A generalized solution of a mixed problem for the wave equation is constructed under minimal conditions on the right side of the equation. The solution is represented as a series from the Fourier method, and its sum is found. The form of a generalized solution of a mixed problem for an inhomogeneous telegraph equation is given.

The Sturm–Liouville operator with a singular potential on an interval with conjugation conditions at the interior point of the interval is considered. The operator potential may have a non-integrable singularity. For a strong solution of the Cauchy problem for an equation with a parameter, asymptotic formulas and estimates are obtained on each of the smoothness segments of the solution.

Under minimal conditions on the initial data of the mixed problem for the telegraph equation, a generalized solution is obtained in two different ways in the form of a rapidly converging function series-an analog of the well-known d'Alembert formula. The first approach is based on the sequential method: a generalized solution of the problem is determined as the limit of classical solutions of a sequence of problems. The second approach is based on the axiomatic method. Classical solutions are not involved in constructing a generalized solution. Euler's theory of divergent series with an augmented system of axioms is used. The specific feature of the problem under consideration is the presence of a nonlocal boundary condition in which the value of the function at an interior point of the interval occurs. Both approaches to constructing a generalized solution lead to the same rapidly (exponentially) converging function series.

The first boundary value problem for a second–order differential operator with a singular potential on a segment with conjugation conditions at an interior point of the segment is studied. For the solution of the problem with a parameter, asymptotic formulae and estimates are obtained on each of the segments of smoothness. A similar formula is obtained for the solution of the associated problem.

We study the first boundary value problem for a second-order differential operator with a singular coefficient on an interval with transmission conditions at an interior point. Asymptotic formulas are obtained for the eigenfunctions and eigenvalues of both the original and the adjoint operator. The completeness and unconditional basis property of the eigenfunction systems of these operators in the space of square integrable functions on the interval are established. Il’in’s method and Il’in’s conditions are applied to establish the Bessel inequality.

We study the first boundary value problem for a second-order differential operator with a singular coefficient on an interval with transmission conditions at an interior point. Asymptotic formulas are obtained for the eigenfunctions and eigenvalues of both the original and the adjoint operator. The completeness and unconditional basis property of the eigenfunction systems of these operators in the space of square integrable functions on the interval are established. Il’in’s method and Il’in’s conditions are applied to establish the Bessel inequality.

A new way of constructing a rapidly converging series is presented. The series is a generalized solution to a mixed problem for a telegraph equation considered in a half-strip. The case of an essentially non-self-adjoint operator with respect to the spatial variable is considered. To construct the solution, we apply the axiomatic model of A.P. Khromov based on the active participation of divergent series. Euler’s ideas of assigning a sum to a divergent series are used. Variable separation, which usually results in slowly converging series, generally cannot be applied to the considered problem. An analog of Lebesgue’s theorem on the integration of trigonometric Fourier series for the considered biorthogonal series is proved.

An algorithm is presented for constructing and calculating a rapidly converging series that is a (generalized or classical) solution to a mixed problem for a telegraph equation considered in a half-strip. The case of an essentially non-self-adjoint operator with respect to the spatial variable is considered. The constructed series is a generalized d’Alembert formula. The proposed approach supersedes traditional variable separation for solving mixed problems, which usually results in slowly converging series.

We consider a boundary value problem for an elliptic differential equation with analytic coefficients that is degenerate in one of the variables in a rectangle. Using the method of spectral separation of singularities, a solution of this problem is constructed in the form of a Poisson series-a series in the eigenfunctions of the second-order limit linear ordinary differential operator with analytic coefficients. Estimates are obtained for the functions of the fundamental system of solutions and the Green's functions of the sequence of boundary value problems corresponding to this operator; this enables one to weaken the previously known conditions for the convergence of the series constructed for the solution, including the case of presence of logarithmic singularities.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021

The paper discusses the basics of the spectral method of V. A. ll'in on an example of a simple second order differential operator on a segment of the number line. The first theorem of ll'in on the unconditional basis property is stated. Its detailed proof is given. A chain of generalizations of this theorem is traced. A recently established a theorem on the unconditional basis property for the differential operators with general integral boundary conditions is formulated. The substantiation of the statements about the uniform convergence of biorthogonal expansions of functions using the ll'in method is presented. The main theorems, including, the recently established theorem for operators with integral boundary conditions are formulated.

