The Jump Problem for Mixed-Type Equations with Defects on the Type Change Line

Boundary Value Problems20102010:681709

DOI: 10.1155/2010/681709

Received: 2 January 2010

Accepted: 4 June 2010

Published: 27 June 2010

Abstract

The jump problem and problems with defects on the type change line for model mixed-type equations in the mixed domains are investigated. The explicit solutions of the jump problem are obtained by the method of integral equations and by the Fourier transformation method. The problems with defects are reduced to singular integral equations. Some results for the solution of the equation under consideration are discussed concerning the existence and uniqueness for the solution of the suggested problem.

1. Introduction

Consider the jump problem and problems with defects on the type change line for the mixed-type equation of the first kind
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ1_HTML.gif
(1.1)
This equation is a model equation among mixed-type equations of the first kind. For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq2_HTML.gif , (1.1) coincides with Lavrent'ev-Bitsadze equation and the Tricomi equation, respectively. For even http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq3_HTML.gif (1.1) coincides with the Gellerstedt equation (see, [19]). Equation (1.1) is elliptic for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq4_HTML.gif and hyperbolic for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq5_HTML.gif . In the formulation of the boundary value problems in the mixed domain, it is usually required that the unknown solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq6_HTML.gif and its normal derivative should be continuous on the type change line http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq7_HTML.gif , that is, the conditions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ2_HTML.gif
(1.2)
should be fulfilled. More generally conjugation conditions with continuous coefficients of the form
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ3_HTML.gif
(1.3)
have been discussed (see [10, 11]). There are defects on the type change line if the conjugation conditions (1.2) are replaced by conditions of another form. For example, if the boundary values of the solution or its normal derivative are given on defect. Such terminology is taken from the boundary value problems of elasticity theory. So problems with defects on the type change line will form special class of boundary value problems for the mixed-type equations with discontinuous coefficients in the conjugation conditions. We say that boundary value problems in the mixed domain with the conjugation conditions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ4_HTML.gif
(1.4)

are the jump problems on the type change line for (1.1). Obviously, the classical Tricomi problem is the jump problem with zero jump. Two methods are used in this papre to solve the jump problem: the method of integral equations and the method of integral Fourier transformation. It is shown that explicit solutions of the jump problem can be used as potentials under researching boundary value problems with defects.

2. The Jump Problem for Lavrent'ev-Bitsadze Equation: The Method of Integral Equations

Let the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq8_HTML.gif be bounded by the line http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq9_HTML.gif with the ends at the points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq11_HTML.gif of the real axis and by the characteristics http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq13_HTML.gif of Lavrent'ev-Bitsadze equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ5_HTML.gif
(2.1)

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq15_HTML.gif be the elliptic and the hyperbolic parts of the mixed domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq16_HTML.gif .

The unknown solution of (2.1) in the jump problem as in the Tricomi problem should satisfy the following boundary conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ6_HTML.gif
(2.2)

(here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq17_HTML.gif is arc abscissa of arc http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq18_HTML.gif being measured from the point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq19_HTML.gif to the point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq20_HTML.gif ).

As it is commonly accepted in the theory of the boundary value problems for the mixed-type equations we denote
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ7_HTML.gif
(2.3)

Assume that on the segment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq21_HTML.gif there is the finite number of points in which functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq22_HTML.gif can have discontinuities of the first kind and functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq23_HTML.gif can have singularities of integrable order. We say that such points are the exclusive points.

Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq24_HTML.gif is the set of points of the segment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq25_HTML.gif which are not the exclusive ones.

We can seek a solution of the problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq26_HTML.gif in the different classes of solutions [2, Section http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq27_HTML.gif ]. The regular solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq28_HTML.gif and satisfies (2.1) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq29_HTML.gif . The generalized solution of the class http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq30_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq31_HTML.gif , satisfies (2.1) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq32_HTML.gif and is the generalized solution of (2.1) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq33_HTML.gif in the sense that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ8_HTML.gif
(2.4)

(the D'Alembert formula) where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq34_HTML.gif . Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq35_HTML.gif is a class of functions which can have singularities of integrable order at the points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq37_HTML.gif , but satisfy Hoelder's condition with some index at any part of the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq38_HTML.gif . As it is known the generalized solution of the class http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq39_HTML.gif will be regular if we assume in addition that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq40_HTML.gif .

