scholarly journals Fractional integrodifferential equations with nonlocal conditions and generalized Hilfer fractional derivative

2019 ◽  
Vol 11 (4) ◽  
pp. 151-170
Author(s):  
Hanan Abdulrahman Wahash ◽  
Mohammed Salem Abdo ◽  
Satish K Panchal
Keyword(s):  
Fractional Derivative ◽  
Nonlocal Conditions ◽  
Integrodifferential Equations ◽  
Hilfer Fractional Derivative

scholarly journals FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS AND Ψ–HILFER FRACTIONAL DERIVATIVE

2019 ◽  
Vol 24 (4) ◽  
pp. 564-584 ◽  
Author(s):  
Mohammed S. Abdo ◽  
Satish K. Panchal ◽  
Hussien Shafei Hussien
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Nonlocal Conditions ◽  
Banach Fixed Point Theorem ◽  
Cauchy Type ◽  
Uniqueness Of Solutions ◽  
Hilfer Fractional Derivative ◽  
Krasnoselskii’S Fixed Point Theorem

Considering a fractional integro-differential equation with nonlocal conditions involving a general form of Hilfer fractional derivative with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation, we also prove the existence, uniqueness of solutions and Ulam-Hyers stability of this problem by employing a variety of tools of fractional calculus including Banach fixed point theorem and Krasnoselskii's fixed point theorem. An example is provided to illustrate our main results.

scholarly journals Integral transforms of the κ-Hilfer fractional derivative

2021 ◽  
Author(s):  
Felix Costa ◽  
Junior Cesar Alves Soares ◽  
Stefânia Jarosz
Keyword(s):  
Fractional Derivative ◽  
Gamma Function ◽  
Fundamental Theorem ◽  
Beta Function ◽  
Integral Transforms ◽  
Fractional Operators ◽  
Riesz Fractional Derivative ◽  
Hilfer Fractional Derivative ◽  
Fundamental Theorem Of Calculus

In this paper, some important properties concerning the κ-Hilfer fractional derivative are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus. Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators.

scholarly journals Stability Results for Implicit Fractional Pantograph Differential Equations via ϕ-Hilfer Fractional Derivative with a Nonlocal Riemann-Liouville Fractional Integral Condition

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 94 ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Kamal Shah ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet ◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Integral Condition ◽  
Hilfer Fractional Derivative ◽  
Fractional Differential ◽  
Generalized Fractional Derivative ◽  
Stability Results

This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.

scholarly journals Fractional Stochastic Differential Equations with Hilfer Fractional Derivative: Poisson Jumps and Optimal Control

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Fathalla A. Rihan ◽  
Chinnathambi Rajivganthi ◽  
Palanisamy Muthukumar
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Mild Solutions ◽  
Poisson Jumps ◽  
Hilfer Fractional Derivative ◽  
Analysis Techniques ◽  
Mild Conditions ◽  
Fractional Stochastic Differential Equations ◽  
Existence Of Optimal Control

In this work, we consider a class of fractional stochastic differential system with Hilfer fractional derivative and Poisson jumps in Hilbert space. We study the existence and uniqueness of mild solutions of such a class of fractional stochastic system, using successive approximation theory, stochastic analysis techniques, and fractional calculus. Further, we study the existence of optimal control pairs for the system, using general mild conditions of cost functional. Finally, we provide an example to illustrate the obtained results.

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