The simplest Cauchy type problems and the boundary value problems connected with them

Abstract

As is well known, the trigonometric systems

$$ \left\{ {\sqrt {\frac{2}{\pi }} \sin \frac{{\pi k}}{\sigma }x} \right\}_1^\infty \;\;\;\;and\;\;\;\;\left\{ {\frac{1}{{\sqrt \sigma }},\left\{ {\sqrt {\frac{2}{\pi }} \cos \frac{{\pi k}}{\sigma }x} \right\}_1^\infty } \right\} $$

(2)

are orthogonal bases of L₂(0, σ) and at the same time they are systems of eigen-functions of definite boundary value problems. These boundary value problems are connected with the simplest differential equation

$$ \frac{{{d^2}}}{{d{x^2}}}y\left( x \right) + \lambda y\left( x \right) = 0,\;\;\;\;\;\;x \in \left( {0,\sigma } \right) $$

(2)

with the suitable boundary conditions at the ends of the interval (0, σ).As was mentioned in the introduction to Chapter 5, the basic biorthogonal systems of Mittag-Leffler type functions, constructed there for L₂(0,σ), also turn out to be systems of eigenfunctions. But in this case they are systems of eigenfunctions of some non-ordinary boundary value problems formulated in terms of integro-differential operators of fractional orders.