scholarly journals Non-instantaneous impulses in Caputo fractional differential equations

2017 ◽  
Vol 20 (3) ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Real World ◽  
Existence Results ◽  
Advantages And Disadvantages ◽  
Fractional Order Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

AbstractRecent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of the solutions. Both approaches are compared and their advantages and disadvantages are illustrated with examples. Also some existence results are derived.

scholarly journals Solutions for Impulsive Fractional Differential Equations via Variational Methods

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Peiluan Li ◽  
Hui Wang ◽  
Zheqing Li
Keyword(s):  
Differential Equations ◽  
Variational Methods ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Critical Points ◽  
Existence Results ◽  
Fractional Order Differential Equations ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential

We investigate the boundary value problems of impulsive fractional order differential equations. First, we obtain the existence of at least one solution by the minimization result of Mawhin and Willem. Then by the variational methods and a very recent critical points theorem of Bonanno and Marano, the existence results of at least triple solutions are established. At last, two examples are offered to demonstrate the application of our main results.

Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions

2017 ◽  
Vol 24 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Donal O’Regan ◽  
Snezhana Hristova
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Difference ◽  
Initial Time ◽  
Advantages And Disadvantages ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

AbstractThe strict stability properties are generalized to nonlinear Caputo fractional differential equations in the case when both initial points and initial times are changeable. Using Lyapunov functions, some criteria for strict stability, eventually strict stability and strict practical stability are obtained. A brief overview of different types of derivatives in the literature related to the application of Lyapunov functions to Caputo fractional equations are given, and their advantages and disadvantages are discussed with several examples. The Caputo fractional Dini derivative with respect to to initial time difference is used to obtain some sufficient conditions.

A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations

2016 ◽  
Vol 19 (2) ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Direct Method ◽  
Sufficient Conditions ◽  
Dini Derivative ◽  
Advantages And Disadvantages ◽  
Impulsive Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

AbstractWe present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fractional differential equations is an appropriate definition of its derivative among the differential equation of fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo fractional Dini derivative, is generalized to impulsive fractional differential equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results.

scholarly journals Regarding on the exact solutions for the nonlinear fractional differential equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 478-482 ◽  
Author(s):  
Melike Kaplan ◽  
Murat Koparan ◽  
Ahmet Bekir
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Simple Equation ◽  
Wave Transformation ◽  
Nonlinear Ordinary Differential Equations ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

AbstractIn this work, we have considered the modified simple equation (MSE) method for obtaining exact solutions of nonlinear fractional-order differential equations. The space-time fractional equal width (EW) and the modified equal width (mEW) equation are considered for illustrating the effectiveness of the algorithm. It has been observed that all exact solutions obtained in this paper verify the nonlinear ordinary differential equations which was obtained from nonlinear fractional-order differential equations under the terms of wave transformation relationship. The obtained results are shown graphically.

scholarly journals APPLICATIONS OF BI-FRAMELET SYSTEMS FOR SOLVING FRACTIONAL ORDER DIFFERENTIAL EQUATIONS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040051 ◽  
Author(s):  
MUTAZ MOHAMMAD ◽  
CARLO CATTANI
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Sparse Matrices ◽  
Computational Method ◽  
Extension Principle ◽  
Vanishing Moments ◽  
Fractional Order Differential Equations ◽  
Fractional Differential

Framelets and their attractive features in many disciplines have attracted a great interest in the recent years. This paper intends to show the advantages of using bi-framelet systems in the context of numerical fractional differential equations (FDEs). We present a computational method based on the quasi-affine bi-framelets with high vanishing moments constructed using the generalized (mixed) oblique extension principle. We use this system for solving some types of FDEs by solving a series of important examples of FDEs related to many mathematical applications. The quasi-affine bi-framelet-based methods for numerical FDEs show the advantages of using sparse matrices and its accuracy in numerical analysis.

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
Author(s):  
Ravi Agarwal ◽  
Snehana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Intervals ◽  
Dini Derivative ◽  
Comparison Results ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

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.

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