Research

# A problem with conditions given on inner characteristics and on the line of degeneracy for a mixed-type equation with singular coefficients

Menglibay Kholtojibaevich Ruziev

Author Affiliations

Institute of Mathematics, National University of Uzbekistan named after Mirzo Ulugbek, Tashkent, Uzbekistan

Boundary Value Problems 2013, 2013:210  doi:10.1186/1687-2770-2013-210

 Received: 11 March 2013 Accepted: 12 August 2013 Published: 12 September 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In an unbounded domain, we consider a problem with conditions given on inner characteristics in a hyperbolic part of the considered domain and on some parts of the line of parabolic degeneracy. We prove the unique solvability of the mentioned problem with the help of the extremum principle. The proof of solvability is based on the theory of singular integral equations, Wiener-Hopf equations and Fredholm integral equations.

### Introduction and formulation of a problem

Swedish mathematician Gellerstedt [1] investigated a boundary value problem for the equation (m is an odd number), in which the values of a sought function are given on two pieces of characteristics, and on curve (), the value of its derivative is given. This problem has applications in transonic gas dynamics [2].

The Gellerstedt problem and related problems for mixed elliptic-hyperbolic equations were studied in the works [3-5]. The work [6] is devoted to studying the Gellerstedt problem with data on one family of characteristics and with nonlocal gluing conditions. In the work [7] the unique solvability of the Gellerstedt problem for a parabolic-hyperbolic equation of the second kind was studied. The Cauchy problem was investigated by Jachmann and Reissig [8]. Flaisher [9] studied a problem with data on characteristics, outgoing from the origin.

In an unbounded domain, Wolfersdorf [10] investigated the Tricomi problem for the Gellerstedt equation , .

Boundary value problems for the wave equation and equations of mixed type were investigated in [11]. In the work [12] the general Tricomi-Rassias problem was investigated for the generalized Chaplygin equation. In the paper, the representation of a solution of the general Tricomi-Rassias problem was given for the first time; moreover, the uniqueness and existence of a solution for the problem were proved by a new method. In the works [13,14], fundamental solutions were found and boundary value problems for degenerate elliptic equations were solved.

Due to applications in gas dynamics, the interest in studying boundary value problems for degenerate elliptic and mixed-type equations with singular coefficients has been growing. Note the latest work [15] on this topic, where the Dirichlet problem for a three-dimensional elliptic equation with singular coefficients was investigated. Let be a domain of the complex plane , where is a half-plane , is a finite domain of the half-plane , bounded by characteristics AC and BC of the equation

(1)

outgoing from the points , , and by the segment AB of the straight line , . In equation (1) assume that m, , are some real numbers such that , , .

Let be a finite domain separated from by the arc of the normal curve , , , , .

We introduce the following denotations: , , are points of intersection of the characteristic with that outgoing from the point , where is an arbitrary fixed number, , is a subdomain in the unbounded domain D.

Consider the diffeomorphism mapping the segment into the segment ; moreover, , , . As an example, we take a linear function , where , , , .

Note that in the Gellerstedt problem the values of a sought function in the hyperbolic part of the mixed domain D are given on the characteristics and : , .

Boundary value problem for equation (1) in the case when , with data on the piece of of the characteristic AC and with inner boundary local shifting condition on AB of the line of degeneracy , was studied in the work [16], and with data on pieces and in the work [17].

In the present work, we study a new boundary problem, where characteristic is free from the conditions, and the needed condition of Gellerstedt is replaced by an inner boundary condition with local shifting on the parabolic line of degeneracy.

### Formulation of the problem

Problem G. In the domain D, find a function satisfying the following conditions:

(1) the function is continuous in any subdomain of the unbounded domain D;

(2) belongs to the space and satisfies equation (1) in this domain;

(3) is a generalized solution from the class () in the domain  [18];

(4) the following equalities are fulfilled:

(2)

(5) satisfies the boundary conditions

(3)

(4)

(5)

and the conjugation condition

(6)

Moreover, these limits at , can have singularity of the order less than , where , , , , are given functions such that , , , , , μ-const., the functions are expressed as in a neighborhood of the points , , and they satisfy Holder’s condition on any intervals , , . For a sufficiently large absolute value , they satisfy the inequality , where δ, M are positive constants.

Note that condition (5) is an inner boundary condition with local shifting on the parabolic line of degeneracy [16-18].

#### The uniqueness of the solution of problem G

Theorem 1Let conditions, , , , be fulfilled. Then problemGhas only a trivial solution.

