Continuous dependence of uncertain fractional differential equations with Caputo’s derivative

2020 ◽  
pp. 1-10
Author(s):  
Ziqiang Lu ◽  
Yuanguo Zhu ◽  
Jiayu Shen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Liu Process ◽  
Uncertain Dynamic Systems ◽  
Caputo’S Derivative ◽  
Fractional Differential ◽  
Lipschitz Conditions

Uncertain fractional differential equation driven by Liu process plays an important role in describing uncertain dynamic systems. This paper investigates the continuous dependence of solution on the parameters and initial values, respectively, for uncertain fractional differential equations involving the Caputo fractional derivative in measure sense. Several continuous dependence theorems are obtained based on uncertainty theory by employing the generalized Gronwall inequality, in which the coefficients of uncertain fractional differential equation are required to satisfy the Lipschitz conditions. Several illustrative examples are provided to verify the validity of the obtained results.

scholarly journals Uniqueness of Iterative Positive Solution to Nonlinear Fractional Differential Equations with Negatively Perturbed Term

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Jingjing Tan ◽  
Meixia Li ◽  
Aixia Pan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Convergence Rate ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Iterative Scheme ◽  
Approximation Error ◽  
Fractional Differential

We prove that there are unique positive solutions for a new kind of fractional differential equation with a negatively perturbed term boundary value problem. Our methods rely on an iterative algorithm which requires constructing an iterative scheme to approximate the solution. This allows us to calculate the estimation of the convergence rate and the approximation error.

scholarly journals Ulam-Hyers Stability for Cauchy Fractional Differential Equation in the Unit Disk

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Unit Disk ◽  
Fractional Differential Equations ◽  
Fractional Operators ◽  
Owa Operators ◽  
Non Linear ◽  
Fractional Differential

We prove the Ulam-Hyers stability of Cauchy fractional differential equations in the unit disk for the linear and non-linear cases. The fractional operators are taken in sense of Srivastava-Owa operators.

scholarly journals SEPARABLE LOCAL FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650021 ◽  
Author(s):  
KIRAN M. KOLWANKAR
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Local Scaling ◽  
Fractal Sets ◽  
Method Of Solution ◽  
Fractional Differential ◽  
Simple Equations

The concept of local fractional derivative was introduced in order to be able to study the local scaling behavior of functions. However it has turned out to be much more useful. It was found that simple equations involving these operators naturally incorporate the fractal sets into the equations. Here, the scope of these equations has been extended further by considering different possibilities for the known function. We have also studied a separable local fractional differential equation along with its method of solution.

scholarly journals Complex Transforms for Systems of Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solutions ◽  
Fractional Differential

We provide a complex transform that maps the complex fractional differential equation into a system of fractional differential equations. The homogeneous and nonhomogeneous cases for equivalence equations are discussed and also nonequivalence equations are studied. Moreover, the existence and uniqueness of solutions are established and applications are illustrated.

On fractional lyapunov exponent for solutions of linear fractional differential equations

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Nguyen Cong ◽  
Doan Son ◽  
Hoang Tuan
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Lyapunov Spectrum ◽  
Bounded Linear ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations

AbstractOur aim in this paper is to investigate the asymptotic behavior of solutions of linear fractional differential equations. First, we show that the classical Lyapunov exponent of an arbitrary nontrivial solution of a bounded linear fractional differential equation is always nonnegative. Next, using the Mittag-Leffler function, we introduce an adequate notion of fractional Lyapunov exponent for an arbitrary function. We show that for a linear fractional differential equation, the fractional Lyapunov spectrum which consists of all possible fractional Lyapunov exponents of its solutions provides a good description of asymptotic behavior of this equation. Consequently, the stability of a linear fractional differential equation can be characterized by its fractional Lyapunov spectrum. Finally, to illustrate the theoretical results we compute explicitly the fractional Lyapunov exponent of an arbitrary solution of a planar time-invariant linear fractional differential equation.

scholarly journals The fractional residual method for solving the local fractional differential equations

2020 ◽  
Vol 24 (4) ◽  
pp. 2535-2542
Author(s):  
Yong-Ju Yang
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Initial Solution ◽  
New Method ◽  
Residual Method ◽  
Fractional Differential ◽  
Efficiency And Reliability

This paper proposes a new method to solve local fractional differential equation. The method divides the studied equation into a system, where the initial solution is obtained from a residual equation. The new method is therefore named as the fractional residual method. Examples are given to elucidate its efficiency and reliability.

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.

COMPARISON THEOREMS FOR CAPUTO–HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2019 ◽  
Vol 27 (03) ◽  
pp. 1950036 ◽  
Author(s):  
LI MA
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Fractional Derivatives ◽  
Comparison Theorems ◽  
Hadamard Fractional Differential Equations ◽  
Fractional Differential ◽  
The One ◽  
Lipschitz Conditions ◽  
Theoretical Results

The main purpose of this paper is to investigate the comparison theorems for fractional differential equations involving Caputo–Hadamard fractional derivatives. First, we indicate the continuous dependence on parameters of solutions for Caputo–Hadamard fractional differential equations (C-HFDEs). Then, the first and second comparison theorems for C-HFDEs are proposed and proved, respectively. In addition, we establish the generalized comparisons for C-HFDEs under the one-side Lipschitz conditions. At last, the corresponding examples are also provided to verify the theoretical results.

Well-Posedness of Fractional Differential Equations

2017 ◽  
Author(s):  
Changpin Li ◽  
Li Ma
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Well Posedness ◽  
Fractional Differential ◽  
Well Posed

Generally speaking, definite conditions of fractional differential equations with Riemann-Liouville, Riesz or Hadamard fractional derivatives are quite different from those of classic differential equations. In this paper, we propose the well-posed conditions for fractional differential equation involving right Riemann-Liouville, Riesz and Hadamard fractional derivatives.

scholarly journals Asymptotically Linear Solutions for Some Linear Fractional Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-8 ◽  
Author(s):  
Dumitru Baleanu ◽  
Octavian G. Mustafa ◽  
Ravi P. Agarwal
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Asymptotically Linear ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations ◽  
Functional Coefficient

We establish here that under some simple restrictions on the functional coefficienta(t)the fractional differential equationD0tα[tx′−x+x(0)]+a(t)x=0,  t>0, has a solution expressible asct+d+o(1)fort→+∞, whereD0tαdesignates the Riemann-Liouville derivative of orderα∈(0,1)andc,d∈ℝ.

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