scholarly journals Existence Solution and Controllability of Sobolev Type Delay Nonlinear Fractional Integro-Differential System

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 79 ◽  
Author(s):  
Hamdy Ahmed ◽  
Mahmoud El-Borai ◽  
Hassan El-Owaidy ◽  
Ahmed Ghanem
Keyword(s):  
Fractional Order ◽  
Differential System ◽  
Diffusion Processes ◽  
Sufficient Conditions ◽  
Null Controllability ◽  
Sobolev Type ◽  
Materials With Memory ◽  
Impulsive Delay ◽  
Physical Phenomena ◽  
With Memory

Fractional integro-differential equations arise in the mathematical modeling of various physical phenomena like heat conduction in materials with memory, diffusion processes, etc. In this manuscript, we prove the existence of mild solution for Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. We establish the sufficient conditions for the approximate controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. In addition, we prove the exact null controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. Finally, an example is given to illustrate the obtained results.

scholarly journals Second-Order Impulsive Delay Differential Systems: Necessary and Sufficient Conditions for Oscillatory or Asymptotic Behavior

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 722
Author(s):  
Shyam Sundar Santra ◽  
Khaled Mohamed Khedher ◽  
Osama Moaaz ◽  
Ali Muhib ◽  
Shao-Wen Yao
Keyword(s):  
Asymptotic Behavior ◽  
Differential System ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Differential Systems ◽  
Sufficient And Necessary Conditions ◽  
Impulsive Delay ◽  
Delay Differential Systems ◽  
Delay Differential ◽  
Necessary And Sufficient

In this work, we aimed to obtain sufficient and necessary conditions for the oscillatory or asymptotic behavior of an impulsive differential system. It is easy to notice that most works that study the oscillation are concerned only with sufficient conditions and without impulses, so our results extend and complement previous results in the literature. Further, we provide two examples to illustrate the main results.

scholarly journals Observability of Fractional-Order Impulsive Control Integro-Differential System with Nonlocal Initial Condition

2020 ◽  
Vol 55 (1) ◽  
Author(s):  
Sameer Qasim Hasan
Keyword(s):  
Fixed Point ◽  
Fractional Order ◽  
Differential System ◽  
Impulsive Control ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Necessary Condition ◽  
Primary Role ◽  
Bounded Linear

The article describes a new concept for initial and exactly observability of nonlocal fractional-order impulsive control integro-differential system. This is based on the concepts of the abstract Cauchy problem, which depended on some necessary and sufficient conditions. These conditions established on the semigroup theory of bounded operators as a dynamical operator system, which generated by bounded linear operators. Moreover, invertible operators play a primary role, and we presented a necessary condition for some nonlinear multi variables functions. Thus, all these operators were treated in nonlinear functional analysis to guaranty the initial observable and exactly observability. Therefore, from the mild solution of the system and exactly homogenous part, we proved the equivalent concepts between the initial observability and exactly the observability. Thus, our approach in this article is to prove the uniqueness of initial nonlocal values with admissible control, which belongs to the second-order Lebesgue integrable. The interest of observability results in this article lies by proving a unique fixed point, which is nonlocal initial values that are described in the proposal formula by using Banach’s fixed point theory. The processing observability for complexly systems (such as this system) with all components and properties was established and can be used for many control system applications.

On linear theory of heat conduction in materials with memory. Existence and uniqueness theorems for the final value problem

1977 ◽  
Vol 76 (2) ◽  
pp. 119-137 ◽  
Author(s):  
H. Grabmüller
Keyword(s):  
Heat Conduction ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Function Algebra ◽  
Fading Memory ◽  
Uniqueness Of Solutions ◽  
Materials With Memory ◽  
Solvability Conditions ◽  
Broad Variety ◽  
With Memory

SynopsisAn improperly posed problem is studied for a linear partial integro-differential equation of convolution type on the semi-axis. The problem originates from a generalised process of heat conduction in materials with fading memory, where the temperature of the material has to be determined for prescribed homogeneous boundary conditions and for a given final temperature distribution. By using eigenfunctions of the n-dimensional Laplacian, the problem is reduced to a family of equivalent initial-value problems for ordinary integro-differential equations; the latter are treated by the method of factorisation in a suitable function algebra. Sufficient conditions for the existence and uniqueness of solutions to the original problem are obtained in terms of the solvability conditions of the reduced problems. The whole analysis is performed simultaneously in a broad variety of spaces consisting of functions with an exponential growth rate (in the time variable) at infinity. One of the main advantages in the present approach is that solutions, if they exist, can always be computed explicitly.

scholarly journals Analysis and Dynamics of Fractional Order Covid-19 Model with Memory Effect

2021 ◽  
pp. 104017
Author(s):  
Supriya Yadav ◽  
Devendra Kumar ◽  
Jagdev Singh ◽  
Dumitru Baleanu
Keyword(s):  
Memory Effect ◽  
Fractional Order ◽  
With Memory

Analysis of a New Class of Impulsive Implicit Sequential Fractional Differential Equations

2020 ◽  
Vol 21 (6) ◽  
pp. 571-587 ◽  
Author(s):  
Akbar Zada ◽  
Sartaj Ali ◽  
Tongxing Li
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sufficient Conditions ◽  
Uniqueness Of Solutions ◽  
Integer Order ◽  
New Class ◽  
Proposed Model ◽  
Fractional Differential ◽  
Sequential Fractional Differential Equations

AbstractIn this paper, we study an implicit sequential fractional order differential equation with non-instantaneous impulses and multi-point boundary conditions. The article comprehensively elaborate four different types of Ulam’s stability in the lights of generalized Diaz Margolis’s fixed point theorem. Moreover, some sufficient conditions are constructed to observe the existence and uniqueness of solutions for the proposed model. The proposed model contains both the integer order and fractional order derivatives. Thus, the exponential function appearers in the solution of the proposed model which will lead researchers to study fractional differential equations with well known methods of integer order differential equations. In the last, few examples are provided to show the applicability of our main results.

scholarly journals Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Qiong Meng ◽  
Zhen Jin ◽  
Guirong Liu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Multiple Delays ◽  
All Solutions ◽  
Sufficient And Necessary Conditions ◽  
Delay Differential ◽  
T Tau ◽  
Sufficient And Necessary Condition

AbstractThis paper studies the linear fractional-order delay differential equation $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ D − α C x ( t ) − p x ( t − τ ) = 0 , where $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ 0 < α = odd integer odd integer < 1 , $p, \tau >0$ p , τ > 0 , ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s . We obtain the conclusion that $$ p^{1/\alpha } \tau >\alpha /e $$ p 1 / α τ > α / e is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.

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