Vol 53, No 9 (2017)

We consider the inverse problem for a functional-differential equation in which the delay function and a function occurring in the source are unknown. The values of the solution and its derivative at x = 0 are given as additional information. We derive a system of nonlinear integral equations for the unknown functions. This system is used to prove a uniqueness theorem for the inverse problem.

We study the existence, uniqueness, and nonnegativity of solutions of a family of delay integral equations used in mathematical models of living systems. Conditions ensuring these properties of solutions on an infinite time interval are obtained. The continuous dependence of solutions on the initial data on finite time intervals is analyzed. Special cases in the form of delay differential and integro-differential equations arising in population dynamics models are presented.

We construct an explicit representation of the solution of a multidimensional Abel integral equation of the second kind with partial fractional integrals in terms of the Wright function.

We prove the existence of a solution of an inhomogeneous generalized Wiener–Hopf equation whose kernel is a probability distribution on R generating a random walk drifting to +∞, while the inhomogeneous term f of the equation belongs to the space L₁(0,∞) or L_∞(0,∞). We establish the asymptotic properties of the solution of this equation under various assumptions about the inhomogeneity f.

We obtain closed-form solutions of singular integral equations with Cauchy kernels in the case of an infinite integration domain. Further, we find natural conditions for the uniqueness of solutions of these equations for the case in which the solution contains arbitrary functions or arbitrary constants.

We study the numerical solution of a linear hypersingular integral equation arising when solving the Neumann boundary value problem for the Laplace equation by the boundary integral equation method with the solution represented in the form of a double layer potential. The integral in this equation is understood in the sense of Hadamard finite value. We construct quadrature formulas for the integral occurring in this equation based on a triangulation of the surface and an application of the linear approximation to the unknown function on each of the triangles approximating the surface. We prove the uniform convergence of the quadrature formulas at the interpolation nodes as the triangulation size tends to zero. A numerical solution scheme for this integral equation based on the suggested quadrature formulas and the collocation method is constructed. Under additional assumptions about the shape of the surface, we prove a uniform estimate for the error in the numerical solution at the interpolation nodes.