2019
Том 71
№ 4

# Lopushanskyi A. O.

Articles: 4
Article (Ukrainian)

### Inverse problem in the space of generalized functions

Ukr. Mat. Zh. - 2016. - 68, № 2. - pp. 241-253

For a linear nonhomogeneous diffusion equation with fractional derivative of order $\beta \in (0, 2)$ with respect to time, we establish a unique solvability of the inverse problem of determination of a pair of functions: the generalized solution u (classical as a function of time) of the first boundary-value problem for the indicated equation with given generalized functions on the right-hand sides and the unknown (depending on time) continuous coefficient of the minor term of the equation under the overdetermination condition $$\bigl( u(\cdot , t), \varphi_0(\cdot ) \bigr) = F(t), t \in [0, T].$$ Here, $F$ is a given continuous function and $(u(\cdot , t), \varphi_0(\cdot ))$ is the value of the unknown generalized function u on a given test function $\varphi_0$ for any $t \in [0, T]$.

Article (Ukrainian)

### Application of the Laplace Transform of Tempered Distributions to the Construction of Functional Calculus

Ukr. Mat. Zh. - 2015. - 67, № 11. - pp. 1498-1511

We use the generalized n-dimensional Laplace transform of tempered distributions whose supports are located in a positive n-dimensional cone to construct functional calculus for the commutative collections of injective generators of n-parameter analytic semigroups of operators acting in a Banach space.

Article (Ukrainian)

### One Inverse Problem for the Diffusion-Wave Equation in Bounded Domain

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 666–678

We prove the theorems on the existence and unique determination of a pair of functions: a(t) >0, t ∈ [0,T], and the solution u(x, t) of the first boundary-value problem for the equation $$\begin{array}{ll}{D}_t^{\beta }u-a(t){u}_{xx}={F}_0\left(x,t\right),\hfill & \left(x,t\right)\in \left(0,l\right)\times \left(0,T\right],\hfill \end{array}$$

with regularized derivative D t β u of the fractional order β ∈ (0, 2) under the additional condition a(t)u x (0, t) = F(t), t ∈ [0,T].

Article (Ukrainian)

### Space-time fractional Cauchy problem in spaces of generalized functions

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1067-1079

We prove a theorem on the existence and uniqueness and obtain a representation using the Green vector function for the solution of the Cauchy problem $$u^{(\beta)}_t + a^2(-\Delta)^{\alpha/2}u = F(x, t), \quad (x, t) \in \mathbb{R} ^n \times (0, T], \quad a = \text{const}$$ $$u(x, 0) = u_0(x), \quad x \in \mathbb{R} ^n$$ where $u^{(\beta)}_t$ is the Riemann-Liouville fractional derivative of order $\beta \in (0,1)$, and $u_0$ and $F$ belong to some spaces of generalized functions. We also establish the character of the singularity of the solution at $t = 0$ and its dependence on the order of singularity of the given generalized function in the initial condition and the character of the power singularities of the function on right-hand side of the equation. Here, the fractional $n$-dimensional Laplace operator $\mathfrak{F}[(-\Delta)^{\alpha/2} \psi(x)] = |\lambda|^{\alpha} \mathfrak{F}[\psi(x)]$.