scholarly journals Analytic Approach for Solving System of Fractional Differential Equations

2021 ◽  
Vol 32 (1) ◽  
pp. 14
Author(s):  
Nabaa N Hasan ◽  
Zainab John
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Initial Conditions ◽  
Analytic Approach ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential

In this paper, Sumudu transformation (ST) of Caputo fractional derivative formulae are derived for linear fractional differential systems. This formula is applied with Mittage-Leffler function for certain homogenous and nonhomogenous fractional differential systems with nonzero initial conditions. Stability is discussed by means of the system's distinctive equation.

ANALYSIS OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH MULTI-ORDERS

Fractals ◽  
2007 ◽  
Vol 15 (02) ◽  
pp. 173-182 ◽  
Author(s):  
WEIHUA DENG ◽  
CHANGPIN LI ◽  
QIAN GUO
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
Rational Numbers ◽  
The Other ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Stability Theorems ◽  
Fractional Differential ◽  
Fractional Orders

In this paper, we study two kinds of fractional differential systems with multi-orders. One is a system of fractional differential equations with multi-order, [Formula: see text], [Formula: see text]; the other is a multi-order fractional differential equation, [Formula: see text]. By the derived technique, such two kinds of fractional differential equations can be changed into equations with the same fractional orders providing that the multi-orders are rational numbers, so the known theorems of existence, uniqueness and dependence upon initial conditions are easily applied. And asymptotic stability theorems for their associate linear systems, [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], [Formula: see text], are also derived.

scholarly journals On Stability of Nonautonomous Perturbed Semilinear Fractional Differential Systems of Order α∈(1,2)

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Mohammed M. Matar ◽  
Esmail S. Abu Skhail
Keyword(s):  
Neural Networks ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Protected Area ◽  
Marine Protected Area ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential ◽  
Stability Results

We study the Mittag-Leffler and class-K function stability of fractional differential equations with order α∈(1,2). We also investigate the comparison between two systems with Caputo and Riemann-Liouville derivatives. Two examples related to fractional-order Hopfield neural networks with constant external inputs and a marine protected area model are introduced to illustrate the applicability of stability results.

LYAPUNOV-TYPE INEQUALITIES FOR LOCAL FRACTIONAL DIFFERENTIAL SYSTEMS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050131
Author(s):  
YONGFANG QI ◽  
LIANGSONG LI ◽  
XUHUAN WANG
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Analytical Method ◽  
Systematic Design ◽  
Differential Systems ◽  
Design Algorithm ◽  
Fractional Differential Systems ◽  
Lyapunov Inequalities ◽  
Fractional Differential

This paper deals with the problem of Lyapunov inequalities for local fractional differential equations with boundary conditions. By using analytical method, a novel Lyapunov-type inequalities for the local fractional differential equations is provided. A systematic design algorithm is developed for the construction of Lyapunov inequalities.

scholarly journals On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms

2020 ◽  
Vol 40 (2) ◽  
pp. 227-239
Author(s):  
John R. Graef ◽  
Said R. Grace ◽  
Ercan Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Nonoscillatory Solutions ◽  
Fractional Differential

This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

scholarly journals A Note on the Stability Analysis of Fuzzy Nonlinear Fractional Differential Equations Involving the Caputo Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Ali El Mfadel ◽  
Said Melliani ◽  
M’hamed Elomari
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Direct Method ◽  
Stability Result ◽  
Nonlinear Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

In this paper, we present and establish a new result on the stability analysis of solutions for fuzzy nonlinear fractional differential equations by extending Lyapunov’s direct method from the fuzzy ordinary case to the fuzzy fractional case. As an application, several examples are presented to illustrate the proposed stability result.

scholarly journals Existence and Integral Representation of Scalar Riemann-Liouville Fractional Differential Equations with Delays and Impulses

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 607
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Lower Limit ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
Integral Representations ◽  
Time Integral ◽  
Fractional Differential ◽  
Set Up ◽  
Banach Principle

Nonlinear scalar Riemann-Liouville fractional differential equations with a constant delay and impulses are studied and initial conditions and impulsive conditions are set up in an appropriate way. The definitions of both conditions depend significantly on the type of fractional derivative and the presence of the delay in the equation. We study the case of a fixed lower limit of the fractional derivative and the case of a changeable lower limit at each impulsive time. Integral representations of the solutions in all considered cases are obtained. Existence results on finite time intervals are proved using the Banach principle.

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