A Boundary Value Problem for the Degenerated Elliptic Equation with Singular Coef ficient and Spectral Parameter

RUZIEV M.Kh.

Institute of Mathematics,Academy of Science of the Republic of Uzbekistan,Mirzo Ulugbek str.81,100170 Tashkent,Uzbekistan.

Abstract.In this paper,the Dirichlet problem in the vertical half-bandfor the degenerated elliptic equation with singular coefficient and spectral parameter is investigated.Using the Hankel transformation and the method of separation of variables,the solution of the Dirichlet problem is presented in the explicit form.

KeyWords:Half-band;Hankeltransformation;series;theFourier method;uniqueness;existence.

1 Introduction and formulation of the problem

Boundary value problems in various unbounded domains for the Gellerstedt equation were studied in[1,2].The Dirichlet problem for the degenerated elliptic equation

where k∈R was considered in[3].Boundary value problems for the elliptic equations were studied in works[4,5].Singular partial differential equations appear at studying various problems of aerodynamics and gas dynamics[6]and irrigation problems[7].A series of interesting results,devoted to studying boundary-value problems for partial differential equations were obtained in works[8–11].

We consider the equation

Problem D.Find a function U(x,y),with the following properties:

2.it satisfies the relation

3.it satisfies the boundary conditions

where ?1(y),?2(y),τ(x)are giving functions,moreover,?1(0)=?2(0)=τ(0)= τ(1)=0.

2 Uniqueness of the solution for the problem

Theorem 2.1.The solution of the Dirichlet problem for equation(1.1)is unique.

Proof.Let Dhbe a finite domain,intercepted from the domain D by the straight line y=h,h＞0,Oh(0,h),Bh(1,h).

Let U(x,y)be the solution of the homogeneous Dirichlet problem.Suppose that U(x,y)/=0 in.Then there exists a domain Dh,such that U(x,y)/=0 in Dh.Then the function U(x,y)can attain its positive maximum and negative minimum in the domainonly on.By virtue of the maximum principle for elliptic equations[12],the function U(x,y)doesn’t attain its positive maximum in inner points of the domain Dh.According to the homogeneous boundary condition it cannot also be attained on∪OOh∪BBh.

3 Existence of the solution for the problem

Theorem 3.1.Let the following conditions be ful filled:

?i(y)vanishes at y→∞,i=1,2, τ(x)∈C2[0,1],τ′′(0)=τ′′(1)=0,τ′′′(x)∈L2(0,1).Then the solution of the Problem D exists.

Proof.We seek the solution of the equation(1.1)in the half-band 0≤x≤1,y≥0,satisfying conditions(1.2)-(1.5)in the form of the sum of two functions

where U1(x,y),U2(x,y)are solutions of(1.1)satisfying the boundary conditions:

respectively.

We seek the function U1(x,y)in the form

where a1(s)and a2(s)are yet unknown functions,Jν(z)is the Bessel function of the first kind[13],β=.By virtue of boundary conditions(3.1),from(3.7)we have

Using the fact that the Hankel transformation

is the reconversion to itself,we obtain

From here after simple transformations we find that

Formulas(3.7)and(3.8)determinate the solution U1(x,y).

Using the asymptotic expansion of Hankel[14]

and by virtue of

it is easy to show that the function U1(x,y)satisfies conditions(3.2),(3.3),where Γ(z)is the Euler integral of the second kind[15].

Further,we find the function U2(x,y)by the method of separation of variables.We seek a solution of equation(1.1)in the form U2(x,y)=X(x)Y(y)satisfying the conditions(3.4)-(3.6).

Substituting this product into equation(1.1),we obtain

whereμis the separation constant.As is known,the solution of equation(3.9)with condition(3.4)has the form

In the equation(3.10)(whereμ=(πn)2)we replace

where Iν(z)and Kν(z)[15]are,respectively,the modified Bessel functions of the first and third kind,C1and C2are arbitrary constants.

Then on the basis of(3.12)and(3.14)the general solution of equation(3.10)is determined by formula

Taking into account the behavior ofthe functionsat z→∞,we get that C1=0.Then we obtain

Then,in view of(3.15)and(3.11),we obtain

We seek a solution of the original problem in the form of a series

In(3.16)we choose the coefficients,so that u(x,y)satisfies the condition u(x,0)=τ(x).Setting the series(3.16)to the condition u(x,0)=τ(x),we have

It is easy to show that

Then

i.e.decompositionof the function τ(x)into a Fourier series with respect to a systemof the function Xn(x)=sin(πnx),orthogonal on the interval[0,1].Then

Consequently,

We find another form for the function u2(x,y).On the basis of the integral representation of the function Kν(z):

we show that

By virtue of(3.18),we rewrite(3.17)in the form

Then we can write(3.19)in the form

Further,using the imaginary Jacobi transformation:

it is easy to show that

According to(3.21),equality(3.20)takes the form

The following formula is valid(see[13])

where Rep＞0,Req＞0.By virtue of this formula,we obtain

Similarly as it is easy to show that

Substituting(3.23)and(3.24)into(3.22)we obtain

where

It is easy to show that the function U2(x,y)satisfies the condition(3.4).

Using the asymptotical expansion of the Macdonald function[15]

one can easily be convinced that the function U2(x,y)satisfies the condition(3.5).

Thus,the unique solution of the Dirichlet problem for equation(1.1)in the vertical half-band D is determined by formulas(3.7),(3.8)and(3.25).

Acknowledgments

This work has supported by the Ministry of Higher and Secondary Special Education of Uzbekistan under Research Grant OT-F4-88.The author would like to thank anonymous referees for their useful suggestions.