# Lopushanskaya G. P.

### Inverse problem in the space of generalized functions

Lopushanskaya G. P., Lopushanskyi A. O., Rapita V.

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 $t \in [0, T]$.

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

Lopushanskaya G. P., Lopushanskyi A. O.

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*]*.*

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

Lopushanskaya G. P., Lopushanskyi A. O.

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 $\mathfrak{F}[(-\Delta)^{\alpha/2} \psi(x)] = |\lambda|^{\alpha} \mathfrak{F}[\psi(x)]$.

### Generalized boundary values of the solutions of semilinear elliptic equations from weighted functional spaces

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 472-493

In weighted *C*-spaces, we establish the solvability of a boundary-value problem for a semilinear elliptic equation of order 2*m* in a bounded domain with generalized functions given on its boundary, strong power singularities at some points of the boundary, and finite orders of singularities on the entire boundary. The behavior of the solution near the boundary of the domain is analyzed.

### Generalized boundary values of solutions of quasilinear elliptic equations with linear principal part

Ukr. Mat. Zh. - 2007. - 59, № 12. - pp. 1674–1688

We establish conditions for the nonlinear part of a quasilinear elliptic equation of order $2m$ with linear principal part under which a solution regular inside a domain and belonging to a certain weighted $L_1$-space takes boundary values in the space of generalized functions.

### Solution of boundary-value problems for elliptic equations in the space of distributions

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 190–203

We extend the well-known approach to solution of generalized boundary-value problems for second-order elliptic and parabolic equations and for second-order strongly elliptic systems of variational type to the case of a general normal boundary-value problem for an elliptic equation of order*2m*. The representation of a distribution from (*C* ^{∞} *(S)*)’ is established and is usedfor the proof of convergence of an approximate method of solution of a normal elliptic boundary-value problem in unnormed spaces of distributions.

### Basic boundary-value problems for one equation with fractional derivatives

Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 48–59

We prove some properties of solutions of an equation $\cfrac{\partial^{2\alpha}u}{\partial x_1^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_2^{2\alpha}} + \cfrac{\partial^{2\alpha}u}{\partial x_3^{2\alpha}} = 0, \quad \alpha \in \left( \cfrac 12\, ; 1 \right ]$, in a domain $\Omega \subset R^3$ which are similar to the properties of harmonic functions. By using the potential method, we investigate principal boundary-value problems for this equation.

### Approximate method for the solution of the generalized Dirichlet problem

Ukr. Mat. Zh. - 1994. - 46, № 10. - pp. 1417–1420

We extend the method for approximate solution of classical boundary-value problems for the Laplace equation suggested in [1–3] to the case of the Poisson equation with generalized functions on the right-hand side of the equation and in the boundary conditions.

### Method of investigating boundary-value problems in distribution spaces and boundary integral equations

Ukr. Mat. Zh. - 1991. - 43, № 5. - pp. 632–639

### Some properties of solutions of nonlocal elliptic problems in the space of generalized functions

Ukr. Mat. Zh. - 1989. - 41, № 11. - pp. 1487–1494

### Solution of the parabolic boundary-value problem in a space of generalized functions with the help of the green matrix

Ukr. Mat. Zh. - 1986. - 38, № 6. - pp. 795–798

### Representation of a solution of a generalized boundary problem for a system of differential equations, elliptic in the sense of Petrovskii

Gupalo A. S., Lopushanskaya G. P.

Ukr. Mat. Zh. - 1985. - 37, № 1. - pp. 116 – 119