scholarly journals On a Hilfer Type Fractional Differential Equation with Nonlinear Right-Hand Side

2021 ◽  
Vol 103 (3) ◽  
pp. 140-155
Author(s):  
T. K. Yuldashev ◽  
◽  
B. J. Kadirkulov ◽  
A. R. Marakhimov ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Initial Value Problem ◽  
Integral Transformation ◽  
Numerical Realization ◽  
Initial Value ◽  
Volterra Type ◽  
Fractional Integral Equation ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

In this article we consider the questions of one-valued solvability and numerical realization of initial value problem for a nonlinear Hilfer type fractional differential equation with maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation. The theorem of existence and uniqueness of the solution of given initial value problem in the segment under consideration is proved. For numerical realization of solution the generalized Jacobi–Galerkin method is applied. Illustrative examples are provided.

scholarly journals On Solvability of an Initial Value Problem for Hilfer type Fractional Differential Equation with Nonlinear Maxima

2020 ◽  
pp. 48-65
Author(s):  
T.K. Yuldashev ◽  
B.J. Kadirkulov
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Initial Value Problem ◽  
Integral Transformation ◽  
Initial Value ◽  
Volterra Type ◽  
Fractional Integral Equation ◽  
Uniqueness Of The Solution ◽  
Fractional Differential ◽  
The Stability

In this article we consider the questions of one-valued solvability of initial value problem for a nonlinear Hilfer type fractional differential equation with nonlinear maxima. By the aid of uncomplicated integral transformation based on Dirichlet formula, this initial value problem is reduced to the nonlinear Volterra type fractional integral equation with nonlinear maxima. It is proved the theorem of existence and uniqueness of the solution of given initial value problem in an interval under consideration. It is proved also the stability of the desired solution with respect to given parameter.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

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.

Controllability of Abstract Systems of Fractional Order

2015 ◽  
Vol 18 (6) ◽  
Author(s):  
Therese Mur ◽  
Hernán R. Henríquez
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Integral Equation ◽  
Control Systems ◽  
Banach Spaces ◽  
Fractional Differential Equation ◽  
Fractional Integral ◽  
First Order ◽  
Fractional Integral Equation ◽  
Fractional Differential

AbstractIn this paper we are concerned with the controllability of control systems governed by a fractional differential equation in Banach spaces. Using the properties of the Mittag-Leffler function we generalize to these systems a result of Korobov and Rabakh, which was established for first order systems. We apply our results to study the controllability of a system modeled by a fractional integral equation in a Hilbert space.

scholarly journals Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Changyou Wang ◽  
Haiqiang Zhang ◽  
Shu Wang
Keyword(s):  
Differential Equation ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Caputo Derivative ◽  
Nonlinear Term ◽  
Upper And Lower Solutions ◽  
Initial Value ◽  
Nonlinear Fractional Differential Equation ◽  
Control Functions ◽  
Fractional Differential

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained.

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