Vol 228, No 4 (2018)

We consider evolution equations with variable parameters. We obtain new representations of solutions and indicate their applications to inverse problems in mathematical physics.

Based on the theory of thermodynamically compatible systems, we formulate the governing equations of a gas-liquid compressible pipe flow and develop the Runge–Kutta–TVD and Runge–Kutta–WENO high accuracy methods. The computational model is used to solve a series of test cases demonstrating its efficiency and capability to be applied to slug flows modeling.

We introduce a class of pairs of operators defining a linear homogeneous degenerate evolution fractional differential equation in a Banach space. Reflexive Banach spaces are represented as the direct sums of the phase space of the equation and the kernel of the operator at the fractional derivative. In a sector of the complex plane containing the positive half-axis, we construct an analytic family of resolving operators that degenerate only on the kernel. The results are used in the study of the solvability of initial-boundary value problems for partial differential equations containing fractional time-derivatives and polynomials in the Laplace operator with respect to the spatial variable.

We consider the equilibrium problem for plates with cracks located along rigid inclusions and analyze the dependence of the solution and the derivative of the energy functional on variations of the inclusion size. We establish the solvability of the optimal control problem, where the quality functional is given by the derivative of the energy functional, whereas the control parameter correspond to the inclusion size. A similar analysis is performed for the equilibrium problems for inhomogeneous plates with rigid delaminated inclusions.

We consider the nonlinear inverse problem for a second order hyperbolic equation with unknown coefficient depending on the time. We establish the existence and uniqueness of regular solutions which are used for constructing a soltuion to the inverse problem under consideration.

We study the Dirichlet problem for the inhomogeneous p-Laplace equation with a nonlinear source and a gradient term. We investigate the influence of the gradient term on the existence of radially symmetric solutions. Sufficient conditions for the existence of solutions are explicetly given in terms of the data of the problem.