Vol 238, No 3 (2019)

We study the conditional symmetry of a system of nonlinear reaction-diffusion equations and establish the existence of operators of conditional symmetry for systems of nonlinear reaction-diffusion equations with any number of independent variables and find these operators in the explicit form. The exact solutions of nonlinear reaction-diffusion equations with exponential nonlinearity are constructed.

We establish sufficient conditions for the existence of solutions of a weakly perturbed linear boundary-value problems for a system of integrodifferential equations. We also establish conditions for the existence and uniqueness of solutions of problems of this kind and propose an iterative procedure for the construction of the required solutions.

We consider weakly perturbed boundary-value problems for the Fredholm integral equations with degenerate kernel in Banach spaces and establish the conditions of bifurcation from the point ε = 0 for the solutions of weakly perturbed boundary-value problems for Fredholm integral equations with degenerate kernel in Banach spaces. A convergent iterative procedure is proposed for finding the solutions in the form of series \( {\sum}_{i=-1}^{+\infty }{\varepsilon}^i{z}_i(t) \) in powers of ε.

We consider a linear parabolic boundary-value problem in a thin 3D starlike joint that consists of a finite number of thin curvilinear cylinders connected through a domain (node) with diameter O(ε). We develop a procedure for the construction of the complete asymptotic expansion of the solution as ε → 0, i.e., in the case where the starlike joint is transformed into a graph. By using the method of matching of the asymptotic series, we deduce the limit problem (ε = 0) on a graph with the corresponding Kirchhoff-type conjugation conditions at the vertex. We also prove asymptotic estimates, which enable us to trace the influence of the geometric shape of the node and the physical processes running in the node on the global asymptotic behavior of the solution.

We present new fixed-point theorems for mappings acting on a metric space and establish conditions for the existence of bounded solutions of difference equations.

We establish necessary and sufficient conditions for the existence of solutions of a linear matrix boundaryvalue problem for a system of matrix differential-algebraic equations with pulsed action. We also construct a generalized Green operator of linear boundary conditions for a system of matrix differentialalgebraic equations with pulse action. To solve the matrix differential-algebraic boundary problem with pulsed action, we use the original conditions of solvability and the structure of the general solution of the linear matrix equation.