scholarly journals DIRICHLET PROBLEM FOR PULKIN’S EQUATION IN A RECTANGULAR DOMAIN

2017 ◽  
Vol 20 (10) ◽  
pp. 91-101
Author(s):  
R.M. Safina
Keyword(s):  
Mixed Type ◽  
Type Equation ◽  
Rectangular Domain ◽  
Regular Solutions ◽  
Singular Coefficient ◽  
Small Denominator ◽  
Small Denominators ◽  
One Dimensional ◽  
Criterion Of Uniqueness ◽  
The Given

In the given article for the mixed-type equation with a singular coefficient the first boundary value problem is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral problem the criterion of uniqueness is established. The solution the problem is constructed as the sum of series of Fourier - Bessel. At justification of convergence of a row there is a problem of small denominators. In connection with that the assessment about apartness of small denominator from zero with the corresponding asymptotic which allows to prove the convergence of the series constructed in a class of regular solutions under some restrictions is given.

scholarly journals Keldysh problem for Pulkin’s equation in a rectangular domain

2017 ◽  
Vol 21 (3) ◽  
pp. 53-63
Author(s):  
R.M. Safina
Keyword(s):  
Uniform Convergence ◽  
Mixed Type ◽  
Type Equation ◽  
Rectangular Domain ◽  
Regular Solutions ◽  
Singular Coefficient ◽  
Small Denominator ◽  
One Dimensional ◽  
Criterion Of Uniqueness ◽  
Keldysh Problem

In this article for the mixed type equation with a singular coefficient Keldysh problem of incomplete boundary conditions is studied. On the basis of property of completeness of the system of own functions of one-dimensional spectral prob- lem the criterion of uniqueness is established. The solution is constructed as the summary of Fourier-Bessel row. At the foundation of the uniform convergence of a row there is a problem of small denominators.Under some restrictions on these tasks evaluation of separation from zero of a small denominator with the corresponding asymptotics was found, which helped to prove the uniform con- vergence and its derivatives up to the second order inclusive, and the existence theorem in the class of regular solutions.

A nonlocal boundary-value problem for a fourth-order mixed-type equation

2020 ◽  
Vol 17 (1) ◽  
pp. 30-40
Author(s):  
Kudratillo Fayazov ◽  
Ikrombek Khajiev
Keyword(s):  
Fourth Order ◽  
Mixed Type ◽  
Nonlocal Boundary ◽  
Type Equation ◽  
Small Denominators ◽  
Mixed Type Equation ◽  
The Stability ◽  
Criterion Of Uniqueness ◽  
Uniqueness Of A Solution ◽  
Stability Of The Solution

The criterion of uniqueness of a solution of the problem with periodicity and nonlocal and boundary conditions is established by the spectral analysis for a fourth-order mixed-type equation in a rectangular region. When constructing a solution in the form of the sum of a series, we use the completeness in the space L_2, the system of eigenfunctions of the corresponding problem orthogonally conjugate. When proving the convergence of a series, the problem of small denominators arises. Under some conditions imposed on the parameters of the data of the problem and given functions, the stability of the solution is proved.

scholarly journals An Inverse Source Non-local Problem for a Mixed Type Equation with a Caputo Fractional Differential Operator

2017 ◽  
Vol 7 (2) ◽  
pp. 417-438 ◽  
Author(s):  
E. Karimov ◽  
N. Al-Salti ◽  
S. Kerbal
Keyword(s):  
Mixed Type ◽  
Type Equation ◽  
Integral Form ◽  
Rectangular Domain ◽  
Regularity Conditions ◽  
Mixed Type Equation ◽  
Fractional Differential Operator ◽  
Non Local ◽  
Fractional Differential ◽  
The Given

AbstractWe consider the unique solvability of an inverse-source problem with integral transmitting condition for a time-fractional mixed type equation in rectangular domain where the unknown source term depends only on the space variable. The solution is based on a series expansion using a bi-orthogonal basis in space, corresponding to a non-self-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. The influence of the transmitting condition on the solvability of the problem is also demonstrated. Two different transmitting conditions are considered — viz. a full integral form and a special case. In order to simplify the bulky expressions appearing in the proof of our main result, we establish a new property of the recently introduced Mittag-Leffler type function in two variables.

scholarly journals ON SOME PROBLEMS FOR A LOADED PARABOLIC-HYPERBOLIC EQUATION

2017 ◽  
Vol 19 (6) ◽  
pp. 201-204
Author(s):  
A.V. Tarasenko
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Mixed Type ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Dimensional Problem ◽  
One Dimensional ◽  
Various Boundary Conditions ◽  
Rectangular Area ◽  
Criterion Of Uniqueness

Some problems with various boundary conditions for the loaded mixed type equation in rectangular area are studied. The criterion of uniqueness is established and theorems of an existence of solutions to the problems are proved. The solutions are constructed as Fourier series with respect to eigenfunctions of a corresponding one-dimensional problem.

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