Boundary Value Problem for the System Equations Mixed Type

In this paper, we consider a system of equations of mixed type and with changing time direction. It is proved that solution of the system is not stable depend from the variation of the data. Theorems of uniqueness and conditional stability proved. The approximate solution constructed and numerical results are given.

The most important example of such type equations are mixed-type equation. A systematic study of them began with the work F.Tricomi and S. Gellerstedt. Shortly because of research S.A. Chaplygin and F.I. Frankl found out that mixed-type equations have important practical applications in the calculation of the flow of gas at around and supersonic speeds. Many important practical applications, such as jet aircraft and astronautics, rocketry, gas-dynamic lasers, caused an avalanche growth in boundary value problems of research for equations of mixed type, as a purely mathematical, and having applied nature.
In this paper we study the conditional correctness, prove the theorem on conditional stability and uniqueness of the solution of problem (1) -(4) and construct an approximate solution by the regularization method. We estimate norm of the difference between exact and regularized solutions.
Definition. By solution of the problem (1)-(4) we understand a pair of functions (u, v) having respective continuous derivatives involved in the system of equations and satisfy the system of equations (1) and the conditions (2) -(4).

The main results
For further discussion, we need in the following lemmas: Lemma 1 [6]. Let v(x, t) satisfies the equation Definition of norm and Proof of Lemma 1 one can find in [6].
Boundary Value Problem for the System Equations Mixed Type Lemma 2. Let u(x, t) satisfies the equation and conditions u(−1, t) = u(1, t) = 0, u(−0, t) = u(+0, t), u x (−0, t) = u x (+0, t), then for u(x, t) for every 0 < t < T the following estimate Proof. Solution of the equation (5) one can present as the sum whereū(x, t) solution of the homogeneous equation u(x, t)a particular solution of the inhomogeneous equation Moreover functionsū(x, t),ũ(x, t) satisfy the boundary conditions and gluing conditions According to the results of [5,6] solutionsū(x, t) andũ(x, t) can be presented in the form hereū ± k (t),ũ ± k (t) for each k = 1, 2, ... satisfy the following problems, respectively: eigenfunctions, corresponding respectively to positive λ + k and negative λ − k eigenvalues, and numbers λ + k , −λ − k constitute non-decreasing sequence. Notice, that X ± k (x) and λ ± k eigenfunctions and eigenvalues of the following problem: It is easy to note [3], that Next, using the logarithmic convexity of the functionsū ± k (t) for each k = 1, 2, ... , we have the following estimates Summing over k, k = 1, 2, ... and using the Holder inequality for the sum we have taking into account (6) from this we obtain required inequality. The Lemma 2 proved. Set of correctness M is defined as follows Theorem 1. Let the solution of the problem (1) -(4) exists and (u(x, t), v(x, t)) ∈ M , then the solution of the problem is unique.
Proof. Let two pair of functions Then pair of functions (u(x, t), v(x, t)) satisfies the system of equations (1) and the homogeneous conditions (2) - (4). From the result of Lemma 1 it is easy to see v(x, t) = 0 then v(x, t) = 0, and on the base of Lemma 2, we get u(x, t) = 0, from here u(x, t) = 0 for any On the base of the conditions of Theorem pair of functions (u(x, t), v(x, t)) satisfies the system of equations and gluing conditions From the initial conditions, we obtain the following Since from (7) v(x, T ) ≤ m, u(x, T ) ≤ m, and from Lemma 2 The results of this work can easily be extended to equation that is more general.

Approximate solution
The approximate solution for accurate data defined as follows N -integer parameter regularization. The approximate solution of the approximate data defined as follows where , t), v(x, t)) ∈ M . Then we estimate the norm of the difference between of exact and approximate solutions by the way (13) 1. Let g(x) = 0. We estimate the second term on the right side of inequality (13) by way From (7) follow v(x, T ) ≤ m, then easy to see that Right sides the array in (14) reaches his maximum under the condition (15), if since the corresponding series for h(x) is convergent. We now estimate the right side of inequality (12). Consider the second term Now, from the definition of the norm We estimate (17) under the condition (18). It is easy to notice that its maximum value reaches under the condition Using this fact we estimate (17) under (16) At the final we get estimate

Conclusions
Results of numerical experiments showed the effectiveness of the proposed approach.
Choosing parameter regularization from the optimality of estimate of the norm of difference between exact and approximate solution give for us possibility to stable calculation. We can see approximate calculation depends from the choosing of correctness set and parameter of regularization.