A new technique for solving Sobolev type fractional multi-order evolution equations

Abstract

A strong inspiration for studying Sobolev type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices.

Perturbation theory for fractional evolution equations in a Banach space

“…Proof. This lemma is a special case of Lemma 3.2 in [29,30] with β = 0 which can be proved via mathematical induction principle. So, the proof of this lemma is omitted here.…”

“…Case 2b: Permutable case (AB = BA). This case is obtained directly from (4.15) using the following identity for Q A,B k,m [29,30]:…”

Asymptotic behavior of solutions to fractional stochastic multi-term differential equation systems involving non-permutable matrices

“…An explicit representation of Q A,B k,m can be found in Table 1 in [30]. In the case of permutable matrices, i.e.…”

Exponential stability and stabilization of fractional stochastic degenerate evolution equations in a Hilbert space: Subordination principle

“…The existence and uniqueness of the mild solution of degenerate evolution equations to infinite dimensional frameworks have been studied by several researchers under Lipschitz and non-Lipschitz conditions. More information on Sobolev-type differential equations and their fractional-order analogues can be found in [6,14,32,43,45,49] in recent decades. However, there are few works on Sobolev-type differential equations in the stochastic setting.…”