Uncertain Fractional Evolution Equations with Non-Lipschitz Conditions Using the Condensing Mapping Approach

2021 ◽  
Author(s):  
Nguyen Thi Kim Son
Keyword(s):  
Evolution Equations ◽  
Fractional Evolution Equations ◽  
Mapping Approach ◽  
Condensing Mapping ◽  
Lipschitz Conditions

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.

scholarly journals Complete controllability for a class of fractional evolution equations with uncertainty

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Nguyen Thi Kim Son ◽  
◽  
Nguyen Phuong Dong ◽  
Le Hoang Son ◽  
Alireza Khastan ◽  
...  
Keyword(s):  
Evolution Equations ◽  
Complete Controllability ◽  
Fractional Evolution Equations

scholarly journals A Novel Analytical Technique to Obtain Kink Solutions for Higher Order Nonlinear Fractional Evolution Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Qazi Mahmood Ul Hassan ◽  
Jamshad Ahmad ◽  
Muhammad Shakeel
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Fractional Evolution Equations ◽  
Analytical Technique ◽  
Fractional Complex Transform ◽  
Fifth Order

We use the fractional derivatives in Caputo’s sense to construct exact solutions to fractional fifth order nonlinear evolution equations. A generalized fractional complex transform is appropriately used to convert this equation to ordinary differential equation which subsequently resulted in a number of exact solutions.

scholarly journals Green’s Function Estimates for Time-Fractional Evolution Equations

2019 ◽  
Vol 3 (2) ◽  
pp. 36
Author(s):  
Ifan Johnston ◽  
Vassili Kolokoltsov
Keyword(s):  
Green’S Function ◽  
Elliptic Operator ◽  
Green's Function ◽  
Green's Functions ◽  
Evolution Equations ◽  
Green’S Functions ◽  
Variable Coefficients ◽  
Second Order ◽  
Fractional Evolution Equations ◽  
Order Elliptic Operator

We look at estimates for the Green’s function of time-fractional evolution equations of the form D 0 + * ν u = L u , where D 0 + * ν is a Caputo-type time-fractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y - 1 - β for β ∈ ( 0 , 1 ) , and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green’s function of D 0 β u = L u in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green’s function of D 0 β u = Ψ ( - i ∇ ) u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α . Thirdly, we obtain local two-sided estimates for the Green’s function of D 0 β u = L u where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green’s functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green’s functions associated with L and Ψ , as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D 0 ( ν , t ) u = L u , where D ( ν , t ) is a Caputo-type operator with variable coefficients.

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