scholarly journals Oscillation for a Class of Fractional Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Zhenlai Han ◽  
Yige Zhao ◽  
Ying Sun ◽  
Chao Zhang
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Real Number ◽  
Fractional Differential Equation ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Nonlinear Fractional Differential Equation ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Fractional Differential

We consider the oscillation for a class of fractional differential equation[r(t)g(D-αy)(t)]'-p(t)f∫t∞‍(s-t)-αy(s)ds=0,fort>0,where0<α<1is a real number andD-αyis the Liouville right-sided fractional derivative of orderαofy. By generalized Riccati transformation technique, oscillation criteria for a class of nonlinear fractional differential equation are obtained.

scholarly journals Oscillation criteria for nonlinear fractional differential equation with damping term

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 119-128 ◽  
Author(s):  
Mustafa Bayram ◽  
Hakan Adiguzel ◽  
Aydin Secer
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Riccati Transformation ◽  
Damping Term ◽  
Nonlinear Fractional Differential Equation ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Liouville Derivative ◽  
Fractional Differential ◽  
Oscillation Of Solutions

AbstractIn this paper, we study the oscillation of solutions to a non-linear fractional differential equation with damping term. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By using a variable transformation, a generalized Riccati transformation, inequalities, and integration average techniquewe establish new oscillation criteria for the fractional differential equation. Several illustrative examples are also given.

scholarly journals Oscillation Behavior for a Class of Differential Equation with Fractional-Order Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shouxian Xiang ◽  
Zhenlai Han ◽  
Ping Zhao ◽  
Ying Sun
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Generalized Riccati Transformation ◽  
Fractional Differential ◽  
Oscillation Theorems ◽  
Positive Integers

By using a generalized Riccati transformation technique and an inequality, we establish some oscillation theorems for the fractional differential equation[atpt+qtD-αxt)γ′ − b(t)f∫t∞‍(s-t)-αx(s)ds = 0, fort⩾t0>0, whereD-αxis the Liouville right-sided fractional derivative of orderα∈(0,1)ofxandγis a quotient of odd positive integers. The results in this paper extend and improve the results given in the literatures (Chen, 2012).

scholarly journals Oscillation for a Class of Right Fractional Differential Equations on the Right Half Line with Damping

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Hui Liu ◽  
Run Xu
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Half Line ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Fractional Differential ◽  
The Right

In this paper, we discuss a class of fractional differential equations of the form D-α+1y(t)·D-αy(t)-p(t)f(D-αy(t))+q(t)h∫t∞(s-t)-αy(s)ds=0.D-αy(t) is the Liouville right-sided fractional derivative of order α∈(0,1). We obtain some oscillation criteria for the equation by employing a generalized Riccati transformation technique. Some examples are given to illustrate the significance of our results.

scholarly journals Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jun-Rui Yue ◽  
Jian-Ping Sun ◽  
Shuqin Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Nonlinear Fractional Differential Equation ◽  
Krasnoselskii Fixed Point Theorem ◽  
Fractional Differential

We consider the following boundary value problem of nonlinear fractional differential equation:(CD0+αu)(t)=f(t,u(t)),  t∈[0,1],  u(0)=0,   u′(0)+u′′(0)=0,  u′(1)+u′′(1)=0, whereα∈(2,3]is a real number, CD0+αdenotes the standard Caputo fractional derivative, andf:[0,1]×[0,+∞)→[0,+∞)is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.

scholarly journals Positive Solution for the Nonlinear Hadamard Type Fractional Differential Equation withp-Laplacian

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Ya-ling Li ◽  
Shi-you Lin
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Continuous Function ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fixed Point Theorems ◽  
Laplacian Operator ◽  
Nonlinear Fractional Differential Equation ◽  
Fractional Differential

We study the following nonlinear fractional differential equation involving thep-Laplacian operatorDβφpDαut=ft,ut,1<t<e,u1=u′1=u′e=0,Dαu1=Dαue=0, where the continuous functionf:1,e×0,+∞→[0,+∞),2<α≤3,1<β≤2.Dαdenotes the standard Hadamard fractional derivative of the orderα, the constantp>1, and thep-Laplacian operatorφps=sp-2s. We show some results about the existence and the uniqueness of the positive solution by using fixed point theorems and the properties of Green's function and thep-Laplacian operator.

scholarly journals Exact Solutions of Bernoulli and Logistic Fractional Differential Equations with Power Law Coefficients

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2231
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Power Law ◽  
Dynamic Processes ◽  
Nonlinear Fractional Differential Equation ◽  
First Order ◽  
Fractional Differential ◽  
Special Case

In this paper, we consider a nonlinear fractional differential equation. This equation takes the form of the Bernoulli differential equation, where we use the Caputo fractional derivative of non-integer order instead of the first-order derivative. The paper proposes an exact solution for this equation, in which coefficients are power law functions. We also give conditions for the existence of the exact solution for this non-linear fractional differential equation. The exact solution of the fractional logistic differential equation with power law coefficients is also proposed as a special case of the proposed solution for the Bernoulli fractional differential equation. Some applications of the Bernoulli fractional differential equation to describe dynamic processes with power law memory in physics and economics are suggested.

Lyapunov-type inequalities for nonlinear fractional differential equation with Hilfer fractional derivative under multi-point boundary conditions

2018 ◽  
Vol 21 (3) ◽  
pp. 833-843 ◽  
Author(s):  
Youyu Wang ◽  
Qichao Wang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Point Boundary ◽  
Hilfer Fractional Derivative ◽  
Nonlinear Fractional Differential Equation ◽  
Early Results ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

Abstract In this work, we establish Lyapunov-type inequalities for the fractional boundary value problems with Hilfer fractional derivative under multi-point boundary conditions, the results are new and generalize and improve some early results in the literature.

scholarly journals The Oscillatory of Linear Conformable Fractional Differential Equations of Kamenev Type

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Hui Liu ◽  
Run Xu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Average Method ◽  
Riccati Transformation ◽  
Transformation Technique ◽  
Oscillation Criteria ◽  
Generalized Riccati Transformation ◽  
Fractional Differential ◽  
Integral Average

In this paper, the oscillatory of the Kamenev-type linear conformable fractional differential equations in the form of ptyα+1tα+yα+1t+qtyt=0 is studied, where t≥t0 and 0<α≤1. By employing a generalized Riccati transformation technique and integral average method, we obtain some oscillation criteria for the equation. We also give some examples to illustrate the significance of our results.

Sign in / Sign up

Export Citation Format

Share Document