In the jump problem on the type change line for Lavrent'ev-Bitsadze equation, we need to seek a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq41_HTML.gif which

(1)is regular or generalized solution of (2.1) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq42_HTML.gif ;

(2)satisfies the boundary conditions (2.2);

(3)has the limiting values http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq43_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq44_HTML.gif and

(4)the conditions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ9_HTML.gif
(2.5)

are fulfilled.

Let us obtain functional correlations at the segment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq45_HTML.gif which connect functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq46_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq47_HTML.gif . The general scheme of reasoning is just the same as under solving the Tricomi problem.

Consider auxilliary boundary value problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq48_HTML.gif in the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq49_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq50_HTML.gif be the Green function of the problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq51_HTML.gif for (2.1) in the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq52_HTML.gif . Then in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq53_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ10_HTML.gif
(2.6)
By this
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ11_HTML.gif
(2.7)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ12_HTML.gif
(2.8)
From D'Alembert's formula (2.4) and condition on the characteristic it follows immediately that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ13_HTML.gif
(2.9)
Consider system of (2.7), (2.9), and (2.5) for functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq54_HTML.gif . Subtracting (2.9) from (2.7) we get
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ14_HTML.gif
(2.10)
It follows from here that function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq55_HTML.gif should be a solution of integral equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ15_HTML.gif
(2.11)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ16_HTML.gif
(2.12)

Similar equation can be obtained for the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq56_HTML.gif .

If the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq57_HTML.gif is a semidisc http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq58_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ17_HTML.gif
(2.13)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ18_HTML.gif
(2.14)
Let us transform the integral equation (2.11) with logarithmic singularity in the kernel into the integral equation with the analogue of the Cauchy kernel. As
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ19_HTML.gif
(2.15)
so the new function to be found,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ20_HTML.gif
(2.16)
should satisfy the equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ21_HTML.gif
(2.17)
Generally speaking (2.17) is the complete singular integral equation with the Cauchy kernel, but special form of the kernel enables us to construct its solution in the explicit form. By this it is advisible to use the method of reduction to the Riemann boundary value problem for automorphic functions [12, Chapter III]. Consider an auxiliary piecewise-holomorphic function
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ22_HTML.gif
(2.18)
satisfying the automorphic type condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ23_HTML.gif
(2.19)
It follows from the analogues of the Sohotski formulas that integral equation (2.17) is equivalent to the Riemann boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ24_HTML.gif
(2.20)

for analytic functions satisfying the condition (2.19). The solutions of problem (2.20) should be limited at the points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq59_HTML.gif and at infinity.

The canonical function of the Riemann problem in class of the automorphic functions has the form
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ25_HTML.gif
(2.21)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ26_HTML.gif
(2.22)

is simple automorphic function of group http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq60_HTML.gif . As it is shown in [13, page 111], there is a unique opportunity to choose numbers http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq61_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq62_HTML.gif being stationary points of group of the homographic transformations).

So index of problem (2.20) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq63_HTML.gif and its unique solution limited at infinity
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ27_HTML.gif
(2.23)
By condition (2.19) an arbitrary constant in the right-hand side is equal to zero. Since the boundary value of the canonical function from the upper half-plane on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq64_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ28_HTML.gif
(2.24)
then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ29_HTML.gif
(2.25)
By this if the elliptic part http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq65_HTML.gif of the mixed domain is a semidisc then by formula (2.25), we can write down the solution of the integral equation (2.11) in the form
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ30_HTML.gif
(2.26)

The function to be found http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq66_HTML.gif can be obtained by differentiation, but as it will be shown later it is not obligatory.

By the main correlation (2.7)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ31_HTML.gif
(2.27)

here it is taken into account that function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq67_HTML.gif satisfies (2.17). Expressions of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq68_HTML.gif can be obtained from conditions (2.5).

The solution of the jump problem in the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq69_HTML.gif can be easily derived by the D'Alembert formula (2.4), and it is not necessary to seek the expression of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq70_HTML.gif for this, it is sufficient to have formula to calculate its primitive. The solution of the jump problem in the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq71_HTML.gif can be obtained by two methods: either as a solution of problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq72_HTML.gif or as a solution of the Dirichlet problem.

Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq73_HTML.gif be a simple connected domain bounded by piecewise-smooth curve and let function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq74_HTML.gif conformally map by variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq75_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq76_HTML.gif onto unit disc in such way that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq77_HTML.gif . Then (see, [14, page 464]) function
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ32_HTML.gif
(2.28)
is the Green function of the Dirichlet problem for the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq78_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq79_HTML.gif is conformal mapping of the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq80_HTML.gif onto unit disc then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ33_HTML.gif
(2.29)

More general statement is formulated in [2, page 30]. If the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq81_HTML.gif maps the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq82_HTML.gif of the plane http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq83_HTML.gif onto the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq84_HTML.gif of the plane http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq85_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq86_HTML.gif is the Green function of the Dirichlet problem for the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq87_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq88_HTML.gif is the Green function of the Dirichlet problem for the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq89_HTML.gif .

In the case of problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq90_HTML.gif it is also possible to use the method of conformal mappings [2, Section http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq91_HTML.gif ]. Let domains http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq93_HTML.gif be bounded by segment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq94_HTML.gif of real axis and by curves http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq95_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq96_HTML.gif placed in the upper half-plane. Let function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq97_HTML.gif map the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq98_HTML.gif onto the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq99_HTML.gif in such way that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq100_HTML.gif goes over into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq101_HTML.gif and ends of this segment remain stationary. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq102_HTML.gif is the Green function of problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq103_HTML.gif for the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq104_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq105_HTML.gif is the Green function of problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq106_HTML.gif for the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq107_HTML.gif .

By this way, the Green function of problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq108_HTML.gif for the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq109_HTML.gif in the jump problem can be derived from the Green function of problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq110_HTML.gif for some simple canonical domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq111_HTML.gif by conformal mapping. In [2] the upper half-plane is chosen as a canonical domain but in our case a semidisc is more convenient to be considered as such domain. Hence if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq112_HTML.gif is the Green function (2.13) of the problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq113_HTML.gif for semidisc and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq114_HTML.gif is a mapping of any other domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq115_HTML.gif onto this semidisc satisfying the above mentioned conditions. Then for the Green function of the problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq116_HTML.gif for the domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq117_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ34_HTML.gif
(2.30)

So the integral equation (2.11) can be transformed into equation of the form (2.17) by substitution of variables.

3. The Jump Problem for Lavrent'ev-Bitsadze Equation: The Method of Fourier Transformation

Let us construct the solution of the jump problem for Lavrent'ev-Bitsadze equation in the unbounded mixed domain by the method of the integral Fourier transformation.

Preliminary, we consider two auxiliary Cauchy problems in the upper and lower half-planes using some results of the works [15, 16]. We will use the following denotions: under Fourier transformation function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq118_HTML.gif goes over into function (distribution) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq119_HTML.gif .

Note that the boundary value problems in the half-space for partial differential equations have been investigated quite adequately (see, [17]). If the Cauchy problem in the half-space is overdetermined, then analysis of the algebraic equation for the Fourier transform of the unknown solution gives necessary and sufficient conditions for the boundary functions.

We seek a solution of (2.1) in the upper half-plane http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq120_HTML.gif satisfying the boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ35_HTML.gif
(3.1)
The Fourier transform of the unknown solution will be a solution of the equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ36_HTML.gif
(3.2)
This solution exists if and only if when the right-hand side of (3.2) vanishes under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq121_HTML.gif , that is, the condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ37_HTML.gif
(3.3)
is fulfilled. Consequently,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ38_HTML.gif
(3.4)
or
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ39_HTML.gif
(3.5)

Equality (3.3) is the main correlation between boundary functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq122_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq123_HTML.gif .

We seek a solution of (2.1) in the lower half-plane http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq124_HTML.gif satisfying the boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ40_HTML.gif
(3.6)
The Fourier transform of the unknown solution satisfies the equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ41_HTML.gif
(3.7)

and boundary functions can be given arbitrary.

It follows from (3.7), that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ42_HTML.gif
(3.8)
(this distribution is obtained by the method of passing to the complex plane). So
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ43_HTML.gif
(3.9)

Note that, if we pass in this formula from the Fourier transforms of the boundary functions to their prototypes we obtain the D'Alembert formula (2.4).

