßi(t )
(a11(t) - (Ai + jUi) ai (t) - o2 (t)) + ^(x, t)
dt} I +
*
-t)]
ß1(t)
(tf(t) - (Ai + Ml) n (t) - 72(t))
dt,
Mil /-^JV^-M exp c \
2W 01 (i) V a/ '
(30) is the second kind Volterra type integral equation. From the representations of the functions n1(x,t), and Mi(x) applying known properties of the Bessel function [10], taking account of (6)-(8) and based on the general theory of integral equations, one can easily be sure that (30) has a unique solution, which is represented as
where
T2(x)= Mi(x) + wi(0)M|(x),
Mi(x) = 1^(0) + ^ I M1(t) + JR(t,s)M1 (s)ds I dt,
M|(x) = J
M2(t)+ / R(t, s)M2(s)ds
dt,
(32)
where R(x,t) is the resolvent of the kernel n1(x,t). Hence, by virtue of the condition (12), t1(1) = $2(0), with respect to a1(1) = 0, w1(0) are determined uniquely. Since the Problem 1 reduced to the equivalent integral equation (30), from the unique solvability of the equation (30) we can conclude that Problem 1 has a unique solution. The solution u2(£,n) of the Problem 1 in is determined by the formula (21), where the function T2(x) can be defined by (33) and the function v2(x) using the representation
0 a/A"
of the integral operator C0f [12], by the formula (26).
For determination function u1(x,y) in domain problem 1 reduce to problem: (4),
u1(x, 0) = T1(x),
for equation
^adX + Cj (w1xx - - A1M1) = Ti (x) + CM1T1 (x).
Introduce new unknown function u(x,y), by inversion formula
U1(x, y) = e-A1yu (x, y). Problem in this equivalent way reduced to Problem A:
d
a— + c ) (Uxx - Uy) = F(x,y),
(33)
(34)
u (0,y) = eA1y 91 (y), U (1,y)= eA1y 92(y)
= eA1y,
x
t
Ux (0, y)= eA1y 03 (y), u (x, 0) = t1 (x),
here F(x,y) = —(x)c^1e^1yT1(x) — well-known function.
Unique solvability problem A proved in [11, §2, chapter 4]. We can conclude from these that u(x,y) e C(Qf) nC1^,- UI) nC3,1^)HC3,2(^), i.e., there exists a regular solution of Problem 1.
Analogously, we can investigate the following problem:
Problem 2. To find a regular solution (3) from the class of functions
W2 = {u(x,y): wx e C(AA0), u e C(Q,-) nuIUBC),, = 1,2}, (36)
satisfying boundary conditions (4),
ufoy)|Bc = faCx^ ^U(X,y)
d n
= <p2(x), 2 < x < 1, (37)
BC 2
together with the gluing conditions (1) and (2), where n is the interior normal, <p1(y), 02(y), (y), fa(x) and ^(x) are given functions, moreover 02(0) = $1(0), a1(x)ft1(x) = 0.
Similarly as in Theorems 2.1, we can obtain the following expression.
Theorem 2.2. Let «1(x),ïï(x) GC1[0;1]nC3(0;1), fr(x),a2(x),7>(x) GC1[0,1]nC2(0;1),
(y) G C1[0;h], 03(y) G C[0;h] nC1[0;h), j = 1,2,
Y1(x) G C1
2;1
nC3( 1; 1 ), ^(x) G C
2;1
n C2(2;1 ),
then there exists a unique solution to the Problem 2 in the domain Q.
References
[1] Gel'fand I. M.,, "Some questions of analysis and differential equations", Uspekhi Mat. Nauk Ser, 14:3(87) (1959), 3-19.
[2] Nakhushev A.M., Equations of mathematical biology, Vishaya shkola, Moscow, 1995, 301 pp.
[3] Wiener, J., Debnath, L., "Partial differential equations with piecewise constant delay", Internat. J. Math. and Math. Scz, 14 (1991), 485-496.
[4] Kozhanov A. I., "Nonlinear loaded equations and inverse problems", Comput. Math. Phys., 44:4 (2004), 657-675.
[5] Nakhushev A.M., "Loaded equations and their applications", Differential equations, 19:1 (1983), 86-94.
[6] Djuraev T. D., Sopuev A., Mamajonov M., Boundary value problems for the parabolic-hyperbolic type equations, Fan, Tashkent, 1986.
[7] Eshmatov B. E., Karimov E. T., "Boundary value problems with continuous and special gluing conditions for parabolic-hyperbolic type equations", Centr. Eur. J. Math., 5(4) (2007), 741-750.
[8] Baltaeva U. I., "Solvability of the analogs of the problem Tricomi for the mixed type loaded equations with parabolic-hyperbolic operators", Boundary Value Problems, 211 (2014), 1-12.
[9] Sabytov K. B., "Construction in an explicit form of solutions of the Darboux problems for the telegraph equation and their application in generalizing integral equations", Differential equations, 25:6 (1990), 1023-1032.
[10] Bateman G., Erdelji A., Higher transcendental functions. Bessel functions, functions of parabolic cylinder, orthogonal polynomials, McGraw Hill, New York, 1953.
[11] Djuraev T. D., Boundary Value Problems for Equations of Mixed and Mixed-Composite Types, Fan, Tashkent, 1979, 240 pp.
[12] Salakhitdinov M.S., Urinov, A.K., Boundary value problems for equations of mixed type with a spectral parameter, Fan, Tashkent, 1997.
[13] Islomov B., Baltaeva U.I., "Boundary value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients", Journal of Differential Equations, 2015 (2015), 1-10.
Поступила в редакцию / Original article submitted: 14.09.2016
