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 On the Distributed Order Fractional Multi-Strain Tuberculosis Model: a Numerical Study

2020 ◽  
Vol 8 (1) ◽  
pp. 175-186
Author(s):  
Nasser Sweilam ◽  
S. M. AL-Mekhlafi ◽  
A. O. Albalawi
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Difference Method ◽  
Numerical Study ◽  
Proposed Model ◽  
Fractional Differential ◽  
Distributed Order ◽  
Nonstandard Finite Difference

In this paper, a novel mathematical distributed order fractional model of multistrain Tuberculosis is presented. The proposed model is governed by a system of distributed order fractional differential equations, where the distributed order fractional derivative is defined in the sense of the Grünwald-Letinkov definition. A nonstandard finite difference method is proposed to study the resulting system. The stability analysis of the proposed model is discussed. Numerical simulations show that the nonstandard finite difference method can be applied to solve such distributed order fractional differential equations simply and eectively.

An Efficient Method for Numerical Solutions of Distributed-Order Fractional Differential Equations

2018 ◽  
Vol 13 (11) ◽  
Author(s):  
N. Jibenja ◽  
B. Yuttanan ◽  
M. Razzaghi
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Approximation Technique ◽  
Algebraic Equations ◽  
Block Pulse Functions ◽  
Fractional Differential ◽  
Taylor Polynomials ◽  
First Time ◽  
Distributed Order

This paper presents an efficient numerical method for solving the distributed fractional differential equations (FDEs). The suggested framework is based on a hybrid of block-pulse functions and Taylor polynomials. For the first time, the Riemann–Liouville fractional integral operator for the hybrid of block-pulse functions and Taylor polynomials has been derived directly and without any approximations. By taking into account the property of this operator, the problem under consideration is converted into a system of algebraic equations. The present method can be applied to both linear and nonlinear distributed FDEs. Easy implementation, simple operations, and accurate solutions are the essential features of the proposed hybrid functions. Illustrative examples are examined to demonstrate the performance and effectiveness of the developed approximation technique, and a comparison is made with the existing results.

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 Stability Analysis of Distributed Order Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
H. Saberi Najafi ◽  
A. Refahi Sheikhani ◽  
A. Ansari
Keyword(s):  
Stability Analysis ◽  
Characteristic Function ◽  
Differential Equations ◽  
Density Function ◽  
Robust Stability ◽  
Fractional Differential Equations ◽  
Stability Condition ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

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.

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