Consider the jump problem for Lavrent'ev-Bitsadze equation in the unbounded mixed domain. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq125_HTML.gif be the upper half-plane, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq126_HTML.gif be the unbounded characteristic triangle bounded by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq127_HTML.gif and by positive semiaxis of the axis http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq128_HTML.gif . We should seek a solution of (2.1) under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq129_HTML.gif which satisfying the boundary conditions on the negative semiaxis http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq130_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ44_HTML.gif
(3.10)
on the characteristic
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ45_HTML.gif
(3.11)
and on the line of type change under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq131_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ46_HTML.gif
(3.12)

In the particular case under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq132_HTML.gif , the jump problem coincides with the Tricomi problem in the unbounded mixed domain. Without loss of generality we can assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq133_HTML.gif .

We will seek a solution of the jump problem in the upper and in the lower half-planes as solutions of the Cauchy problems. Let us continue the unknown solution in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq134_HTML.gif onto the whole lower half-plane so that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ47_HTML.gif
(3.13)
The Fourier transforms of values of the unknown solution on the axis http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq135_HTML.gif should satisfy the transformed conditions (3.12)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ48_HTML.gif
(3.14)

and the condition (3.3). Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq136_HTML.gif are the Fourier transforms of functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq137_HTML.gif being completed by zero up to the whole axis.

Denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq138_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq139_HTML.gif and represent each of these functions as a sum of the Fourier transforms of one-side-functions. Hence
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ49_HTML.gif
(3.15)
By condition (3.10),
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ50_HTML.gif
(3.16)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq140_HTML.gif is the Fourier transform of the boundary function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq141_HTML.gif being completed by zero up to the whole axis.

Condition (3.11) on the characteristic can be written down in the form
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ51_HTML.gif
(3.17)
where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq142_HTML.gif is the Fourier transform of the boundary function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq143_HTML.gif being completed by zero up to the whole axis. Actually, by the D'Alembert formula
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ52_HTML.gif
(3.18)
After Fourier transformation subject to the evident identity
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ53_HTML.gif
(3.19)

we obtain (3.17).

Condition (3.3) in the new denotions has the form
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ54_HTML.gif
(3.20)
Excluding function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq144_HTML.gif from (3.17) and (3.20) we get in view of (3.16)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ55_HTML.gif
(3.21)

Equality (3.21) is the condition of the Riemann boundary value problem with discontinuous coefficient given on the real axis (this equality is being understood as the equality of distributions).

Note that solution of the jump problem in the whole plane without condition on the characteristic (3.11) is not unique but is determined within the arbitrary function.

The canonical function has the form
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ56_HTML.gif
(3.22)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq145_HTML.gif is a single-valued branch of the power function which is chosen in the plane with a cut along positive semiaxis http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq146_HTML.gif of real axis and takes on the real values http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq147_HTML.gif on the upper side of the cut.

Denote by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ57_HTML.gif
(3.23)
Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ58_HTML.gif
(3.24)
By the Sohotski formulas
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ59_HTML.gif
(3.25)

Now we can easily obtain the expressions of the other auxiliary functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq148_HTML.gif and consequently, the solution of the jump problem in the domains http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq149_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq150_HTML.gif .

The technique of the integral Fourier transformation can be used also in the cases when the mixed domain in the jump problem has another form.

If the elliptic part of the mixed domain is, for example, a semidisc then the Fourier transformation method can be modificated in the following way. Assume that the unknown solution of the jump problem on the semidisc is equal to zero. Continue the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq151_HTML.gif to the whole upper half-plane symmetrically about http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq152_HTML.gif , that is, in such way that values of function are equal at the points symmetrical about semidisc. Besides the solution and its normal derivative should be continuous on the semidisc. Then all formulas obtained at the beginning of the section remain valid but after substitution of variable integrals on infinite intervals can be transformed into integrals on segment. This method can be used in more general case when the elliptic part of the mixed domain is a half of the symmetrical about real axis fundamental domain of group of homographic transformations [12, Chapter III].

4. The Boundary Value Problems with Defect on the Line of Type Change for Lavrent'ev-Bitsadze Equation

Let the mixed domain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq153_HTML.gif be bounded by the line http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq154_HTML.gif with the ends at the points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq156_HTML.gif of the real axis and by characteristics http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq158_HTML.gif of Lavrent'ev-Bitsadze equation (2.1). Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq159_HTML.gif be a set of disjoint segments placed inside the segment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq160_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq161_HTML.gif be a complement of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq162_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq163_HTML.gif .

