Evolution Equations in Hilbert Spaces via the Lacunae Method

Abstract

In this paper we consider evolution equations in the abstract Hilbert space under the special conditions imposed on the operator at the right-hand side of the equation. We establish the method that allows us to formulate the existence and uniqueness theorem and find a solution in the form of a series on the root vectors of the right-hand side. As an application we consider fractional differential equations of various kinds. Such operators as the Riemann-Liouville fractional differential operator, the Riesz potential, the difference operator have been involved.

A Comparison of Parallel Algorithms for Numerical Solution of Parabolic Problems with Fractional Power Elliptic Operators

“…Next we consider an efficient approach for how to construct operators B h to solve the implicit non-local systems (10) and ( 13) and ( 14).…”

“…The construction of B h is based on the so-called AAA algorithm proposed in [15]. We will apply this general algorithm for implementation of the discrete scheme (10). Let us consider a function…”

“…At the end of this short review we note one more important approach. There exist very interesting analytical methods that can solve the same equations in the form of a series on the root vectors of the right-hand [9,10]. They look very promising for the development of not only theoretical results on the existence and uniqueness of solutions, but also for the numerical solution of real world applied problems.…”