Fractional-order Delay Differential Equations Research Articles

Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses

AbstractIn this paper, we mainly consider the existence and Hyers-Ulam stability of solutions for a class of fractional differential equations involving time-varying delays and non-instantaneous impulses. By the Krasnoselskii’s fixed point theorem, we present the new constructive existence results for the addressed equation. In addition, we deduce that the equations have Hyers-Ulam stable solutions by utilizing generalized Grönwall’s inequality. Some results in this literature are new and improve some early conclusions.

Approximate Solution of Multi-Term Fractional Order Delay Differential Equations Using Homotopy Perturbation Method

In this paper the approximate solution of the non-linear equations of multi-term fractional order delay differential equations by using the homotopy perturbation method is considered. The fractional order derivative is communicated in the Caputo sense. In this methodology, the solutions are found in the form of a convergent power series with easily computed components. Finally, some examples are given to illustrate the obtained results, and then a comparison between the exact and the approximate results were given and they are presented in order to show the reliability and the accuracy of the proposed method.

Solutions of linear uncertain fractional-order delay differential equations

Uncertain fractional-order delay differential equation is a class of fractional-order functional differential equations driven by Liu process. This paper devotes to studying linear uncertain fractional-order delay differential equation. The explicit representation and iterative formula of the solution to linear uncertain fractional-order delay differential equations are obtained. Meanwhile, the inverse uncertainty distribution of the solution to linear uncertain fractional-order delay differential equation by the $$\alpha $$ -path is presented.

Numerical Treatment of Delay Salmonella Fractional Model with Atangana–Baleanu Derivative

Mathematical modelling of Salmonella disease played a significant role in the understanding of the transmission of this epidemic. In this article, we study numerically a fractional dynamical, with time-delay, Salmonella disease model. The introduced model is written as a system of fractional-order delay differential equations. The fractional operator is defined in the Atangana–Baleanu–Caputo sense, where the parameters are varied regarding the fractional-order derivative. We study, for any time delay, the stability of the disease-free equilibrium point and the endemic equilibrium point. Implicit nonstandard finite difference method is introduced to simulate numerically the proposed model. The obtained schema was unconditionally stable. Some comparison with our numerical simulations is introduced to show the effectiveness and applicability of the introduced approach for approximating the solutions of such stiff problems of fractional delay differential equations. Our simulations confirm the theoretical studies in this paper.

Green function method for a fractional–order delay differential equation

Green function method for a fractional–order delay differential equation

Qualitative Analysis of Multi-Terms Fractional Order Delay Differential Equations via the Topological Degree Theory

With the help of the topological degree theory in this manuscript, we develop qualitative theory for a class of multi-terms fractional order differential equations (FODEs) with proportional delay using the Caputo derivative. In the same line, we will also study various forms of Ulam stability results. To clarify our theocratical analysis, we provide three different pertinent examples.

Novel operational matrices-based method for solving fractional-order delay differential equations via shifted Gegenbauer polynomials

Accurate solutions of nonlinear multi-dimensional delay problems of fractional-order arising in mathematical physics and engineering recently have been found to be a challenging task for the research community. This paper witnesses that an efficient fully spectral operational matrices-based scheme is developed and successfully applied for stable solutions of time-fractional delay differential equations (DDEs). Monomials are introduced in order to proposed the novel operational matrices for fractional-order integration Iν and derivative Dν by means of shifted Gegenbauer polynomials. Some ordinary and partial delay differential equations of fractional-order are considered to show reliability, efficiency and appropriateness of the proposed method. In order to approximate the delay term in DDEs a novel delay operational matrix Θba is introduced with the help of shifted Gegenbauer polynomials. The proposed algorithm transform the problem understudy into a system of algebraic equations which are easier to tackle. Analytical solutions of the mentioned problem are effectively obtained, and an inclusive comparative study is reported which reveals that the proposed computational scheme is effective, accurate and well-matched to investigate the solutions of aforementioned problems. Error bound analysis is enclosed in our investigation to reveal the consistency and support the mathematical formulation of the algorithm. This proposed scheme can be extended to explore the solution of more dervisfy problem of physical nature in complex geometry.

On the Novel Finite-Time Stability Results for Uncertain Fractional Delay Differential Equations Involving Noninstantaneous Impulses

In this paper, we mainly study the finite-time stability for a kind of uncertain fractional-order delay differential equations with noninstantaneous impulses. By using the Lyapunov functions along with the generalized Grönwall inequality, we present the new stability results for the considered equations. Finally, two examples are given to demonstrate the effectiveness of our theoretical results.

