involution

Equiconvergence Theorem for Integral Operator with Involution

Authors: Nazarova E. V., Khalova V. A.

In the paper, the integral operator with kernel having discontinuities of the first kind at the lines t = x and t = 1 − x is studied. The equiconvergence of Fourier expansions for arbitrary integrable function f(x) in eigenfunctions and associated functions of the considered operator and expansions of linear combination of functions f(x) and f(1 − x) in trigonometric system is proved. The equiconvergence is studied using the method based on integration of the resolvent using spectral value. Methods, developed by A. P.

Substantiation of Fourier Method in Mixed Problem with Involution

Authors: Burlutskaya M. S., Khromov A. P.

In this paper the mixed problem for the first order differential equation with involution is investigated. Using the received specified asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to transform a series representing the formal solution on Fourier method, and to prove the possibility of its term by term differentiation. At the same time on the initial problem data minimum requirements are imposed.

The Mixed Problem for the Differential Equation with Involution and Potential of the Special Kind

Authors: Khromov A. P.

For the solution of some mixed problem with involution and real symmetrical potential, explicit analytical formula has been found with the use of the Fourier method. Techniques allowing to avoid term-byterm differentiation of the functional series and impose the minimum conditions for initial problem data, are used.

On the Same Theorem on a Equiconvergence at the Whole Segment for the Functional Differential Operators

Authors: Burlutskaya M. S., Khromov A. P.

The equiconvergence of expansions in eigen- and adjoint functions of functional-differential operator with involution, containing the potentials, and simplest functional-differential operator at the whole segment of Fourier series is established.

Operator Integration with an Involution Having a Power Singularity

Authors: Kornev V. V., Khromov A. P.

Spectral properties of the integral operator with an involution of special type in the upper limit are studied and an equiconvergence theorem for its generalized eigenfunction expansions is obtained.

The Theorem on Equiconvergence for the Integral Operator on Simplest Graph with Cycle

Authors: Burlutskaya M. S.

The paper deals with integral operators on the simplest geometric two-edge graph containing the cycle. The class of integral operators with range of values satisfying continuity condition into internal node of graph is described. The equiconvergence of expansions in eigen and adjoint functions and trigonometric Fourier series is established.

On Riescz Bases of Eigenfunction of 2-nd Order Differential Operator with Involution and Integral Boundary Conditions

Authors: Kurdyumov V. P.

Riesz basisness with brackets of the eigen and associated function is proved for a 2-nd order differential operator with involution in the derivatives and with integral boundary conditions. To demonstrate this the spectral problem of the initial operator is reduced to the spectral problem of a 1-st order operator without involution in the 4-dimensional vector-function space.

Integral Operators with Non-smooth Involution

Authors: Khalova V. A., Khromov A. P.

The equiconvergence of expansions in eigen- and associated functions of integral operators with non-smooth involution and trigonometric Fourier series are established.

Classical solution by the Fourier method of mixed problems with minimum requirements on the initial data

Authors: Khromov A. P., Burlutskaya M. S.

The article gives a new short proof the V. A. Chernyatin theorem about the classical solution of the Fourier method of the mixed problem for the wave equation with fixed ends with minimum requirements on the initial data. Next, a similar problem for the simplest functional differential equation of the first order with involution in the case of the fixed end is considered, and also obtained definitive results. These results are due to a significant use of ideas A. N. Krylova to accelerate the convergence of series, like Fourier series.

Mixed problem for simplest hyperbolic first order equations with involution

Authors: Burlutskaya M. S., Khromov A. P.

In this paper investigates the mixed problem for the first order differential equation with involution at the potential and with periodic boundary conditions. Using the received refined asymptotic formulas for eigenvalues and eigenfunctions of the corresponding spectral problem, the application of the Fourier method is substantiated. We used techniques, which allow to avoid investigation of the uniform convergence of the series, obtained by term by term differentiation of formal solution on method of Fourier.

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