2019
Том 71
№ 10

Ashyralyev A.

Articles: 3
Article (English)

The structure of fractional spaces generated by the two-dimensional difference operator on the half plane

Ukr. Mat. Zh. - 2018. - 70, № 8. - pp. 1019-1032

We consider a difference operator approximation $A^x_h$ of the differential operator $A^xu(x) = a_{11}(x)u_{x_1 x_1}(x) - a_{22}(x)u_{x_2x_2} (x) + \sigma u(x),\; x = (x_1, x_2)$ defined in the region $R^{+} \times R$ with the boundary condition $u(0, x_2) = 0,\; x_2 \in R$. Here, the coefficients $a_{ii}(x), i = 1, 2$, are continuously differentiable, satisfy the uniform ellipticity condition $a^2_{11}(x) + a^2_{22}(x) \geq \delta > 0$. We investigate the structure of the fractional spaces generated by the analyzed difference operator. Theorems on well-posedness in a Holder space of difference elliptic problems are obtained as applications.

Article (Ukrainian)

Well-Posedness of the Right-Hand Side Identification Problem for a Parabolic Equation

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 147–158

We study the inverse problem of reconstruction of the right-hand side of a parabolic equation with nonlocal conditions. The well-posedness of this problem in Hölder spaces is established.

Article (English)

On the problem of determining the parameter of a parabolic equation

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1200–1210

We study the boundary-value problem of determining the parameter p of a parabolic equation $$v′(t)+Av(t)=f(t)+p,\;0⩽t⩽1,v(0)=φ,\;v(1)=ψ,$$ with strongly positive operator $A$ in an arbitrary Banach space $E$. The exact estimates are established for the solution of this problem in Hölder norms. In applications, the exact estimates are obtained for the solutions of the boundary-value problems for parabolic equations.