Boundary-value problems for differential equations of fractional order

Abstract

Abstract. We carry out spectral analysis of one class of integral operators associated with fractional order differential equations that arise in mechanics. We establish a connection between the eigenvalues of these operators and the zeros of Mittag-Leffler type functions. We give sufficient conditions for complete nonselfadjointness and completeness of systems of the eigenvalues. We prove the existence and uniqueness of the solutions for several kinds of two-point boundary value problems of fractional differential equations with Caputo's derivatives or Riemann-Liouville Derivatives, and design the single shooting methods to solve them numerically.Chapter 1. Regional problems for differential equations of fractional order 1. The basic concepts The spectral analysis of operators of the form

On one class of persymmetric matrices generated by boundary value problems for differential equations of fractional order

“…Now, we will give a very important result that indicates that matrix T n−1 (μ) has a simple structure [5] for any n. In particular, from above follows the completeness of the system of eigenfunctions of the operator A α,β γ for α = β = γ = 1/ρ, 0 < ρ < 2, which was established by other methods (see [1,2,8]). …”

“…For f(x) ∈ L 1 (0, 1), the function 1) is called the fractional integral of order α > 0 on [0, x], and the function 1) is called the fractional integral of order α > 0 on [x, 1] (see [1]). Here, (α) is the Gamma function of Euler.…”

“…Here, (α) is the Gamma function of Euler. As it is known, (see [1]), the function g(x) ∈ L 1 (0, 1) given by…”

“…It was shown in [1] that the operator generated by the expression (0.2) and boundary value conditions of Sturm-Liouville type has oscillatory properties. Operators such as…”

On numerical solution of fractional order boundary value problem with shooting method

“…Numerous publications relevant with fractional differential equations have been published in time. Some of them are [1]- [4], [7], [8], [10]. Also some important books on fractional field can remark as [3] and [7].…”