On General Solution for Fractional Differential Equations with Not Instantaneous Impulses

2017 ◽  
Vol 151 (1-4) ◽  
pp. 355-369
Author(s):  
Xianmin Zhang ◽  
Xianzhen Zhang ◽  
Hui Cao
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Differential

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

scholarly journals General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative

2017 ◽  
Vol 6 (2) ◽  
pp. 49 ◽  
Author(s):  
Zainab Ayati ◽  
Jafar Biaar ◽  
Mousa Ilei
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Ordinary Differential Equations ◽  
Fractional Differential Equations ◽  
Riccati Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Conformable Fractional Derivative ◽  
Numerical Examples ◽  
Fractional Differential

This paper is aimed to develop two well-known nonlinear ordinary differential equations, Bernoulli and Riccati equations to fractional form. General solution to fractional differential equations are detected, based on conformable fractional derivative. For each equation, numerical examples are presented to illustrate the proposed approach.  

scholarly journals On the fractional differential equations with not instantaneous impulses

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 676-684 ◽  
Author(s):  
Xianmin Zhang ◽  
Praveen Agarwal ◽  
Zuohua Liu ◽  
Xianzhen Zhang ◽  
Wenbin Ding ◽  
...  
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Differential Equations ◽  
Fractional Order Systems ◽  
Fractional Differential

AbstractBased on some previous works, an equivalent equations is obtained for the differential equations of fractional-orderq ∈(1, 2) with non-instantaneous impulses, which shows that there exists the general solution for this impulsive fractional-order systems. Next, an example is used to illustrate the conclusion.

scholarly journals General solution of second order fractional differential equations

2018 ◽  
Vol 7 (2) ◽  
pp. 56 ◽  
Author(s):  
Mousa Ilie ◽  
Jafar Biazar ◽  
Zainab Ayati
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Second Order ◽  
Conformable Fractional Derivative ◽  
High Importance ◽  
Particular Solution ◽  
Fractional Differential ◽  
Linear Fractional Differential Equations ◽  
General Method

Fractional differential equations are often seeming perplexing to solve. Therefore, finding comprehensive methods for solving them sounds of high importance. In this paper, a general method for solving second order fractional differential equations has been presented based on conformable fractional derivative. This method realizes on determining a general solution of homogeneous and a particular solution of a second order linear fractional differential equations. Furthermore, a general solution has been developed for fractional Euler’s equation. For more explanation of each part, some examples have been solved. 

scholarly journals Analysis of fractional duffing oscillator

2020 ◽  
Vol 66 (2 Mar-Apr) ◽  
pp. 187 ◽  
Author(s):  
S.C. Eze
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Fractional Differential Equations ◽  
Analytical Method ◽  
Duffing Oscillator ◽  
Laplace Method ◽  
Nonlinear Fractional Differential Equations ◽  
Non Linear ◽  
Fractional Differential

In this contribution, a simple analytical method (which is an elegant combination of a well known methods; perturbation method and Laplace method) for solving non-linear and non-homogeneous fractional differential equations is pro- posed. In particular, the proposed method was used to analysed the fractional Duffing oscillator.The technique employed in this method can be used to analyse other nonlinear fractional differential equations, and can also be extended to non- linear partial fractional differential equations.The performance of this method is reliable, effective and gives more general solution.

scholarly journals Fractional Fourier Series with Separation of Variables Technique and Its Application on Fractional differential Equations

2021 ◽  
Vol 20 ◽  
pp. 461-469
Author(s):  
Ahmed Bouchenak ◽  
Khalil Roshdi ◽  
Alhorani Mohammed
Keyword(s):  
Fourier Series ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Approximation ◽  
Separation Of Variables ◽  
Heat Equations ◽  
Fourier Series Approximation ◽  
Fractional Differential ◽  
A New Technique

When using some classical methods, such us separation of variables; it is impossible to find a general solution for some differential equations. Therefore, we suggest adding conformable fractional Fourier series to get a new technique to solve fractional Benjamin Bana Mahony and Heat Equations. Furtheremore, we give new numerical approximation for functions using mathematica coding called conformable fractional Fourier series approximation

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

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