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THE AH PROBLEM FOR A LOADED EQUATION OF A PARABOLIC-HYPERBOLIC TYPE, DEGENERATING INSIDE THE REGION Juraev F.M. Email: [email protected]
Juraev Furkat Muhitdinovich - Senior Lecturer, DEPARTMENT OF DIFFERENTIAL EQUATIONS, FACULTY OF PHYSICS AND MATHEMATICS, BUKHARA STATE UNIVERSITY, BUKHARA, REPUBLIC OF UZBEKISTAN
Abstract: the following article deals with the provision of a unique solution of a problem similar tu the Hellerstedt problem for a loaded parabolic-hyperbolic equation with a breakdown within the field. The uniqueness of the solution of the AH problem is proved using the extremum principle, and existence is proved by the method of integral equations. At present, the range of problems under consideration for nondegenerate loaded equations of hyperbolic, parabolic, hyperbolic-parabolic, and elliptic-parabolic types has expanded significantly, also considered inverse problems posed equations of mixed type. Methods for numerical solutions of these problems are given.
Keywords: a degenerate loaded equation, boundary value problems, the Hellerstedt problem, the existence and uniqueness of a solution, equations of mixed types.
ЗАДАЧА АГ ДЛЯ НАГРУЖЕННОГО УРАВНЕНИЯ ПАРАБОЛО-ГИПЕРБОЛИЧЕСКОГО ТИПА, ВЫРОЖДАЮЩЕГОСЯ ВНУТРИ ОБЛАСТИ Жураев Ф.М.
Жураев Фуркат Мухитдинович - старший преподаватель, кафедра дифференциальных уравнений, физико-математический факультет, Бухарский государственный университет, г. Бухара, Республика Узбекистан
Аннотация: в данной статье доказана однозначно разрешимость решения задачи Геллерстедта для нагруженного уравнения параболо-гиперболического типа, вырождающегося внутри области. Единственность решения задачи АГ доказывается с помощью принципа экстремума, а существование - методом интегральных уравнений. В настоящее время круг рассматриваемых задач для невырождающихся нагруженных уравнений гиперболического, параболического, гиперболо-параболического и эллиптико-параболического типов значительно расширился, также рассматриваются обратные задачи, поставленные уравнениям смешанного типа. Даётся метод численных решений этих задач.
Ключевые слова: вырождающие нагруженные уравнения, краевая задача, задача типа Геллерстедта, существование и единственность решение, уравнения смешанного типа.
UDC 517.956.6
Boundary value problems for nondegenerate loaded equations of mixed type of the second and third order, when the loaded part contains a trace or a derivative of the desired function, were studied in the works of [1]-[4]. A three-dimensional analogue of the Tricomi problem for a loaded parabolic-hyperbolic type equation was studied in the work of [5].
19
As far as we know, boundary problems such as the Tricomi and Hellerstedt problems for a degenerate loaded equation of mixed type of the second order have been studied relatively little. We have considered the work of [6]-[9].
Now they study inverse problems posed to equations of mixed type [14], [17]. Their numerical solution is widely used [9], [11]- [13], [15]. In some cases, by showing the equivalence of equations of mixed type to systems of hyperbolic type, one can prove the existence and uniqueness of a solution to the original problem [10], [16], [18].
Based on this, the present work is devoted to the formulation and study of a boundary value problem such as the Hellerstedt problem for a loaded equation of parabolic-hyperbolic type, degenerating inside the domain.
1. Statement of the AH problem Lets consider the equation
IP
L
y
0 =
U — \x\P U — JLIM (x,0), y > 0,
xx I I y V /
m (1) - (—yT Uyy + Lj+2U (x,0), y < 0
m
U
xx
where m, p, J, Jl-+2 - m, p, J, Jj+2 - any real numbers, moreover
m < 0, p > 0, j > 0, jj+2 < 0, (/ = 1,2; j = 1,4) (2,)
We introduce the following notation: Q01 = Qq o {(x, y): x > 0, y > 0}, Q02 = Q0 o{(x, y): x < 0, y > 0}, Q3 = Q01 ^Q02 ^ J3, J3 ={(x, y) : x = 0,0 < y < 1}
Ju =|(x,y): 0 <x <x°, y = 0J, J12 =j(x,y): ^ < x < x0, y = o},
J21 =|(x,y): x0 <x<, y = 0}, J22 =|(x,y): <X<!, J = 0}, Jq ={(x,y): 0<x< 1, y = 0}, J = {(x,y): 0<x<x0, y = 0}, J2={(x,y): x0<x< 1, y = 0}, Q = Qn ^Q^ ^Q31 ^ J0,
J 1n =|(x,y): — f < x< 0, y = 0}, J'12 =j(^ y) : — x0 < x < , y = o} ,
J121 =|(x,y): — xo2+1 <x<xo, y = 0}, J^ = j(x,y): — 1 <x<— ^, y = o},
J1o ={(x,y): —1 <x<0, y = 0}, J\ ={(x,y): — xq <x<0, y = 0},
J'2 = {(x,y): —1 <x<x0, y = 0}, Q2 = Q12 ^Q22 ^Q^ ^ J10,
Q = Q3 Q1 ^ Q2
After some designations, in the field Q for the equation, (1, ) the following problem is studied:
;
Task AH. Is required to find a function u (X, y ) that has the following properties:
1) u (X, y) e C (Q) n Cjj (Q01 uQ02) n C2 (Q1 uQ2);
2) u (X, y) is a regular solution to the equation (l/) in areas Q ^ and Q j2 (j _ 0~3);
3) uyeC(Qz)that Uy (X, y) can handle infinity less order in l - 2p |x| ^ 0, andv|x| ^ 1, is
, . Fl ^
limited;
4) ux E C(Q3) on the line of degeneration J3 bonding conditions are met
limux (x y ) = xlim0 ux ( x y ) '(0' y) E J3 (3)
5) U ( X, y ) satisfies boundary conditions
U
I A,B,
u
E Cn
= ( (y)0 ^ y ^ 1 (4, )
V, (x) ' N e J,2, (5,)
u\e2C12 =VI (x), NeJ,2; (6,)
where (( (y ), Vf ( X) , Vi ( )) ,(i = 1,2)- defined functions, moreover
Wx ( •) = W 2 ( •) > wl1 ( ) = ¥2 ()
йМеОДПС1^з), (7,)
m
Here 2fi= , that
m - 2
0 <P< 2. (9)
2. The study of the AH problem for the equation (^ )
Theorem. If the conditions are met (2i), (7.), (8г.) and (9), in the areas Q solution of the
task AG exists and entire.
The uniqueness of the solution of the AG problem is proved using the extremum principle, and existence is proved by the method of integral equations.
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