By an example of two problems it is shown that the regularization method of singular perturbations, developed for the construction of regularized asymptotic solutions of singularly perturbed problems, can be successfully applied to the construction of solutions of irregularly degenerate elliptic problems. In both cases, the spectrum of the limit operator is used to describe the characteristics of the problem. New variables (countable many) are introduced and a new problem is written in a space of infinite dimension. The resulting task will already be regular. Narrowing its solution is the solution of the original problem. In the case of a problem with a small parameter at the highest derivative, the solution of the newly obtained problem is sought by the method of the classical perturbation theory in a special space of nonresonant solutions. Theorems on existence of formal and asymptotic solution of the problem are given. In the case of a degenerate elliptic equation, an extended problem is solved. Statements about existence of formal and classical solutions of the considered problem are given. Estimates of the rate of decrease of the components of solutions are given.

for a second-order ordinary differential operator with integral boundary conditions on an interval of the real line, we derive conditions for the uniform convergence of the spectral expansion of a function in a series in the system of eigenfunctions and associated functions of the operator. We obtain estimates of the rate of convergence of the series and the rate of equiconvergence of such an expansion of a function and its expansion in the trigonometric Fourier series. We also study the uniform convergence of the expansion of a function in the biorthogonal system.

We solve a boundary value problem (problem E in the sense of M.V. Keldysh) for an irregularly degenerate elliptic operator in a rectangle. The exact solution of the problem is constructed as a series in the eigenfunctions of the limit operator. The method of spectral isolation of singularities, which generalizes the method of regularization of singular perturbations to the case of degenerate elliptic equations, is developed.

Conditions are established under which the Bessel inequality, the Riesz basis property theorem, and the theorem about the unconditional basis property of the system of eigenfunctions and associated functions hold true for an ordinary second-order differential operator on an interval of the real line with integral boundary conditions.

We consider a second-order differential operator on an interval of the real line with integral boundary conditions and the adjoint of this operator. We obtain a priori estimates of the eigenfunctions and associated functions of the adjoint operator.

We consider a second-order differential operator on an interval of the real line with integral boundary conditions. We show how to construct the adjoint operator. The differential operation of the adjoint operator can be loaded, and the domain of that operator can contain functions that, together with their derivatives, have jump discontinuities at countably many points. For the root functions of the adjoint operator, we obtain integral representations, in particular, a mean-value formula.

For a loaded second-order differential operator on a finite interval of the real line, we estimate the equiconvergence rate of the spectral expansion of a function with the trigonometric Fourier series expansion of the same function both on an interior compact set and on the entire interval.

The present review contains results of V. A. Il'in and his pupils concerning an assessment of speed of convergence and equiconvergence with a trigonometrical series of Fourier of spectral decomposition of functions on root functions of linear ordinary differential operators both self-conjugate, and not self-conjugate, set on a final piece of a numerical straight line. The first theorem of V. A. Ilyin of equiconvergence of spectral decomposition for the differential operator of any order is provided. Theorems of the speed of equiconvergence of spectral decomposition at first for any self-conjugate expansions of the one-dimensional operator Schrodinger are formulated. Thus the potential of the operator can have any features on interval border. This allows us to receive new results even for all classical orthogonal polynomials. Further results for not self-conjugate operators are formulated. The review for the so-called loaded differential operators comes to the end with the theorem of equiconvergence speed. Estimates of speed of equiconvergence of decomposition are received both on any internal compact of an interval, and on the whole interval. Dependence of an assessment of speed of equiconvergence of decomposition on any compact of the main interval from distance of this compact to interval border is established.