Proof Solution of the modified Cauchy problem for equation (1) in the domain , satisfying initial conditions , , , , is given by the formula [18]

(7)

where , , is Euler’s gamma function [19].

Satisfying (7) to condition (4), after some evaluations, we obtain

(8)

where

, is a differential operator of fractional order in a sense of Riemann-Liouville [19].

Equality (8) is the first functional relation between functions and , on an interval of the axis reduced from the domain .

Prove that if , , , , , then the solution of problem G in the domain is trivial by virtue of equality (2).

Let be a point of positive maximum of the function in the domain . By virtue of (2) for , there exists such that at

(9)

Considering designation , , condition (5) can be written as

(10)

Hence, at (where ) we get . Then, taking into account, we have , i.e., . By the Hopf principle [20], the function cannot reach its positive maximum and negative minimum on inner points of the domain . By virtue of , from (10) (where ) it follows that there are no points of extremum in the interval of the axis .

Assume that the sought function reaches its positive maximum and negative minimum on points of the interval of the axis .

Let (where ) be a point of positive maximum (negative minimum) of the function . Then [21]

(11)

It is well known that on the point of positive maximum (negative minimum) of the function for the differential operators of fractional order, the following inequality () holds. Hence, considering (8) (where ), we deduce

(12)

Inequalities (11) and (12) contradict the gluing condition (6), therefore . Hence, there is no point of positive maximum (negative minimum) of the function in the interval AB. Let . By the Hopf principle and the statements obtained above, it follows that , and by virtue of (9), . Consequently, for . From here, due to arbitrariness of ε at , we conclude that in the domain . Then

(13)

Considering (13), due to continuity of the solution in and conjugation condition (6), we restore the desired function in the domain as a solution of the modified Cauchy problem with homogeneous data and get in the domain . The proof of Theorem 1 is complete. □

#### The existence of the solution of problem G

Theorem 2Let the following conditions be fulfilled: , , , , , where, , , , . Then problemGhas a solution.

Proof The solution of the Dirichlet problem satisfying condition (3) and the requirement , , can be represented in the form

(14)

where ,

Differentiating (14) along y and considering the equality

we get

Integrating by parts (taking , into account), after some evaluations, we have

(15)

Multiplying both sides of (15) to , and passing to the limit at , we obtain

(16)

where .

Equation (16) is the second functional relation between unknown functions , in the interval I of the axis deduced from the upper half-plane.

Note that relation (16) is valid for the whole I. Breaking into the intervals and , then into the integral with bounds , we make change of variables . By virtue of (10), from (16) we obtain

(17)

Considering (6), excluding the function from (8) and (17), we deduce

(18)

Applying the operator to the both sides of (18), considering , we have

(19)

Further, one can easily prove that

(20)

(21)

(22)

Substituting (20)-(22) into (19), after some calculations, we get the singular integral equation regarding the function :

(23)

where , , . The integral operator on the right-hand side of (23) is not regular since expression under the integral has isolated singularity of the first order at , , and this is why this item is written separately. Setting the right-hand side of (23) temporarily as a known function, we rewrite it as follows:

(24)

where

(25)

Setting , , we rewrite equation (24) as

(26)

We search for a solution of equation (26) in the class of functions, satisfying Holder’s condition on and bounded at , and at they can tend to infinity with order less than . Index χ of this class is equal to zero, and solution can be explicitly found by the method of Carleman-Vekua [22]:

where , , . From here, according to designation, we have

(27)

Substituting (25) into (27), after some evaluations, we obtain

(28)

where

Equation (28), by virtue of , where , , can be rewritten as

(29)

Further, in (29) we calculate the inner integral

Expanding the rational factor of the integrand in simple fractions and completing simple calculations, we have

(30)

Substituting (30) into (29), considering , , after some calculations, we get

(31)

Considering formulas , , we rewrite equality (31) as

(32)

In (32) allocating a characteristic part, we get

(33)

where

is a regular operator. We rewrite equation (33) as

(34)

Making change of variables , and designating , we write equation (34) as

(35)

where

We introduce the following notations:

(36)

By virtue of (36), equation (35) has the form

(37)

Equation (37) is an integral equation of Wiener-Hopf [23]. Functions , are indicative of decrease at infinity; moreover, , . Consequently, , and a solution of equation (37) will be sought in the class [23]. Using Fourier transformation, equation (37) is deduced to the Riemann problem and is solved in quadratures. Fredholm’s theorems are only valid in one particular case, when the index of these equations is equal to zero.

