scholarly journals Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales

2016 ◽  
Vol 2016 ◽  
pp. 1-21 ◽  
Author(s):  
Yanning Wang ◽  
Jianwen Zhou ◽  
Yongkun Li
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Differential Equation ◽  
Time Scales ◽  
Fractional Derivatives ◽  
Point Theory ◽  
Existence Results ◽  
Recent Approach ◽  
Equation Boundary ◽  
Fractional Differential

Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap-Laplacian conformable fractional differential equation boundary value problem on time scaleT:  Tα(Tαup-2Tα(u))(t)=∇F(σ(t),u(σ(t))),Δ-a.e.  t∈a,bTκ2,u(a)-u(b)=0,Tα(u)(a)-Tα(u)(b)=0,whereTα(u)(t)denotes the conformable fractional derivative ofuof orderαatt,σis the forward jump operator,a,b∈T,  0<a<b,  p>1, andF:[0,T]T×RN→R. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.

scholarly journals Positive Solutions for a Higher-Order Semipositone Nonlocal Fractional Differential Equation with Singularities on Both Time and Space Variable

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
Author(s):  
Kemei Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Differential Equation ◽  
Point Theorem ◽  
Fractional Derivatives ◽  
Space Variable ◽  
Higher Order ◽  
Existence Results ◽  
Fractional Differential

In this paper, we consider the following higher-order semipositone nonlocal Riemann-Liouville fractional differential equation D0+αx(t)+f(t,x(t),D0+βx(t))+e(t)=0,  0<t<1,D0+βx(0)=D0+β+1x(0)=⋯=D0+n+β-2x(0)=0, and D0+βx(1)=∑i=1m-2ηiD0+βx(ξi), where D0+α and D0+β are the standard Riemann-Liouville fractional derivatives. The existence results of positive solution are given by Guo-krasnosel’skii fixed point theorem and Schauder’s fixed point theorem.

scholarly journals Application of topological degree method in quantitative behavior of fractional differential equations

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 421-432
Author(s):  
Rahman ur ◽  
Saeed Ahmad ◽  
Fazal Haq
Keyword(s):  
Differential Equation ◽  
Existence And Uniqueness ◽  
Topological Degree ◽  
Order Differential Equation ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Existence Results ◽  
Research Article ◽  
Fractional Differential ◽  
Topological Degree Method

In the present manuscript we incorporate fractional order Caputo derivative to study a class of non-integer order differential equation. For existence and uniqueness of solution some results from fixed point theory is on our disposal. The method used for exploring these existence results is topological degree method and some auxiliary conditions are developed for stability analysis. For further elaboration an illustrative example is provided in the last part of the research article.

scholarly journals Mathematical modelling of the mass-spring-damper system - A fractional calculus approach

2012 ◽  
Vol 22 (5) ◽  
pp. 5-11 ◽  
Author(s):  
José Francisco Gómez Aguilar ◽  
Juan Rosales García ◽  
Jesus Bernal Alvarado ◽  
Manuel Guía
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Mathematical Modelling ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Time Derivative ◽  
System A ◽  
Fractional Differential ◽  
Mass Spring ◽  
Derivatives Of

In this paper the fractional differential equation for the mass-spring-damper system in terms of the fractional time derivatives of the Caputo type is considered. In order to be consistent with the physical equation, a new parameter is introduced. This parameter char­acterizes the existence of fractional components in the system. A relation between the fractional order time derivative and the new parameter is found. Different particular cases are analyzed

Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives

2020 ◽  
Vol 23 (1) ◽  
pp. 103-125 ◽  
Author(s):  
Latif A-M. Hanna ◽  
Maryam Al-Kandari ◽  
Yuri Luchko
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Fractional Derivatives ◽  
Fractional Integrals ◽  
Operational Method ◽  
Initial Value ◽  
Right Hand ◽  
Fractional Differential ◽  
Left And Right ◽  
Fractional Integrals And Derivatives

AbstractIn this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the four-parameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

Initialization Issues of the Caputo Fractional Derivative

2005 ◽  
Author(s):  
B. N. Narahari Achar ◽  
Carl F. Lorenzo ◽  
Tom T. Hartley
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Past History ◽  
The Past ◽  
History Of ◽  
Fractional Differential ◽  
Generally True

The importance of proper initialization in taking into account the history of a system whose time evolution is governed by a differential equation of fractional order, has been established by Lorenzo and Hartley, who also gave the method of properly incorporating the effect of the past (history) by means of an initialization function for the Riemann-Liouville and the Grunwald formulations of fractional calculus. The present work addresses this issue for the Caputo fractional derivative and cautions that the commonly held belief that the Caputo formulation of fractional derivatives properly accounts for the initialization effects is not generally true when applied to the solution of fractional differential equations.

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