Integro-differential equation with fractional order Caputo operators and spectral parameters

Abstract

The questions of unique solvability of a boundary value problem for an integro-differential equation with two Caputo fractional operators and spectral parameters are considered. The fractional Caputo operator’s order is bigger than one and smaller than two. Using the method of Fourier series, countable systems of ordinary fractional integro-differential equations with degenerate kernel is obtained. Further, is used a method of degenerate kernel. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system the regular and irregular values of the spectral parameters were calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. In proofing of the convergence of Fourier series the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. We also studied the continuous dependence of the solution of the problem on small parameter for regular values of spectral parameters. The results obtained in this paper are formulated as a theorem.