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.

scholarly journals Wavelet Collocation Method for Solving Multiorder Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
M. H. Heydari ◽  
M. R. Hooshmandasl ◽  
F. M. Maalek Ghaini ◽  
F. Mohammadi
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Fractional Differential Equations ◽  
Initial Conditions ◽  
General Procedure ◽  
Wavelet Expansion ◽  
Operational Matrices ◽  
Chebyshev Wavelets ◽  
Block Pulse Functions ◽  
Fractional Differential

The operational matrices of fractional-order integration for the Legendre and Chebyshev wavelets are derived. Block pulse functions and collocation method are employed to derive a general procedure for forming these matrices for both the Legendre and the Chebyshev wavelets. Then numerical methods based on wavelet expansion and these operational matrices are proposed. In this proposed method, by a change of variables, the multiorder fractional differential equations (MOFDEs) with nonhomogeneous initial conditions are transformed to the MOFDEs with homogeneous initial conditions to obtain suitable numerical solution of these problems. Numerical examples are provided to demonstrate the applicability and simplicity of the numerical scheme based on the Legendre and Chebyshev wavelets.

scholarly journals Existence of Positive Solutions for a Class of Delay Fractional Differential Equations with Generalization to N-Term

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Banach Fixed Point Theorem ◽  
Existence Of Positive Solutions ◽  
Fractional Differential

We established the existence of a positive solution of nonlinear fractional differential equationsL(D)[x(t)−x(0)]=f(t,xt),t∈(0,b]with finite delayx(t)=ω(t),t∈[−τ,0], wherelimt→0f(t,xt)=+∞, that is,fis singular att=0andxt∈C([−τ,0],ℝ≥0). The operator ofL(D)involves the Riemann-Liouville fractional derivatives. In this problem, the initial conditions with fractional order and some relations among them were considered. The analysis rely on the alternative of the Leray-Schauder fixed point theorem, the Banach fixed point theorem, and the Arzela-Ascoli theorem in a cone.

scholarly journals Construction a distributed order smoking model and its nonstandard finite difference discretization

2021 ◽  
Vol 7 (3) ◽  
pp. 4636-4654
Author(s):  
Mehmet Kocabiyik ◽  
◽  
Mevlüde Yakit Ongun ◽  
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Health Problems ◽  
Potential Health ◽  
Fractional Differential ◽  
Distributed Order ◽  
Nonstandard Finite Difference

<abstract><p>Smoking is currently one of the most important health problems in the world and increases the risk of developing diseases. For these reasons, it is important to determine the effects of smoking on humans. In this paper, we discuss a new system of distributed order fractional differential equations of the smoking model. With the use of distributed order fractional differential equations, it is possible to solve both ordinary and fractional-order equations. We can make these solutions with the density function included in the definition of the distributed order fractional differential equation. We construct the Nonstandard Finite Difference (NSFD) schemes to obtain numerical solutions of this model. Positivity solutions are preserved under positive initial conditions with this discretization method. Also, since NSFD schemes can preserve all the properties of the continuous models for any discretization parameter, the method is successful in dynamical consistency. We use the Schur-Cohn criteria for stability analysis of the discretized model. With the solutions obtained, we can understand the effects of smoking on people in a short time, even in different situations. Thus, by knowing these effects in advance, potential health problems can be predicted, and life risks can be minimized according to these predictions.</p></abstract>

scholarly journals Oscillation of impulsive conformable fractional differential equations

2016 ◽  
Vol 14 (1) ◽  
pp. 497-508 ◽  
Author(s):  
Jessada Tariboon ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Equivalence Transformation ◽  
Fractional Differential ◽  
Riccati Techniques

AbstractIn this paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form$$\left\{ \begin{array}{l} {t_k}{D^\alpha }\left( {p\left( t \right)\left[ {{t_k}{D^\alpha }x\left( t \right) + r\left( t \right)x\left( t \right)} \right]} \right) + q\left( t \right)x\left( t \right) = 0,\quad t \ge {t_0},\;t \ne {t_k},\\ x\left( {t_k^ + } \right) = {a_k}x(t_k^ - ),\quad {t_k}{D^\alpha }x\left( {t_k^ + } \right) = {b_{k\;{t_{k - 1}}}}{D^\alpha }x(t_k^ - ),\quad \;k = 1,2, \ldots. \end{array} \right.$$Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques.

Improvements in a method for solving fractional integral equations with some links with fractional differential equations

2018 ◽  
Vol 21 (1) ◽  
pp. 174-189 ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Differential Equations ◽  
Integral Equations ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Source Term ◽  
Initial Conditions ◽  
Standard Procedure ◽  
Relaxation Equation ◽  
Constant Coefficients ◽  
Fractional Differential

Abstract In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure. In this procedure, there were two major obstacles that did not allow to obtain a full result. These were the assumptions over the smoothness of the source term and the assumption about the rationality of the orders. So, one of the main topics of this document is to describe a modification of the procedure presented in [4], when the source term is not smooth enough to differentiate the required amount of times. Furthermore, we will also study the fractional integral equations with non-rational orders by a limit process of fractional integral equations with rational orders. Finally, we will connect the previous material with some fractional differential equations with Caputo derivatives described in [7]. For instance, we will deal with the fractional oscillation equation, the fractional relaxation equation and, specially, its particular case of the Basset problem. We also expose how to compute these solutions for the Riemann-Liouville case.

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