scholarly journals Stability of scalar nonlinear fractional differential equations with linearly dominated delay

2020 ◽  
Vol 23 (1) ◽  
pp. 250-267 ◽  
Author(s):  
Hoang The Tuan ◽  
Stefan Siegmund
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Equilibrium Points ◽  
Asymptotic Behavior Of Solutions ◽  
Asymptotically Stable ◽  
Fractional Delay ◽  
Fractional Differential ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

AbstractIn this paper, we study the asymptotic behavior of solutions to a scalar fractional delay differential equations around the equilibrium points. More precise, we provide conditions on the coefficients under which a linear fractional delay equation is asymptotically stable and show that the asymptotic stability of the trivial solution is preserved under a small nonlinear Lipschitz perturbation of the fractional delay differential equation.

scholarly journals Numerical Study for Time Delay Multistrain Tuberculosis Model of Fractional Order

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Nasser Hassan Sweilam ◽  
Seham Mahyoub Al-Mekhlafi ◽  
Taghreed Abdul Rahman Assiri
Keyword(s):  
Time Delay ◽  
Differential Equations ◽  
Fractional Order ◽  
Delay Differential Equations ◽  
Numerical Study ◽  
Equilibrium Points ◽  
Fractional Delay ◽  
The Stability ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

A novel mathematical fractional model of multistrain tuberculosis with time delay memory is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the sense of the Grünwald–Letinkov definition. Modified parameters are introduced to account for the fractional order. The stability of the equilibrium points is investigated for any time delay. Nonstandard finite deference method is proposed to solve the resulting system of fractional-order delay differential equations. Numerical simulations show that nonstandard finite difference method can be applied to solve such fractional delay differential equations simply and effectively.

scholarly journals Hermite Wavelet Method for Fractional Delay Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Umer Saeed ◽  
Mujeeb ur Rehman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Wavelet Method ◽  
Numerical Examples ◽  
Method Of Steps ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

We proposed a method by utilizing method of steps and Hermite wavelet method, for solving the fractional delay differential equations. This technique first converts the fractional delay differential equation to a fractional nondelay differential equation and then applies the Hermite wavelet method on the obtained fractional nondelay differential equation to find the solution. Several numerical examples are solved to show the applicability of the proposed method.

Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis

2021 ◽  
Vol 6 (1) ◽  
pp. 10
Author(s):  
İbrahim Avcı 
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Fractional Integration ◽  
Operational Matrix ◽  
Absolute Error ◽  
Algebraic Equations ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures.

scholarly journals Computer Simulation and Iterative Algorithm for Approximate Solving of Initial Value Problem for Riemann-Liouville Fractional Delay Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 477
Author(s):  
Snezhana Hristova ◽  
Kremena Stefanova ◽  
Angel Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Finite Interval ◽  
Initial Value ◽  
Monotone Iterative ◽  
Practical Usefulness ◽  
Fractional Delay ◽  
Fractional Differential ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

The main aim of this paper is to suggest an algorithm for constructing two monotone sequences of mild lower and upper solutions which are convergent to the mild solution of the initial value problem for Riemann-Liouville fractional delay differential equation. The iterative scheme is based on a monotone iterative technique. The suggested scheme is computerized and applied to solve approximately the initial value problem for scalar nonlinear Riemann-Liouville fractional differential equations with a constant delay on a finite interval. The suggested and well-grounded algorithm is applied to a particular problem and the practical usefulness is illustrated.

Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays

2019 ◽  
Vol 69 (3) ◽  
pp. 583-598 ◽  
Author(s):  
Ravi Agarwal ◽  
Snezhana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Razumikhin Technique ◽  
Fractional Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Delay ◽  
Fractional Differential ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

Abstract Caputo fractional delay differential equations with non-instantaneous impulses are studied. Initially a brief overview of the basic two approaches in the interpretation of solutions is given. A generalization of Mittag-Leffler stability with respect to non-instantaneous impulses is given and sufficient conditions are obtained. Lyapunov functions and the Razumikhin technique will be applied and appropriate derivatives among the studied fractional equations is defined and applied. Examples are given to illustrate our results.

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