scholarly journals Numerical Solutions of Conformable Fractional Differential Equations by Taylor and Finite Difference Methods

2019 ◽  
pp. 850-863
Author(s):  
Şuayip TOPRAKSEVEN
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Finite Difference Methods ◽  
Fractional Differential ◽  
Difference Methods

FINITE DIFFERENCE METHODS FOR FRACTIONAL DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (04) ◽  
pp. 1230014 ◽  
Author(s):  
CHANGPIN LI ◽  
FANHAI ZENG
Keyword(s):  
Numerical Methods ◽  
Finite Difference ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Finite Difference Methods ◽  
Kinetic Equations ◽  
Fourier Method ◽  
Time Space ◽  
Fractional Differential ◽  
Difference Methods

In this review paper, the finite difference methods (FDMs) for the fractional differential equations are displayed. The considered equations mainly include the fractional kinetic equations of diffusion or dispersion with time, space and time-space derivatives. In some way, these numerical methods have similar form as the case for classical equations, some of which can be seen as the generalizations of the FDMs for the typical differential equations. And the classical tools, such as the von Neumann analysis method, the energy method and the Fourier method are extended to numerical methods for fractional differential equations accordingly. At the same time, the techniques for improving the accuracy and reducing the computation and storage are also introduced.

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>

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