Solvability of the analogs of the problem Tricomi for the mixed type loaded equations with parabolic-hyperbolic operators
© Baltaeva; licensee Springer 2014
Received: 30 November 2013
Accepted: 28 August 2014
Published: 25 September 2014
In this paper the unique solvability of the analog of the Tricomi problem for the third order loaded differential and integro-differential equations with parabolic-hyperbolic operators is proved. The existence of a solution is proved by the reduction to the integral equations with a shift of the second kind. We obtain necessary and sufficient conditions for the existence of a unique solution.
The first fundamental research on the theory of mixed type equations are the works of F Tricomi, S Gellerstedt, which were published in the 1920s. Due to the research of FI Frankl, IN Vekua, MA Lavrent‘ev, and AN Bitsadze, P Germain, R Bader, M Protter, K Morawets, MS Salakhidinov, TD Djuraev, AM Nakhushev, and many other authors, this theory became one of the main directions of the modern theory of partial differential equations.
The necessity of the consideration of the parabolic-hyperbolic type equation was specified in 1956 by Gel‘fand . He gave an example connected to the movement of the gas in a channel surrounded by a porous environment. Inside the channel the movement of the gas was described by the equation; outside by the diffusion equation.
Recently, loaded equations have been of great interest owing to intensive studying the problems of optimal control of the agro-economical system, of long-term forecasting, and regulating the subsoil waters layer and soil moisture. First, the most general definition of the loaded equation was given by Nakhushev in 1978 . He also gave notions and a detailed classification for different loaded differential, loaded integral, loaded functional equations and considered their numerical applications.
where is the density grain in the point at time flying with speed c under an angle θ, , to the straight line . Moreover, the functions , , and are given.
in the phase domain . Here is an unknown function, which defines the density of particles flying into the direction of y, , from the point . The function is positive and bounded, which characterizes the absorption of the medium; λ is the spectral parameter; and are given functions.
Basic questions of the theory of boundary value problems for partial equations are the same for the boundary value problems for the loaded equations. However, the existence of the loaded operator does not always make it possible to apply directly the known theory of boundary value problems.
What puts the considered problems in a class by itself is that the equation of the third order with the most generalized parabolic-hyperbolic and loaded operator is studied. An investigation is in order of the existence and uniqueness of the solution of the stayed boundary value problems for the loaded third order equations mixed type. One investigates the proof of the existence and uniqueness of the solution of the boundary value problem for the loaded differential and integro-differential equations of the mixed type for the analog problems of Tricomi.
In this section we give some formulas which will be used in what follows. We note that the regular solutions of some hyperbolic equations are directly connected with Riemann-Hadamard functions. Therefore, the basic properties such as the decomposition formulas, formulas of the analytical continuation, the formula of the differentiation for the Riemann-Hadamard function are necessary for studying of the regular solutions .
where is the Bessel function .
as function is a solution equation to the conjugate equation and as a function is a solution’s characteristic equation ;
at and ;
at , where and denote the Riemann-Hadamard functions for and .
3 Formulating the problem and main functional relations
in the domain Ω.
Ω is a simple connected domain located in the plane of independent variables x and y, bounded by with segments , , (, , , ) and by with characteristics , of (1).
In (1) , () are given real parameters.
We investigate the following problem.
satisfies equation (1) in and ;
- (3)satisfies the boundary conditions:(2)
where n is the interior normal, , , , , and are given real-valued functions, such that .
Let , , , .
here , are arbitrary continuous functions.
where is an unknown constant, which will be found later on.
4 Main results
then there exists a unique solution to the problem T0.
where is the resolvent kernel of .
For proving the unique solution equation (26) in class it is enough to prove that the corresponding homogeneous equation has only a trivial solution in this class.
Reducing the foregoing estimate for it follows that .
Thus the proof of (26) exists as a unique solution which is equivalent with problem T0.
After finding according to the condition , we can find the value of .
here , is defined by (31).
The proof of the unique solvability of problem can be found in . Hence, problem T0 has also unique solvability. Theorem 1 is proved. □
Analogously we can prove unique solvability for problem T0 where , and , .
5 Problem T1 for integro-differential equation
is continuous up to ;
- (3)the sewing condition(33)
- (4)the boundary conditions(34)
where n is the interior normal, , , , , and are given real-valued functions, moreover, .
then there exists a unique solution to problem T1.
Proof of Theorem 2
here , are arbitrary continuous functions.
Here is determined by (45).
where is an unknown constant to be defined.
here , , , depend on the given function, for the index implied summation from 1 to n. is an unknown constant to be defined.
where is the resolvent of the kernel ().
After finding and , the solution of problem T1 defined in by a formula due to Cauchy  or Darboux (see (43)). For the construction of the solution BVP A in we pass to the auxiliary problem  for (39) and similarly to  we prove the uniqueness of the solution.
Problem T1 is uniquely solvable. Theorem 2 is proved. Analogously, one can prove the uniqueness of the solution problem T1 in the case where , . □
The author read and approved the final manuscript.
The author is grateful to Professor B Islamov for drawing attention to these problems; she would also like to thank the anonymous reviewers for their valuable suggestions.
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