# Cherepennikov V. B.

### One method for the investigation of linear functional-differential equations

Cherepennikov V. B., Vetrova E. V.

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 594-600

We consider the scalar linear retarded functional differential equation $$\dot{x}(t) = ax(t - 1)+ bx \left( \frac tq \right) + f(t), \quad q > 1.$$ The study of linear retarded functional differential equations deals mainly with two initial-value problems: an initial-value problem with initial function and an initial-value problem with initial point (when one seeks a classical solution whose substitution into the original equation reduces it to an identity). In the present paper, an initial-value problem with initial point is investigated by the method of polynomial quasisolutions. We prove theorems on the existence of polynomial quasisolutions and exact polynomial solutions of the considered linear retarded functional differential equation. The results of a numerical experiment are presented.

### Polynomial quasisolutions of linear second-order differential-difference equations

Cherepennikov V. B., Ermolaeva P. G.

Ukr. Mat. Zh. - 2008. - 60, № 1. - pp. 140–152

The second-order scalar linear difference-differential equation (LDDE) with delay $$\ddot{x}(t) + (p_0+p_1t)\dot{x}(t) = (a_0 +a_1t)x(t-1)+f(t)$$ is considered. This equation is investigated with the use of the method of polynomial quasisolutions based on the presentation of an unknown function in the form of polynomial $x(t)=\sum_{n=0}^{N}x_n t^n.$ After the substitution of this function into the initial equation, the residual $\Delta(t)=O(t^{N-1}),$ appears. The exact analytic representation of this residual is obtained. The close connection is demonstrated between the LDDE with varying coefficients and the model LDDE with constant coefficients whose solution structure is determined by roots of a characteristic quasipolynomial.