Differencial'nye uravneniya

The question of existence and uniqueness of the classical solution of the mixed problem in a half-strip for the nonlinear Liouville equation is investigated. The solution is presented in implicit form as a solution of an integral equation, the solvability of which is proved using the Leray-Schauder theorem, and the corresponding a priori estimate is obtained using energy methods.

A model pseudo-differential equation in Sobolev–Slobodetskii space is considered in a cone which is a direct product of low dimensional cones. Under existence of a special factorization for symbol a general solution can be written, it includes arbitrary functions from certain Sobolev–Slobodetskii space. A certain example in three-dimensional space is considered, the unknown functions can be determined with help of Dirichlet conditions on a piece of a boundary by reducing to a system of linear integral equations.

The first boundary value problems (internal and external) of a two-dimensional filtration flow of a liquid in a non-uniform porous layer of variable thickness and permeability are investigated. The given discrete sources of the flow are located in the flow region and are modeled by singularities (isolated special points) of a complex potential. The boundary of the flow region is modeled by an arbitrary smooth (piecewise smooth) closed contour. A function is specified on the boundary that characterizes the pressure distribution on it and has discontinuities of the first kind. A method for regularizing (smoothing) the boundary condition is proposed, which allows reducing the problems to a boundary singular integral equation with a weak singularity and a smooth right-hand side. This regularization method is applied to the solution of a boundary value problem, simulating the operation of a system of wells in a heterogeneous layer (formation) of soil, on the supply contour of which a given pressure distribution (generalized potential) suffers discontinuities of the first kind.

A two-dimensional hypersingular integral equation in a convex bounded domain whose boundary is a smooth curve is considered. The equation contains an integral operator with an integral understood in the sense of a finite part according to Hadamard. The question of the existence of solutions having a power singularity in the neighborhood of the boundary of the domain is considered: the solution is sought in a class of functions represented as the ratio of a smooth function and the root of the distance from a point to the boundary. It is proved that when an integral operator with a power polar singularity of the third order acts on a function from the class in which the solution is sought, a function arises that is continuous in the sense of Helder on the entire domain. Further, the existence of solutions of the hypersingular equation having the specified power singularity in the neighborhood of the boundary of the domain is proved. A boundary condition is presented under which such a solution is unique.

We consider the Cauchy problem for a system of integro-differential equations of the first order with difference kernels in finite-dimensional Hilbert spaces. This class of equations arises in the mathematical modeling of a wide range of nonstationary processes taking into account memory effects, including, in particular, the system of Maxwell's equations. For numerical solution, a method of reducing the original nonlocal problem to an equivalent system of local differential equations of the first order on the basis of approximation of kernels by a finite sum of exponential functions is proposed. Two-level operator-difference schemes are proposed, for which the stability of the initial data and the right-hand side is analyzed. The performed theoretical analysis demonstrates the correctness of the proposed approach.