scholarly journals Fractional Evolution Equations Governed by Coercive Differential Operators

2009 ◽  
Vol 2009 ◽  
pp. 1-14 ◽  
Author(s):  
Fu-Bo Li ◽  
Miao Li ◽  
Quan Zheng
Keyword(s):  
Differential Operator ◽  
Fractional Order ◽  
Evolution Equations ◽  
Differential Operators ◽  
Fractional Evolution Equations ◽  
Resolvent Family ◽  
Well Posed

This paper is concerned with evolution equations of fractional orderDαu(t)=Au(t);u(0)=u0,u′(0)=0,whereAis a differential operator corresponding to a coercive polynomial taking values in a sector of angle less thanπand1<α<2. We show that such equations are well posed in the sense that there always exists anα-times resolvent family for the operatorA.

On the Influence of Fractional Derivative on Chaos Control of a New Fractional-Order Hyperchaotic System

2020 ◽  
pp. 115-132
Author(s):  
Ahmed Ezzat Mohamed Matouk
Keyword(s):  
Differential Operator ◽  
Fractional Order ◽  
Differential Operators ◽  
Equilibrium Points ◽  
Hyperchaotic System ◽  
Fractional Differential Operator ◽  
Quadratic Nonlinearities ◽  
Potential Applications ◽  
Non Local ◽  
Fractional Differential

The non-local fractional differential operators have potential applications in many fields of science and technology but especially in the field of dynamical systems. This chapter introduces a new hyperchaotic dynamical system involving non-local fractional differential operator with singular kernel (the Caputo type). The system involves three quadratic nonlinearities and also three equilibrium points. Existence of chaotic and hyperchaotic attractors has been illustrated. Based on Matouk's stability theory of four-dimensional fractional-order systems, the influence of the fractional differential operator on stabilizing the proposed system to its three steady states has been shown. Numerical results have been provided to verify the theoretical analysis. This kind of study is expected to add useful applications to chaos-based secure communications and text encryption.

scholarly journals On a fractional-order p-Laplacian boundary value problem at resonance on the half-line with two dimensional kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. F. Imaga ◽  
S. A. Iyase
Keyword(s):  
Differential Operator ◽  
Boundary Value Problem ◽  
Fractional Order ◽  
Boundary Value ◽  
Coincidence Degree ◽  
Half Line ◽  
At Resonance ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Problem At Resonance

AbstractIn this work, we consider the solvability of a fractional-order p-Laplacian boundary value problem on the half-line where the fractional differential operator is nonlinear and has a kernel dimension equal to two. Due to the nonlinearity of the fractional differential operator, the Ge and Ren extension of Mawhin’s coincidence degree theory is applied to obtain existence results for the boundary value problem at resonance. Two examples are used to validate the established results.

scholarly journals Multidimensional van der Corput-Type estimates involving Mittag-Leffler functions

2020 ◽  
Vol 23 (6) ◽  
pp. 1663-1677
Author(s):  
Michael Ruzhansky ◽  
Berikbol T. Torebek
Keyword(s):  
Cauchy Problem ◽  
Exponential Function ◽  
Evolution Equations ◽  
Oscillatory Integrals ◽  
Schrödinger Equations ◽  
Fractional Evolution Equations ◽  
Type Function ◽  
Fractional Schrödinger Equations ◽  
The Cauchy Problem ◽  
Klein Gordon

Abstract The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions E α, β (i λ ϕ(x)), x ∈ ℝ N and E α, β (i α λ ϕ(x)), x ∈ ℝ N for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

scholarly journals Study of Hilfer fractional evolution equations by the properties of controllability and stability

2021 ◽  
Vol 60 (4) ◽  
pp. 3741-3749
Author(s):  
Pallavi Bedi ◽  
Anoop Kumar ◽  
Thabet Abdeljawad ◽  
Aziz Khan
Keyword(s):  
Evolution Equations ◽  
Fractional Evolution Equations

scholarly journals A study on controllability of impulsive fractional evolution equations via resolvent operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Haide Gou ◽  
Yongxiang Li
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Resolvent Operator ◽  
Evolution Equations ◽  
Validity Study ◽  
Resolvent Operators ◽  
Fractional Evolution Equations ◽  
Mönch Fixed Point Theorem ◽  
Alpha Beta

AbstractIn this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the $(\alpha ,\beta )$ ( α , β ) -resolvent operator, we concern with the term $u'(\cdot )$ u ′ ( ⋅ ) and finding a control v such that the mild solution satisfies $u(b)=u_{b}$ u ( b ) = u b and $u'(b)=u'_{b}$ u ′ ( b ) = u b ′ . Finally, we present an application to support the validity study.

A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer†

1978 ◽  
Vol 82 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Operator ◽  
Positive Integer ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Ordinary Differential Operators ◽  
Half Axis ◽  
Complex Coefficients

SynopsisFormulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

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