Open Access

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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq1_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq4_HTML.gif and hyperbolic for https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq7_HTML.gif , that is, the conditions
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq8_HTML.gif be bounded by the line https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq9_HTML.gif with the ends at the points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq10_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq13_HTML.gif of Lavrent'ev-Bitsadze equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ5_HTML.gif
(2.1)

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq14_HTML.gif and https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ6_HTML.gif
(2.2)

(here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq17_HTML.gif is arc abscissa of arc https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq18_HTML.gif being measured from the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq19_HTML.gif to the point https://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
https://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 https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq27_HTML.gif ]. The regular solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq28_HTML.gif and satisfies (2.1) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq29_HTML.gif . The generalized solution of the class https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq30_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq31_HTML.gif , satisfies (2.1) in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq33_HTML.gif in the sense that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq34_HTML.gif . Here https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq36_HTML.gif and https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq43_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq44_HTML.gif and

(4)the conditions
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq45_HTML.gif which connect functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq46_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq48_HTML.gif in the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq49_HTML.gif . Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq51_HTML.gif for (2.1) in the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq52_HTML.gif . Then in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq53_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ10_HTML.gif
(2.6)
By this
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ11_HTML.gif
(2.7)
https://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
https://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 https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ15_HTML.gif
(2.11)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq56_HTML.gif .

If the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq57_HTML.gif is a semidisc https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq58_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ17_HTML.gif
(2.13)
https://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
https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ20_HTML.gif
(2.16)
should satisfy the equation
https://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
https://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
https://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
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ25_HTML.gif
(2.21)
where
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq61_HTML.gif ( https://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) https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq64_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ28_HTML.gif
(2.24)
then
https://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 https://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
https://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 https://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)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq67_HTML.gif satisfies (2.17). Expressions of functions https://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 https://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 https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq74_HTML.gif conformally map by variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq75_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq77_HTML.gif . Then (see, [14, page 464]) function
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq78_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq79_HTML.gif is conformal mapping of the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq80_HTML.gif onto unit disc then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq81_HTML.gif maps the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq82_HTML.gif of the plane https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq83_HTML.gif onto the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq84_HTML.gif of the plane https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq85_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq87_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq89_HTML.gif .

In the case of problem https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq91_HTML.gif ]. Let domains https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq92_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq93_HTML.gif be bounded by segment https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq94_HTML.gif of real axis and by curves https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq95_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq97_HTML.gif map the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq98_HTML.gif onto the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq99_HTML.gif in such way that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq100_HTML.gif goes over into https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq102_HTML.gif is the Green function of problem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq103_HTML.gif for the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq104_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq105_HTML.gif is the Green function of problem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq106_HTML.gif for the domain https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq108_HTML.gif for the domain https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq110_HTML.gif for some simple canonical domain https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq113_HTML.gif for semidisc and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq116_HTML.gif for the domain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq117_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq118_HTML.gif goes over into function (distribution) https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq120_HTML.gif satisfying the boundary conditions
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq121_HTML.gif , that is, the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ37_HTML.gif
(3.3)
is fulfilled. Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ38_HTML.gif
(3.4)
or
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq122_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq124_HTML.gif satisfying the boundary conditions
https://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
https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq125_HTML.gif be the upper half-plane, https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq130_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ44_HTML.gif
(3.10)
on the characteristic
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq131_HTML.gif
https://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 https://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 https://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 https://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
https://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 https://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)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq136_HTML.gif are the Fourier transforms of functions https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq138_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ49_HTML.gif
(3.15)
By condition (3.10),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ50_HTML.gif
(3.16)

where https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ51_HTML.gif
(3.17)
where https://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 https://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
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ54_HTML.gif
(3.20)
Excluding function https://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)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ56_HTML.gif
(3.22)

where https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ57_HTML.gif
(3.23)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ58_HTML.gif
(3.24)
By the Sohotski formulas
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq149_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq153_HTML.gif be bounded by the line https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq154_HTML.gif with the ends at the points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq155_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq157_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq160_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq161_HTML.gif be a complement of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq162_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq163_HTML.gif .

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

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

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

(4) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq169_HTML.gif satisfies on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq171_HTML.gif is empty and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq174_HTML.gif and the Goursat problem in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq176_HTML.gif there is only one segment https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq177_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq179_HTML.gif the values of the derivative https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq183_HTML.gif almost everywhere on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq184_HTML.gif (except for only points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq185_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq186_HTML.gif probably) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq187_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq188_HTML.gif and on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq189_HTML.gif . In the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq190_HTML.gif function https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ61_HTML.gif
(4.2)
By formula (2.27)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ62_HTML.gif
(4.3)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq192_HTML.gif outside interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq193_HTML.gif the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq194_HTML.gif should satisfy the integral equation
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ65_HTML.gif
(4.6)
by this https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq195_HTML.gif under https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq196_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq197_HTML.gif under https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq198_HTML.gif , where
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ67_HTML.gif
(4.8)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ68_HTML.gif
(4.9)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_Equ69_HTML.gif
(4.10)
By the auxiliary function
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq200_HTML.gif . Since the index of the problem https://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
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq203_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq204_HTML.gif , by this the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq205_HTML.gif will be determined and the function https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F681709/MediaObjects/13661_2010_Article_946_IEq208_HTML.gif and https://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

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