Well-Posedness of the Dirichlet Problem for a Degenerating Many-Dimensional Equation of Mixed Type

2018 ◽  
Vol 229 (4) ◽  
pp. 354-366
Author(s):  
S. A. Aldashev
Keyword(s):  
Dirichlet Problem ◽  
Mixed Type ◽  
Well Posedness ◽  
Dimensional Equation ◽  
Equation Of Mixed Type

scholarly journals Polynomial Solutions of the Dirichlet Problem for the Tricomi Equation in a Strip

2018 ◽  
pp. 1-12
Author(s):  
O. D. Algazin
Keyword(s):  
Dirichlet Problem ◽  
Half Plane ◽  
Mixed Type ◽  
Polynomial Growth ◽  
Polynomial Solution ◽  
Transform Method ◽  
Polynomial Solutions ◽  
Tricomi Equation ◽  
Equation Of Mixed Type ◽  
Class Of Functions

In the paper we consider the Tricomi equation of mixed type. This equation is elliptic in the upper half-plane, hyperbolic in the lower half-plane and parabolically degenerate on the boundary of half-planes. Equations of a mixed type are used in transonic gas dynamics. The Dirichlet problem for an equation of mixed type in a mixed domain is, in general, ill- posed. Many papers has been devoted to the search for conditions for the well-posednes of the Dirichlet problem for a mixed-type equation in a mixed domain.This paper is devoted to finding exact polynomial solutions of the inhomogeneous Tricomi equation in a strip with a polynomial right-hand side. The Fourier transform method shows that the Dirichlet boundary value problem with polynomial boundary conditions has a polynomial solution. An algorithm for constructing this polynomial solution is given and examples are considered. If the strip lies in the ellipticity region of the equation, then this solution is unique in the class of functions of polynomial growth. If the strip lies in a mixed domain, then the solution of the Dirichlet problem is not unique in the class of functions of polynomial growth, but it is unique in the class of polynomials.

Note on a maximum principle and a uniqueness theorem for an elliptic-hyperbolic equation

1956 ◽  
Vol 236 (1204) ◽  
pp. 141-144 ◽  
Keyword(s):  
Dirichlet Problem ◽  
Maximum Principle ◽  
Elliptic Equation ◽  
Hyperbolic Equation ◽  
Uniqueness Theorem ◽  
Mixed Type ◽  
Boundary Curve ◽  
Type K ◽  
Equation Of Mixed Type

A maximum principle is proved for the function ψ = J [ — 2u x u y dx + (Ku 2 x — u 2 y ) dy], where u is a solution of the equation of mixed type K(y)u xx + u yy = 0 with K(y) ≷ 0 for y ≷ 0. The proof rests in showing that iff satisfies an elliptic equation for y > 0 and that it is a non-decreasing function of y for y ⩽ 0. This maximum principle leads to a uniqueness theorem for the appropriate analogue to the Dirichlet problem for mixed equations under some conditions on the shape of the boundary curve. Very weak restrictions are imposed on K(y).

scholarly journals Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped

2020 ◽  
Vol 30 (1) ◽  
pp. 31-48
Author(s):  
K.T. Karimov
Keyword(s):  
Mixed Type ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Existence Of A Solution ◽  
One Dimensional ◽  
Singular Coefficients ◽  
Dimensional Equation ◽  
Equation Of Mixed Type ◽  
Keldysh Problem ◽  
The Uniqueness Theorem

This article studies the Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped. Based on the completeness property of eigenfunction systems of two one-dimensional spectral problems, the uniqueness theorem is proved. To prove the existence of a solution to the problem, the Fourier spectral method based on the separation of variables is used. The solution to this problem is constructed in the form of a sum of a double Fourier-Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions.

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