scholarly journals Determination of order in linear fractional differential equations

2018 ◽  
Vol 21 (4) ◽  
pp. 937-948 ◽  
Author(s):  
Mirko D’Ovidio ◽  
Paola Loreti ◽  
Alireza Momenzadeh ◽  
Sima Sarv Ahrab
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Decay Rate ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Numerical Investigations ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

Abstract The order of fractional differential equations (FDEs) has been proved to be of great importance in an accurate simulation of the system under study. In this paper, the orders of some classes of linear FDEs are determined by using the asymptotic behaviour of their solutions. Specifically, it is demonstrated that the decay rate of the solutions is influenced by the order of fractional derivatives. Numerical investigations are conducted into the proven formulas.

Systems of Nonlinear Fractional Differential Equations

2015 ◽  
Vol 18 (1) ◽  
Author(s):  
Tadeusz Jankowski
Keyword(s):  
Differential Equations ◽  
Iterative Method ◽  
Unique Solution ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
The Right ◽  
Linear Fractional Differential Equations

AbstractUsing the iterative method, this paper investigates the existence of a unique solution to systems of nonlinear fractional differential equations, which involve the right-handed Riemann-Liouville fractional derivatives $D^{q}_{T}x$ and $D^{q}_{T}y$. Systems of linear fractional differential equations are also discussed. Two examples are added to illustrate the results.

scholarly journals New Method for Solving Linear Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
S. Z. Rida ◽  
A. A. M. Arafa
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Function Method ◽  
Fractional Derivatives ◽  
Linear Differential Equations ◽  
Fractional Differential ◽  
Linear Differential ◽  
Linear Fractional Differential Equations

We develop a new application of the Mittag-Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.

Numerical solution of fractional differential equations by using fractional B-spline

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Hossein Jafari ◽  
Chaudry Khalique ◽  
Mohammad Ramezani ◽  
Haleh Tajadodi
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Spline Collocation ◽  
B Spline ◽  
Computational Work ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

AbstractIn this paper, we present fractional B-spline collocation method for the numerical solution of fractional differential equations. We consider this method for solving linear fractional differential equations which involve Caputo-type fractional derivatives. The numerical results demonstrate that the method is efficient and quite accurate and it requires relatively less computational work. For this reason one can conclude that this method has advantage on other methods and hence demonstrates the importance of this work.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

scholarly journals Positive solutions for a system of fractional differential equations with p-Laplacian operator and multi-point boundary conditions

2018 ◽  
Vol 23 (5) ◽  
pp. 771-801 ◽  
Author(s):  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Existence Results ◽  
Laplacian Operator ◽  
Point Boundary ◽  
Existence And Nonexistence ◽  
Fractional Differential

>We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations with parameters and p-Laplacian operator subject to multi-point boundary conditions, which contain fractional derivatives. The proof of our main existence results is based on the Guo–Krasnosel'skii fixed-point theorem.

scholarly journals EXISTENCE AND STABILITY RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO FRACTIONAL DERIVATIVES

2019 ◽  
pp. 341
Author(s):  
Mohamed Houas ◽  
Mohamed Bezziou
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Contraction Mapping ◽  
Nonlocal Boundary ◽  
Stability Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Existence And Stability ◽  
Fractional Differential ◽  
Stability Results

In this paper, we discuss the existence, uniqueness and stability of solutions for a nonlocal boundary value problem of nonlinear fractional differential equations with two Caputo fractional derivatives. By applying the contraction mapping and O’Regan fixed point theorem, the existence results are obtained. We also derive the Ulam-Hyers stability of solutions. Finally, some examples are given to illustrate our results.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

On the Application of Fractional Derivatives to the Study of Memristor Dynamics

2020 ◽  
pp. 342-363
Author(s):  
Rawid Banchuin
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Phase Portraits ◽  
Fractional Differential

In this chapter, the authors report their work on the application of fractional derivative to the study of the memristor dynamic where the effects of the parasitic fractional elements of the memristor have been studied. The fractional differential equations of the memristor and the memristor-based circuits under the effects of the parasitic fractional elements have been formulated and solved both analytically and numerically. Such effects of the parasitic fractional elements have been studied via the simulations based on the obtained solutions where many interesting results have been proposed in the work. For example, it has been found that the parasitic fractional elements cause both charge and flux decay of the memristor and the impasse point breaking of the phase portraits between flux and charge of the memristor-based circuits similarly to the conventional parasitic elements. The effects of the order and the nonlinearity of the parasitic fractional elements have also been reported.

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