Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

2020 ◽  
Vol 37 (1) ◽  
pp. 707-731 ◽  
Author(s):  
Khosrow Maleknejad ◽  
Jalil Rashidinia ◽  
Tahereh Eftekhari
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Time Domain ◽  
Numerical Solutions ◽  
Legendre Wavelets ◽  
Fractional Differential ◽  
The Time Domain ◽  
Distributed Order

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>

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.

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 Memory effects on the proliferative function in the cycle-specific of chemotherapy

2021 ◽  
Author(s):  
Najma Ahmed ◽  
Dumitru Vieru ◽  
Fiazud Din Zaman
Keyword(s):  
Mathematical Model ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Classical Model ◽  
Cell Mass ◽  
Time Derivative ◽  
Fractional Model ◽  
Fractional Differential ◽  
Mathcad Software

A generalized mathematical model of the breast and ovarian cancer is developed by considering the fractional differential equations with Caputo time-fractional derivatives. The use of the fractional model shows that the time-evolution of the proliferating cell mass, the quiescent cell mass, and the proliferative function are significantly influenced by their history. Even if the classical model, based on the derivative of integer order has been studied in many papers, its analytical solutions are presented in order to make the comparison between the classical model and the fractional model. Using the finite difference method, numerical schemes to the Caputo derivative operator and Riemann-Liouville fractional integral operator are obtained. Numerical solutions to the fractional differential equations of the generalized mathematical model are determined for the chemotherapy scheme based on the function of "on-off" type. Numerical results, obtained with the Mathcad software, are discussed and presented in graphical illustrations. The presence of the fractional order of the time-derivative as a parameter of solutions gives important information regarding the proliferative function, therefore, could give the possible rules for more efficient chemotherapy.

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