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Methods of construction and study of Frankl system self-similar solutions in the hyperbolic case

Self-similar solution of the Frankl system in the hyperbolic case was found. The Frankl system is a system of mixed type equations. Under certain conditions, it describes a model of the membrane theory of shells. The Frankl system describes a stationary irrotational motion of an ideal gas in the transition vicinity from subsonic to supersonic speeds. We find a sufficient condition on the initial data that guarantees existence of a global classical solution continued from a local solution. The proof of the nonlocal solvability of the problem in the original variables is based on the additional argument method. It allowed justify and construct a numerical solution. Numerical experiments were carried out for model examples of the Frankl system.


Introduction
The main object of research is the  . The Frankl system is a system of two quasi-linear differential equations of the first order. It is a system of mixed type. Equations with opposite signs represent a system of elliptic type. Equations with the same signs represent a system of hyperbolic type.
For the first time the famous physicist F. I. Frankl [1] has combined the study of the system of equations of mixed type with the study of stationary problems of transonic gas dynamics. Some particular cases of the Frankl system can be found in neutron transport theory, theory of thermoelasticity, in describing of the distribution of electrons in the electric field of the sprite, and so on. Bibliography for mixed type equations can be found in many monographs, in particular J. M. Krikunov [2]. S. A. Chaplygin, A. V. Bitsadze, F. Tricomi, S. V. Falkovich, B. L. Rozhdestvensky and N. N. Yanenko are the famous scientists. They studied problems for systems of mixed type equations in particular cases, when the equation system coefficients , PQ possess values of the constants or independent variables. Most often, the study came to the study of differential equations of the second order. Mathematical theory to study a model system of mixed type equations is systematically presented in the monograph by T. V. Chekmarev [3]. He considered the application of this theory to the solution of problems for mixed type systems.
A variety of approaches to the study of mixed systems of differential equations have their advantages and disadvantages. As a rule, the method of research is to convert the nonlinear equations to linear equations in the hodograph plane. Then a linear system of equations is solved. But in the general case, a return to the original variables is a more difficult task than the original task. Application of method of characteristics for quasi-linear equation systems introduces a superposition of unknown functions in the relevant integral equations. In the method of characteristics inverse transformation existence is a condition for original problem solvability. In many cases the problem of inverse function determination is so difficult that it is not solved. The existence of inverse variable transformations is taken as a condition.
To overcome these difficulties we investigate the problem using the method of additional argument [4 -6]. We consider the Frankl system in quite a general form. The method we propose does not replace other known methods. It amplifies them and allows to determine more precisely the conditions for the solvability in the original variables. In this work [6] we found sufficient conditions for a nonlocal solvability of the Cauchy problem for a system of two quasi-linear differential equations of first order. At that, the system of equations is represented in characteristic form. However, for the Frankl system with arbitrarily given initial data, conditions for the existence of the global solution established in work [6], are not fulfilled.
In this paper, we prove the global existence of self-similar solutions of the Frankl system in hyperbolic case, subject to the coordination of the initial data. We have done numerical calculations for model problems of the Frankl system. We found numerical Frankl system solution by grid method. We found self-similar solution of the Cauchy problem by the method of additional argument. We have built a numerical solution by the method of successive approximations.

The statement of the problem
We consider the Frankl system in the hyperbolic case: The Cauchy problem for the system of equations (1) is: Formulation of the problem is as follows. Find an unknown functions ( , ), ( , ) u x y v x y , satisfying the system of equations (1) and the initial conditions (2) in the domain:

Method of problem solving
The problem (1), (2) is solved by the method of additional argument. The starting system is transformed to the system of symmetric quasi-linear equations with help of Riemann invariants: The result is a characteristic shape in the form of a system of differential equations for functions 1  ( There exists a trivial solution 2 ( , ) 0. z x y  Thus, we get the coordination of the initial data for the Frankl system:   x, y , but they depend on an additional argument s . In the framework of an additional argument in [4][5] we determined the conditions of the local existence of a bounded solution for the Frankl system in the hyperbolic case.  We have . Integrating (4) with respect to s , we get the solution of system of ordinary differential equations of the first order: We found the solution of the characteristic form as follows: x y e u v z x y Thus, the problem is reduced to the study of global solvability of a system of two algebraic equations: The standard method of successive approximations is used to prove the existence of continuous constrained solutions together with derivatives of the first order.
We introduce the notation: We demand the implementation of inequalities:  , not to a function 1 () z x, y become infinite. Then for the derivatives of initial functions, we obtain the following conditions: Derivatives of functions     , u x, y v x, y have the following form: x y x u(x, y v(x, y yy       The first experiment. The problem was solved by grid method (Godunov scheme). Boundary conditions have the form:  ci  . In this example we found constants, but the formulas of the functions ( ), ( ) u x, y v x, y are very tedious.
We do not show them.

Conclusion
In the present work we proposed one more method to study the Frankl system of hyperbolic case. This is the method of additional argument. This method made it possible to prove the global existence of self-similar solution, subject to the approval of the initial data. Requirements for the initial data are quite simple. In addition, we found the algorithm for finding the numerical solution of the problem. Numerical calculations were performed on model problems of the Frankl system. Application of 11th International Conference on "Mesh methods for boundary-value problems and applications" IOP Publishing IOP Conf. Series: Materials Science and Engineering 158 (2016) 012085 doi:10.1088/1757-899X/158/1/012085 6 additional argument method has simplified numerical calculations and increased accuracy of estimation evaluation. In further studies we suggest the use of hybrid methods, based on methods of grids and additional argument.