An Approximation Technique for Fractional Order Delay Differential Equations

In this research article, an Iterative Decomposition Method is applied to approximate linear and non-linear fractional delay differential equation. The method was used to express the solution of a Fractional delay differential equation in the form of a convergent series of infinite terms which can be effortlessly computable.The method requires neither discretization nor linearization. Solutions obtained for some test problems using the proposed method were compared with those obtained from some methods and the exact solutions. The outcomes showed the proposed approach is more efficient and correct.

Ulam\u2013Hyers\u2013Mittag-Leffler stability for \u03c8-Hilfer fractional-order delay differential equations

In this paper, we present results on the existence, uniqueness, and Ulam–Hyers–Mittag-Leffler stability of solutions to a class of ψ-Hilfer fractional-order delay differential equations. We use the Picard operator method and a generalized Gronwall inequality involved in a ψ-Riemann–Liouville fractional integral. Finally, we give two examples to illustrate our main theorems.

Chebyshev operational matrix for solving fractional order delay-differential equations using spectral collocation method

In this article, the solution for general form of fractional order delay-differential equations (GFDDEs) is introduced. The proposed GFDDEs have multi-term of integer and fractional order derivatives for delayed or non delayed terms. An operational matrix is presented for all terms. The spectral collocation method is used to solve the proposed GFDDEs as a matrix discretization method. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments.

Modified Laguerre wavelet based Galerkin method for fractional and fractional-order delay differential equations

The application of modified Laguerre wavelet with respect to the given conditions by Galerkin method to an approximate solution of fractional and fractional-order delay differential equations is studied in this paper. For the concept of fractional derivative is used Caputo sense by using Riemann-Liouville fractional integral operator. The presented method here is tested on several problems. The approximate solutions obtained by presented method are compared with the exact solutions and is shown to be a very efficient and powerful tool for obtaining approximate solutions of fractional and fractional-order delay differential equations. Some tables and figures are presented to reveal the performance of the presented method.

Characteristic Orthogonal Polynomial based Galerkin Method for Solving Fractional Order Delay Differential Equations

Characteristic Orthogonal Polynomial based Galerkin Method for Solving Fractional Order Delay Differential Equations

Oscillation criteria for a class of fractional delay differential equations

This paper is devoted to the oscillatory problem in the fractional-order delay differential equations. First, we prove the convergence of the Laplace transform of a fractional operator by the generalized Gronwall inequality with singularity and fractional calculus technique. Then we show that it exhibits oscillation dynamics if the corresponding characteristic equation has no real roots. We further provide other direct and effective criteria depending on the system parameters and fractional exponent. Finally, we carry out some numerical simulations to illustrate our results.

On existence and stability results for nonlinear fractional delay differential equations

We establish existence and uniqueness results for fractional order delay differential equations. It is proved that successive approximation method can also be successfully applied to study Ulam--Hyers stability, generalized Ulam--Hyers stability, Ulam--Hyers--Rassias stability, generalized Ulam--Hyers--Rassias stability, $ \mathbb{E}_{\alpha}$--Ulam--Hyers stability and generalized $ \mathbb{E}_{\alpha}$--Ulam--Hyers stability of fractional order delay differential equations.

Parameter identification and optimisation for a class of fractional-order chaotic system with time delay

The fractional-order chaotic systems have more complex dynamic characteristics than the integer order chaotic systems, which can more reflect the physical properties of the actual system and more practical values, whereas it is difficult to control the synchronisation for fractional chaotic systems. Chaotic system identification is the basis of chaos control and performance analysis. In order to identify the parameters of the chaotic systems with time delay, a novel particle swarm optimisation with increasing inertia weight is proposed and then the issue is settled by solving an optimisation problem. The identification of parameters mainly includes the system order, the time delay parameter and the coefficient parameters. An estimation-correction algorithm based on linear interpolation method is used to solve the fractional-order delay differential equation. The Mackey-Glass chaotic system is conducted and comparisons with other two widely used particle swarm optimisations and the differential evolution algorithm indicate the effectiveness of the proposed method and an improvement in identification accuracy as well as convergence speed.

On numerical solution of fractional order delay differential equation using Chebyshev collocation method

On numerical solution of fractional order delay differential equation using Chebyshev collocation method

English

English

Numerical Study for Time Delay Multistrain Tuberculosis Model of Fractional Order

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.

A Modified Method for Solving Delay Differential Equations of Fractional Order

In this paper, generalized Hat functions operational matricesare proposed and combined with the method of steps to solve linear and nonlinear delay differential equations of fractional order.We convert the delay differential equationsof fractional order to non-delay differential equationsof fractional orderon a given intervalby apply the method of steps, and then apply the operational matrices for generalized Hat function on the obtained non-delay differential equationsof fractional order to transform linear and nonlinear non-delay differential equationsof fractional order into a system of algebraic equations and then find the solution.Two illustrative examples will be presented to show the accuracy and efficiencyof the proposed method.