ON THE APPROXIMATION OF THE FIRST EIGENVALUE OF SOME BOUNDARY VALUE PROBLEMS

The paper studies the representation of eigenfunctions as scalar series for a two-point boundary value problem of the type (𝑛 − 1, 1) under the assumption that there exists a functional concentrated at one point such that the first 𝑛−1 of the original boundary conditions and ℓ̃𝑥 = 1 become the Cauchy conditions at this point. The eigenfunction of the boundary value problem under consideration, corresponding to the eigenvalue λ∗, is represented as a series in powers of λ∗. The equation Φ(λ) = 0, where Φ(λ) is the sum of the series in powers of λ, is considered for finding the eigenvalues of the original problem. Examples of calculating the first eigenvalue of some boundary value problems are given. Various estimates are obtained for the coefficients of such power series. A certain function of two variables 𝑡 and λ is defined, a partial differential equation is obtained for it, and conditions are obtained that it satisfies. The zeros of the “section” of this function coincide with the eigenvalues of the original boundary value problem, which can be used for their approximate calculation.