scholarly journals On definition of solution of initial value problem for fractional differential equation of variable order

2021 ◽  
Vol 6 (7) ◽  
pp. 6845-6867
Author(s):  
Shuqin Zhang ◽  
◽  
Jie Wang ◽  
Lei Hu ◽  
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equation ◽  
Initial Value Problem ◽  
Variable Order ◽  
Initial Value ◽  
Fractional Differential ◽  
Definition Of

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.

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.

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.

Operational method for solving multi-term fractional differential equations with the generalized fractional derivatives

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Myong-Ha Kim ◽  
Guk-Chol Ri ◽  
Hyong-Chol O
Keyword(s):  
Differential Equation ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Fractional Derivatives ◽  
Operational Calculus ◽  
Operational Method ◽  
Initial Value ◽  
Constant Coefficients ◽  
Generalized Fractional Derivatives ◽  
Fractional Differential

AbstractThis paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski’s type. We prove that the initial value problem has the solution if and only if some initial values are zero.

scholarly journals Initial Value Problem for Stochastic Hyprid Hadamard Fractional Differential Equation

2019 ◽  
Vol 16 ◽  
pp. 8288-8296
Author(s):  
Mahmoud Mohammed Mostafa El-Borai ◽  
Wagdy G. El-sayed ◽  
A. A. Badr ◽  
Ahmed Tarek Sayed
Keyword(s):  
Differential Equation ◽  
Fractional Calculus ◽  
Initial Value Problem ◽  
Stochastic Analysis ◽  
Fractional Integral ◽  
Existence Of Solutions ◽  
Initial Value ◽  
Analysis Techniques ◽  
Fractional Differential ◽  
Hadamard Fractional Integral

In this paper, we discuss the existence of solutions for a stochastic initial value problem of Hyprid fractional dierential equations of Hadamard type given by                            where HD is the Hadamard fractional derivative, and is the Hadamard fractional integral and be such that are investigated. The fractional calculus and stochastic analysis techniques are used to obtain the required results. 

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