Non-local Boundary Value Problem for the Third Order Mixed Type Equation in Double-connected Domain

ABDULLAYEV O.Kh

National University of Uzbekistan,Tahskent 700-174,Uzbekistan.

Received 7 March 2014;Accepted 31 August 2014

Non-local Boundary Value Problem for the Third Order Mixed Type Equation in Double-connected Domain

ABDULLAYEV O.Kh?

National University of Uzbekistan,Tahskent 700-174,Uzbekistan.

Received 7 March 2014;Accepted 31 August 2014

.In the present paper an existence and uniqueness of solution of the nonlocal boundary value problem for the third order loaded elliptic-hyperbolic type equation in double-connected domain have been investigated.At the proof of unequivocal solvability of the investigated problem,the extremum principle for the mixed type equations and method of integral equations have been used.

Loaded equation;elliptic-hyperbolic type;double-connected domain;an extremum principle;existence of solution;uniqueness of solution;method of integral equations.

1 Introduction

Boundary value problems(BVP)for mixed type equations have numerous applications in physics,biology and in other material sciences.F.I.Frankl[1]found critical applications of the problem Tricomi and others related problems in transonic gas dynamics. In the works by A.V.Bitsadze[2],the analogue of problem Tricomi for the first time is formulated and investigated in double-connected domain for the equations of the mixed type,whichwas newdirectionofthetheorypartial differential equations.Afterthiswork various problems for the equation of the mixed elliptic-hyperbolic type of the second order in double connected domain were investigated.For example,we note works by M. S.Salahitdinov and A.K.Urinov[3],B.Islomov and O.Kh.Abdullaev[4]and etc.

We as well note that with intensive research on problem of optimal control of the agroeconomical system,regulating thelabel ofgroundwatersand soilmoisture,it hasbecome necessary to investigate a new class of equations called“LOADED EQUATIONS”.Such equations were investigated in first in the works of N.N.Nazarov and N.Kochin, but they didn’t use the term“LOADED EQUATIONS”.For the first time was given the most general definition of a LOADED EQUATIONS and various loaded equations are classified in detail by A.M.Nakhushev[5].After this work has appeared very interesting results of the theory of boundary value problems for the loaded equations parabolic, parabolic-hyperbolic and elliptic-hyperbolic types.We note works of:A.M.Nakhushev[6],K.B.Sabitov and E.R.Melisheva[7],E.R.Melisheva[8]and O.Kh.Abdullaev[9],[10].

Local and non-local problems for the third order loaded equations elliptic-hyperbolic type in double-connected domains as well were not investigated.

In the present paper,the uniqueness and the existence of the solution of non-local boundary value problem for the third order loaded equation of elliptic-hyperbolic type in double-connected domain was proved.

2 The statement of the problem

Consider the equation

in double-connected domain ?,bounded with two lines:

We will introduce designations:

Problem I.To find a function u(x,y)with the following properties:

1)u(x,y)∈C(?);

3)u(x,y)satisfies gluing conditions on lines of changing type:

4)u(x,y)satisfies boundary conditions:

and

Theorem 2.1.If conditions(2.8)-(2.12)and

are satisfied,than the solution of the Problem I exists and unique.

Proof.Note,that at the proof of the theorem the following lemma is important:

Lemma 2.1.Any regular solution of the Eq.(2.1)at the xy6=0,x+y6=±q can be represented in the form of[11]:

where z(x,y)is regular solution of the equation

Without restricting generality it is possible to assume the following conditions

Based on(2.14)and(2.15)considering(2.16),theProblem Iwill be reduced to theProblem I＊:To find a solution of the Eq.(2.15)in the domain of ?,satisfying conditions:

Note that the solution of the Eq.(2.15)in hyperbolic domain looks like[10]:

Owing to(2.23)we will receive,that the solution of the Cauchy problem for the Eq.(2.15) satisfying conditions:

in the domain of Δ2,has a form:

From here,by virtue,(2.21)we obtain:

Further,owing to(2.20)from(2.26)is found:

Hence:

From(2.28)and(2.27)we will receive a functional relation between τ1(y)and ν1(y)from

From the solution of problem Cauchy for the Eq.(2.15)satisfying conditions z(x,0)= τ2(x),zy(x,0)=ν2(x)in the domain of Δ1,and with the account of conditions(2.20)and (2.22),we will obtain a functional relation between τ2(x)and ν2(x):

and we will find

3 The uniqueness of solution of the problem I?.

It is known that,if the homogeneous problem has only trivial solution then a solution of the accordingly non-uniform problem will be unique.

