Sumudu Decomposition Method for Solving Fractional Delay Differential Equations

We study the numerical solutions of nonlinear fractional delay differential equations (DEs) and nonlinear fractional pantograph DEs. We introduce a new class of functions called fractional‐order generalized Taylor wavelets (FOGTW). We provide an exact formula for computing the Riemann‐Liouville fractional integral operator for FOGTW by using the regularized beta functions. By applying the formula and collocation method, we reduce the given nonlinear fractional delay DEs and nonlinear fractional pantograph DEs to a system of algebraic equations. The FOGTW method together with the exact formula is very efficient for solving the nonlinear fractional delay DEs and nonlinear fractional pantograph DEs and give very accurate results. Several examples are given to demonstrate the effectiveness of the present method.

PurposeIn this article, the authors aims to introduce a novel Vieta–Lucas wavelets method by generalizing the Vieta–Lucas polynomials for the numerical solutions of fractional linear and non-linear delay differential equations on semi-infinite interval.Design/methodology/approachThe authors have worked on the development of the operational matrices for the Vieta–Lucas wavelets and their Riemann–Liouville fractional integral, and these matrices are successfully utilized for the solution of fractional linear and non-linear delay differential equations on semi-infinite interval. The method which authors have introduced in the current paper utilizes the operational matrices of Vieta–Lucas wavelets to converts the fractional delay differential equations (FDDEs) into a system of algebraic equations. For non-linear FDDE, the authors utilize the quasilinearization technique in conjunction with the Vieta–Lucas wavelets method.FindingsThe purpose of utilizing the new operational matrices is to make the method more efficient, because the operational matrices contains many zero entries. Authors have worked out on both error and convergence analysis of the present method. Procedure of implementation for FDDE is also provided. Furthermore, numerical simulations are provided to illustrate the reliability and accuracy of the method.Originality/valueMany engineers or scientist can utilize the present method for solving their ordinary or Caputo–fractional differential models. To the best of authors’ knowledge, the present work has not been used or introduced for the considered type of differential equations.

Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations

This research deals with the numerical solution of fractional differential equations with delay using the method of steps and shifted Legendre (Chebyshev) collocation method. This article presents a new formula for the fractional derivatives (in the Caputo sense) of shifted Legendre polynomials. With the help of this tool and previous work of the authors, efficient numerical schemes for solving nonlinear continuous fractional delay differential equations are proposed. The proposed schemes transform the nonlinear fractional delay differential equations to a non-delay one by employing the method of steps. Then, the approximate solution is expanded in terms of Legendre (Chebyshev) basis functions. Furthermore, the convergence analysis of the proposed schemes is provided. Several practical model examples are considered to illustrate the efficiency and accuracy of the proposed schemes.

Most of the physical phenomena located around us are nonlinear in nature and their solutions are of great significance for scientists and engineers. In order to have a better representation of these physical phenomena, fractional calculus is developed. Some of these nonlinear physical models can be represented in the form of delay differential equations of fractional order. In this article, a new method named Gegenbauer Wavelets Steps Method is proposed using Gegenbauer polynomials and method of steps for solving nonlinear fractional delay differential equations. Method of steps is used to convert the fractional nonlinear fractional delay differential equation into a fractional nonlinear differential equation and then Gegenbauer wavelet method is applied at each iteration of fractional differential equation to find the solution. To check the accuracy and efficiency of the proposed method, the proposed method is implemented on different nonlinear fractional delay differential equations including singular-type problems also.

In this paper, we consider a Cauchy problem for a Caputo‐type time delay linear system of fractional differential equations with permutable matrices. First, we provide a new representation of solutions to linear homogeneous fractional differential equations using the Laplace integral transform and variation of constants formula via a newly defined delayed Mittag‐Leffler type matrix function introduced through a three‐parameter Mittag‐Leffler function. Second, with the help of a delayed perturbation of a Mittag‐Leffler type matrix function, we attain an explicit formula for solutions to a linear nonhomogeneous time delay fractional order system using the superposition principle. Furthermore, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations using the contraction mapping principle. Finally, we present an example to illustrate the applicability of our results.

The fractional delay differential equation (FDDE) exhibits an adaptable framework for modeling and analyzing systems where the traditional integer-order model falls short. Its ability to mingle fractional-order dynamics and delay effects makes it a significant tool across distinct fields. The key objective of this article is to manifest the significance of solving fractional delay differential equations numerically by utilizing the Legendre collocation method. The FDDE has been reduced to a system of algebraic equations. Interpretative numerical examples have been demonstrated to display the effectiveness and preciseness of the potential method for solving FDDE. The prospective method has been verified via error analysis and comparison with other methods.

In this paper, an approximate analytical method, New Variational Iteration Method (NVIM) is introduced in this paper for the approximate analytical solution of Fractional Delay Differential Equations (FDDE). The algorithm is illustrated by studying initial value linear and nonlinear problems. The results obtained are presented and show that only few terms are required to get an approximate solution.

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.

<abstract><p>In this article, we solved pantograph delay differential equations by utilizing an efficient numerical technique known as Chebyshev pseudospectral method. In Caputo manner fractional derivatives are taken. These types of problems are reduced to linear or nonlinear algebraic equations using the suggested approach. The proposed method's convergence is being studied with particular care. The suggested technique is effective, simple, and easy to implement as compared to other numerical approaches. To prove the validity and accuracy of the presented approach, we take two examples. The solutions we obtained show greater accuracy as compared to other methods. Furthermore, the current approach can be implemented for solving other linear and nonlinear fractional delay differential equations, owing to its innovation and scientific significance.</p></abstract>

We introduce a theorem on linearized asymptotic stability for nonlinear fractional delay differential equations (FDDEs) with a Caputo order α∈(1,2), which can be directly used for fractional delay neural networks. It relies on three technical tools: a detailed root analysis for the characteristic equation, estimation for the generalized Mittag-Leffler function, and Lyapunov's first method. We propose coefficient-type criteria to ensure the stability of linear FDDEs through a detailed root analysis for the characteristic equation obtained by the Laplace transform. Further, under the criteria, we provide a wise expression of the generalized Mittag-Leffler functions and prove their polynomial long-time decay rates. Utilizing the well-established Lyapunov's first method, we establish that an equilibrium of a nonlinear Caputo FDDE attains asymptotically stability if its linearization system around the equilibrium solution is asymptotically stable. Finally, as a by-product of our results, we explicitly describe the asymptotic properties of fractional delay neural networks. To illustrate the effectiveness of our theoretical results, numerical simulations are also presented.

This paper presents an approximate method for solving a kind of fractional delay differential equations defined in terms of Caputo fractional derivatives. The approximate method is based on the application of the Bernstein’s operational matrix of fractional differentiation. First, Bernstein operational matrix of fractional differentiation is presented generalizing the idea of Bernstein’s operational matrix of derivative for integer orders, and then applied to solve the nonlinear fractional delay differential equations. The operational matrix method combined with the typical tau method reduces the fractional delay differential equation into system of nonlinear equations. Solving these nonlinear equations the desired solution is achieved. Two different cases of the fractional delay differential equations are illustrated and solved using the presented method. Numerical results and discussions demonstrate the applicability of the proposed method.

Numerical simulation of nonlinear fractional delay differential equations with Mittag-Leffler kernels

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.

A new Chelyshkov matrix method to solve linear and nonlinear fractional delay differential equations with error analysis