We should seek the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq164_HTML.gif with the following properties:

(1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq165_HTML.gif satisfies (2.1) in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq166_HTML.gif under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq167_HTML.gif (classical or generalized solution);

(2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq168_HTML.gif ;
  1. (3)
    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ60_HTML.gif
    (4.1)
     

(4) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq169_HTML.gif satisfies on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq170_HTML.gif the conjugation condition (1.4).

If the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq171_HTML.gif is empty and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq172_HTML.gif (there are no defects), then the problem under consideration coincides with the classical Tricomi problem. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq173_HTML.gif , then we have two independent boundary value problems: the Dirichlet problem for the Laplace equation in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq174_HTML.gif and the Goursat problem in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq175_HTML.gif .

Later on for simplicity we will assume that in the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq176_HTML.gif there is only one segment http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq178_HTML.gif .

If in the problem with defect the values of the unknown solution are given on the type change line, then we say that such defect is the defect of the first kind. If on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq179_HTML.gif the values of the derivative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq180_HTML.gif of the unknown solution are given (the defect of the 2nd kind), then by the main correlation (2.9) nothing changes in fact. By the same reason the problem with defect of the 3d kind (when on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq181_HTML.gif the linear combination of the solution and its derivative are given) can be reduced to the problem with defect of the 1st kind. Note that defect can be considered as a cut and independent boundary conditions can be given on every side of the cut.

We will seek a solution of the problem with defect on the line of type change as a solution of the jump problem (see Section 1). Let the elliptic part of the mixed domain be a semidisc. Without loss of generality we can assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq182_HTML.gif .

It follows from the boundary conditions on the type change line that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq183_HTML.gif almost everywhere on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq184_HTML.gif (except for only points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq185_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq186_HTML.gif probably) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq187_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq188_HTML.gif and on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq189_HTML.gif . In the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq190_HTML.gif function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq191_HTML.gif is still unknown in the meantime. This function can be found from the boundary condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ61_HTML.gif
(4.2)
By formula (2.27)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ62_HTML.gif
(4.3)
Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq192_HTML.gif outside interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq193_HTML.gif the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq194_HTML.gif should satisfy the integral equation
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ63_HTML.gif
(4.4)
The integral equation (4.4) is the integral equation with logarithmic kernel represented in the form of the integral with the analogue of the Cauchy kernel with variable limit. Introduce new unknown function
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ64_HTML.gif
(4.5)
Then (4.4) can be transformed into the integral equation with the analogue of the Cauchy kernel
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ65_HTML.gif
(4.6)
by this http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq195_HTML.gif under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq196_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq197_HTML.gif under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq198_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ66_HTML.gif
(4.7)

is supplementary unknown constant.

Let us construct the explicit solution of the integral equation (4.6). Denote by
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ67_HTML.gif
(4.8)
Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ68_HTML.gif
(4.9)
where
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ69_HTML.gif
(4.10)
By the auxiliary function
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ70_HTML.gif
(4.11)
pass to the Riemann boundary value problem with condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ71_HTML.gif
(4.12)
solutions of which we should seek in the class of functions bounded at the points http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq199_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq200_HTML.gif . Since the index of the problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq201_HTML.gif its solution exists if and only if when the solvability condition
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ72_HTML.gif
(4.13)

is fulfilled. From the equality (4.13) the constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq202_HTML.gif will be determined and so the Riemann problem (4.12) will have the unique solution.

Further operations are evident. The difference of the limiting values of the solution of the Riemann problem gives the unknown function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq203_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq204_HTML.gif , by this the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq205_HTML.gif will be determined and the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq206_HTML.gif if it is necessary. But as it was mentioned above, it is sufficient to have only the expression of primitive of the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq207_HTML.gif but not of this function itself.

If the mixed domain is unbounded it is convenient to use under solving the problem with defect on the type change line the results of Section 2 obtained by the Fourier transformation method. Depending on the kind of defect one of the auxiliary functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq208_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq209_HTML.gif will be identically equal to zero and the values of another function on the defect will remain unknown. Immediately from the formula (4.4) it is easy to get the integral equation equivalent to the problem with defect.

Declarations

Acknowledgment

The author wish to thank Professor N. B. Pleshchinskii at Kazan University (Russian) for his critical reading of the manuscript and his valuable comments.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Assiut University
(2)
Department of Mathematics, The Teacher's College in Makkah, Umm Al-Qura University

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Copyright

© Ahmed Maher. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.