We will assume that bj(x)≡μj(t)≡0 then from(2.29)-(2.32)accordingly we will receive:

Note that an extremum principle for the hyperbolic equations[12]will be applicable to the investigated problem if ω1(y)≡0,and it is obtained from(3.2),at τ1(y)≡0,i..at first it is required to prove that τ1(y)≡0.

Let the function τ1(y)reach a positive maximum(a negative minimum)in some point of y0∈A2B2,then from(3.2)based on(2.13)we obtain

To conclude that if bj(x)≡μj(t)≡0 and a1(y)＞1/2,then we have ω1(y)≡0,i.e. an extremum principle for the hyperbolic equations became applicable to investigated problem I?.

The following principle of the extremum is acceptable:

Proof.Lemma 3.2 will be proved similarly as the Lemma 3.1.

Thus,the solution of homogeneous problem I＊is identically equal to zero in the domain of ?.The uniqueness of solution of theProblemI?is proved.Therefore,by virtue of Lemma 2.1 we will conclude that the solution of theProblem Iis unique,too.

4 The existence of solution of the problem I.

where G(ξ,η;x,y)is the Green function of the problem N for the Laplace equation in the domain ?0,which looks like[3]:

and

Further,after differentiating(4.3)by x and(4.4)by y,we will get

where

It is easy to see that

As well,from(2.30)and(2.31)accordingly we can get:

where|M(t,y,λ)|6const,and

where

Note that the unequivocal solvability of the system integral Eqs.(4.9)follows from the uniqueness of solution of the Problem I and from the theory integral equations[14].

Theorem 2.1 is proved.

[1]Frankl F.I.,Selected Works of the Gas Dynamics.M.1973.(In Russian)

[2]Bitsadze.A.V.,To the Problem of the Mixed Type Equations.Trudi.Instit.Matem.AN SSSR., 1953.(In Russian)

[3]Salakhitdinov M.S.,Urinov A.K.,Problems for the general Lavrentyev-Bitsadze equation in doubly connected domain.Izv.N Uz SSR ser.fiz-mat.nauk.,6(1990),29-36.(In Russian)

[4]Islomov B.,Abdullaev O.Kh.,Boundary value problem of type of the problem Bitsadze for the equation of the elliptic-hyperbolic type of the third order in double-connected domain. Dokl.Adigskoy AN.,7(1)(2004),42-47.(In Russian)

[5]Nakhushev A.M.,Loaded equations and their applications.Differe.Equa.,19(1)(1983),74-81.

[6]Nakhushev A.M.,N-the Darboux problem for the degenerate loaded integral differential equation of the second order.Differe.Uravn.,12(1)(1976),103-108.

[7]Sabitov K.B.,Melisheva E.P.,The Dirichlet problem for the loaded mixed type equations in a rectangularity domain.Russi.Math.,57(2013),53-65.

[8]Melisheva E.P.,The Dirichlet problem for the loaded equations of the Lavrentyev-Bitsadze. Vestnik SamGU.,6(80)(2010).(In Russian)

[9]Abdullayev O.Kh.,About uniqueness of solution of the boundary value problem for the loaded equation of the elliptic-hyperbolic type in double-connected domain.Doklady A.N. RUz.,3(2011),7-11.(In Russian)

[10]Abdullayev O.Kh.,Boundary value problem for the loaded equation of the elliptic-hyperbolic type in doubly connected domain.Vestnik.KRAUNS.Russian.Phyz-Math-science,1(8) (2014),33-48.(In Russian)

[11]Salakhitdinov M.S.,Mixed-Compound Type Equations.Fan.1974.(In Russian)

[12]Agmon S.,Nirenberg L.,Protter M.H.,A maximum principle for a class of hyperbolic equation and application to equations elliptic-hyperbolic type.Comm.Pure Appl.Math.,1(1953), 455-470.

[13]Bitsadze A.V.,Differential Equations of Mixed Type.Mac.Millan.Co.,New York.1964.

[14]Muskheleshvili N.I.,Physics,Singular Integral Equations:Boundary Problems of Function Theory and Their Application to Mathematical Physics,Dover Publications,2008.

?Corresponding author.Email addresses:obidjon.mth@gmail.com(O.Kh.Abdullayev)

10.4208/jpde.v27.n4.1 December 2014

AMS Subject Classifications:35M10

Chinese Library Classifications:O175.2