Calculate the index of the expression , where

(38)

Using the theory of residues, we find

(39)

Substituting (39) into (38), we have

By virtue of the condition , and since

then . Therefore, the index of equation (37) , i.e., the variation of the argument of the expression on the real axis expressed in complete revolutions equals zero [23]. Hence, taking into account the fact that the solution to problem G is unique, we deduce the unique solvability of equation (37) and, consequently, that of problem G. Theorem 2 is proved. □

### Competing interests

The author declares that he has no competing interests.

### References

1. Gellerstedt, S: Quelques problèmes mixtes pour l’équation . Arkiv f. M.A.O.F. Bd. 26A. 3, 1–32 (1938)

2. Bers, L: Mathematical Aspects of Subsonic and Transonic Gas Dynamics, Wiley, New York (1958)

3. Krikunov, YM: On one Gellerstedt problem. Izv. Vysš. Učebn. Zaved., Mat.. 6, 56–59 (1975)

4. Solodovnikov, YM: Generalized Gellerstedt problem for a particular case of the equation . Tr. Sem. Po Kr. Zad. Kazan, Vyp.. 10, 130–139 (1973)

5. Khairullin, RS: On the Gellerstedt problem for the second kind equation. Izv. Vysš. Učebn. Zaved., Mat.. 10, 72–77 (2005)

6. Mirsaburov, M, Eshonkulov, BO: The Gellerstedt problem with data on characteristics of one family and with nonlocal gluing conditions. Izv. KBNTs. RAN. 1(18), 48–53 (2002)

7. Mamadaliev, NK: The Gellerstedt problem for a parabolic-hyperbolic equation of the second kind. Int. J. Dyn. Syst. Differ. Equ.. 1(2), 102–108 (2007)

8. Jachmann, K, Reissig, M: Cauchy problem for linear thermo-elastic systems. Part 1: A unified approach. Int. J. Dyn. Syst. Differ. Equ.. 2(3/4), 170–201 (2009)

9. Flaisher, NM: Some problems with data on characteristics for mixed-type equations. Rev. Roum. Math. Pures Appl.. 12(8), 1053–1058 (1967)

10. Wolfersdorf, L: Abelshe Integralgleichungen und Randwertprobleme fur die verallgemeinerte Tricomi-Gleichung. Math. Nachr.. 29(H3/4), 161–178 (1965)

11. Protter, MH: New boundary value problems for the wave equation and equations of mixed type. J. Ration. Mech. Anal.. 3(4), 435–446 (1954)

12. Wen, G, Chen, D, Cheng, X: General Tricomi-Rassias problem and oblique derivative problem for generalized Chaplygin equations. J. Math. Anal. Appl.. 333, 679–694 (2007). Publisher Full Text

13. Hasanov, A, Rassias, JM: Fundamental solutions of two degenerated elliptic equations and solutions of boundary value problems in infinite area. Int. J. Appl. Math. Stat.. 8(M07), 87–95 (2007)

14. Hasanov, A, Rassias, JM, Turaev, M: Fundamental solution for the generalized elliptic Gellerstedt equation. Functional Equations, Difference Inequalities and Ulam Stability Notions, pp. 73–83. Nova Science Publishers, New York (2010) No. 6

15. Karimov, ET, Nieto, JJ: The Dirichlet problem for a 3D elliptic equation with two singular coefficients. Comput. Math. Appl.. 62(1), 214–224 (2011). Publisher Full Text

16. Mirsaburov, M, Ruziev, MK: A boundary value problem for a class of mixed-type equations in an unbounded domain. Differ. Equ.. 47(1), 112–119 (2011)

17. Ruziev, MK: Problems with shifts for mixed elliptic-hyperbolic equations. Russ. Math.. 56(1), 66–75 (2012). Publisher Full Text

18. Salakhitdinov, MS, Mirsaburov, M: Nonlocal Problems for Equations of Mixed Type with Singular Coefficients, Universitet, Tashkent (2005)

19. Smirnov, MM: Equations of Mixed Type, Vyssh. Shkola, Moscow (1985)

20. Bitsadze, AV: Some Classes of Partial Differential Equations, Nauka, Moscow (1981)

21. Ruziev, MK: A boundary value problems for equation of mixed type with singular coefficients in an unbounded domains. Diss., Tashkent (1999)

22. Muskhelishvili, NI: Singular Integral Equations. Boundary-Value Problems of Function Theory and Some of Their Applications to Mathematical Physics, Nauka, Moscow (1968)

23. Gakhov, FD, Cherskii, YI: Convolution Type Equations, Nauka, Moscow (1978)