A problem without initial conditions for a nonlinear ultraparabolic equation with degeneration

We consider the mixed problem for a second-order nonlinear degenerate ultraparabolic equation. We investigate the existence of generalized solutions of this problem in a bounded domain as well as of weak solutions (in the sense of a limit of sequences) of the problem without initial conditions for this equation.     

Branching of solutions for the problem of mean-square approximation of a real finite function of two variables by the modulus of double Fourier transformation

We investigate the branching of solutions of a Hammerstein-type nonlinear two-dimensional integral equation that arises in the problems of mean-square approximation of a real finite nonnegative function of two variables by the modulus of double Fourier integral, depending on two parameters [Mat. Metody Fiz.-Mekh. Polya, 51, No. 1, 53–64; No. 4, 80–85 (2008)]. We have derived analytical expressions for eigenfunctions of the corresponding linear homogeneous integral equation, necessary for the construction of branched-off solutions, and obtained systems of transcendental equations for finding the points of their branching. We also present, in the first approximation, the analytical representations of complex solutions branched-off from the real solution for the two-dimensional case of branching.     

On the best mean-square approximation of a real nonnegative finite continuous function of two variables by the modulus of a double Fourier integral. II

We continue the investigation of the nonlinear problem of mean-square approximation of a real finite nonnegative continuous function of two variables by the modulus of a double Fourier integral depending on two parameters begun in the first part of this work [J. Math. Sci., 160, No. 3, 343–356 (2009)]. Finding the solutions of this problem is reduced to the solution of a nonlinear two-dimensional integral equation of the Hammerstein type. We construct and justify numerical algorithms for determination of branching lines and branched solutions of this equation. Numerical examples are presented.     

Implicit-function methods for the solution of two-parameter linear spectral problems

We consider a method for the solution of linear two-parameter spectral algebraic problems, based on the implicit-function methods. The use of the implicit-function methods enables one to determine the properties and structure of an existing spectrum and to find all components of the spectrum that are contained in a certain bounded convex set of the space \mathbbC² {\mathbb{C}^2} or \mathbbR² {\mathbb{R}^2} .     

Variational Ultraparabolic Inequalities

In a bounded domain of the space ℝ ^{n +2}, we consider variational ultraparabolic inequalities with initial condition. We establish conditions for the existence and uniqueness of a solution of this problem. As a special case, we establish the solvability of mixed problems for some classes of nonlinear ultraparabolic equations with nonclassical and classical boundary conditions.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1616–1628, December, 2004.     

Properties of a solution of the mixed problem for an ultraparabolic equation with memory term

The mixed problem for an ultraparabolic equation is considered. The uniqueness and the existence of a solution of the problem are established. Some estimates of the solution that depend on the kernel of an integral operator are found.     

Properties of solutions of a mixed problem for a nonlinear ultraparabolic equation

Mixed problems for a nonlinear ultraparabolic equation are considered in domains bounded and unbounded with respect to the space variables. Conditions for the existence and uniqueness of solutions of these problems are established and some estimates for these solutions are obtained.     

On the best mean-square approximation of a real nonnegative finite continuous function of two variables by the modulus of a double Fourier integral. I

We study the nonlinear problem of mean-square approximation of a real finite nonnegative continuous function of two variables by the modulus of a double Fourier integral depending on two parameters. The solution of this problem is reduced to the solution of a nonlinear two-dimensional integral equation of the Hammerstein type. Numerical algorithms for determination of branching lines and branched solutions of equation are constructed and substantiated. Some numerical examples are given. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 53–64, January–March, 2008.