Problem For Mixed-type Equation Research Articles

APPROXIMATE SOLUTION OF AN INITIAL-BOUNDARY VALUE PROBLEM FOR AN NONHOMOGENEOUS SECOND-ORDER DIFFERENTIAL EQUATION OF MIXED TYPE WITH TWO DEGENERATE LINES

This work is devoted to the study of an approximate solution of the initial-boundary value problem for the second order mixed type nonhomogeneous differential equation with two degenerate lines. Similar equations have many different applications, for example, boundary value problems for mixed type equations are applicable in various fields of the natural sciences: in problems of laser physics, in magneto hydrodynamics, in the theory of infinitesimal bindings of surfaces, in the theory of shells, in predicting the groundwater level, in plasma modeling, and in mathematical biology. In this paper, based on the idea of A.N. Tikhonov, the conditional correctness of the problem, namely, uniqueness and conditional stability theorems are proved, as well as approximate solutions that are stable on the set of correctness are constructed. In obtaining an apriori estimate of the solution of the equation, we used the logarithmic convexity method and the results of the spectral problem considered by S.G. Pyatkov. The results of the numerical solutions and the approximate solutions of the original problem were presented in the form of tables. The regularization parameter is determined from the minimum estimate of the norm of the difference between exact and approximate solutions.

The Bitsadze–Samarskii Type Problem for Mixed Type Equation in the Domain with the Deviation from the Characteristics

For a mixed elliptic-hyperbolic type equation in a mixed domain, when the elliptic part is a sector of the circle, and the hyperbolic part consists of two characteristic triangles, the Bitsadze–Samarskii type nonlocal problem is investigated. A feature of such problems is that the boundary conditions in the hyperbolic parts of the boundary are determined by a first-order differential operator and pointwise link the values of the partial derivatives of the desired solution on the characteristics with the partial derivatives on arbitrary monotone curves lying inside the characteristic triangles of the equation. To solve the problem, the methods of the theory of partial differential equations and the theory of singular integral equations as well as the methods of energy integrals and complex analysis were used, with the help of which the existence and uniqueness theorem for the solution of the investigated problem is proved. Also, a method of reducing the investigated problem to an equivalent system of singular integral equations is shown, also, a method for solving this system is proposed, which allows to obtain a solution to the problem in explicit form.

Dirichlet Problem for Mixed Type Equation with Characteristic Degeneration and Singular Coefficient

For a mixed-type equation of the second kind with a singular coefficient by the spectral expansions uniqueness criterion is installed for solving the first boundary-value problem. The solution was constructed as the sum of a Fourier–Bessel series. In justifying the uniform convergence appeared the problem of small denominators. In connection with this evaluation is set on a small separation from zero denominator with the corresponding asymptotic behavior that allowed to justify the convergence of the series in the class of regular solutions.

Keldysh problem for mixed type equation with strong characteristic degeneration and singular coefficient

We study a mixed-type equation of the second kind with a singular coefficient. With the help of the spectral analysis method we establish a uniqueness criterion for a solution of the problem with incomplete boundary data. The solution represents the sum of the Fourier–Bessel series. The substantiation of its uniform convergence is based on an estimate of the separation from zero of the small denominator with the corresponding asymptotic behavior. This allows us to prove the convergence of the series in the class of regular solutions.

Tricomi problem for q-difference analog of anticipatory-retarding equation of mixed type

We investigate a boundary-value problem for mixed-type equation with the Lavrent’ev–Bitsadze operator in the main term and q-difference deviations of the argument in the lowest terms. We construct a general solution to the equation and prove a uniqueness theorem without restrictions on the deviation value. Then we show that the problem is uniquely solvable and find integral representations of the solution in the elliptic and hyperbolic domains.

A generalization of Bitsadze–Samarskii problem for mixed type equation

We study a boundary-value problem with Bitsadze–Samarskii conditions on boundary characteristic on a special inner curve and on a segment of degeneration of mixed type equation. Its solvability is proved by method of integral equations, and uniqueness of solution is established by means of the maximum principle.

On a difference scheme of the second order of accuracy for elliptic-parabolic equations

The second order of accuracy difference scheme generated by Crank-Nicholson difference scheme for approximately solving multipoint nonlocal boundary value problem is considered. Well-posedness of this difference scheme in Holder spaces is established. Furthermore, as applications, coercivity estimates in Holder norms for approximate solutions of the multipoint nonlocal boundary value problems for mixed type equations are obtained. Moreover, the method is illustrated by numerical examples.

On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ3

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.

Boundary value problems for mixed type equations and applications

In this paper we outline a general method for finding well-posed boundary value problems for linear equations of mixed elliptic and hyperbolic type, which extends previous techniques of Berezanskii, Didenko, and Friedrichs. This method is then used to study a particular class of fully nonlinear mixed type equations which arise in applications to differential geometry.

On the Theory of Mixed-Type Equations

We briefly review different well-posed boundary-value problems for mixed-type equations and their applications in transonic gas dynamics. We present barotropic relations for plane-parallel flow of a compressible gas that leads to mixed-type model equations on the hodograph plane. We also discuss the question on the existence of transonic solutions of profile flow problems, which is closely related to Dirichlet problems in mixed-type domains.

A Boundary Value Problem for Mixed-Type Equations in Domains with Multiply Connected Hyperbolicity Subdomains

Unlike the familiar problems for mixed-type equations, we pose a boundary value problem for the Lavrent'ev–Bitsadze equation in domains with unbounded multiply connected hyperbolicity subdomains. We prove unique solvability in explicit form. We also study this problem for the general Lavrent'ev–Bitsadze equation.