Fractional Delay Differential Equations Research Articles

Chebyshev Polynomials of Sixth Kind for Solving Nonlinear Fractional PDEs with Proportional Delay and Its Convergence Analysis

This work devotes to solving a class of delay fractional partial differential equations that arises in physical, biological, medical, and climate models. For this, a numerical scheme is implemented that applies operational matrices to convert the main problem into a system of algebraic equations; then, solving the resultant system leads to an approximate solution. The two-variable Chebyshev polynomials of the sixth kind, as basis functions in the proposed method, are constructed by the one-variable ones, and their operational matrices are derived. Error bounds of approximate solutions and their fractional and classical derivatives are computed. With the aid of these bounds, a bound for the residual function is estimated. Three illustrative examples demonstrate the simplicity and efficiency of the proposed method.

A Specific Method for Solving Fractional Delay Differential Equation via Fraction Taylor’s Series

It is well known that the appearance of the delay in the fractional delay differential equation (FDDE) makes the convergence analysis very difficult. Dealing with the problem with the traditional reproducing kernel method (RKM) is very tricky. The feature of this paper is to gain a more credible approximate solution via fractional Taylor’s series (FTS). We use the FTS to deal with the delay for improving the accuracy of the approximate solutions. Compared with other methods, the five numerical examples demonstrate the accuracy and efficiency of the proposed method in this paper.

Analysis of illegal drug transmission model using fractional delay differential equations

<abstract><p>The global burden of illegal drug-related death and disability continues to be a public health threat in developed and developing countries. Hence, a fractional-order mathematical modeling approach is presented in this study to examine the consequences of illegal drug usage in the community. Based on epidemiological principles, the transmission mechanism is the social interaction between susceptible and illegal drug users. A pandemic threshold value ($ \Lambda $) is provided for the illegal drug-using profession, which determines the stability of the model. The Lyapunov function is employed to determine the stability conditions of illegal drug addiction equilibrium point of society. Finally, the proposed model has been extended to include time lag in the delayed illegal drug transmission model. The characteristic equation of the endemic equilibrium establishes a set of appropriate conditions for ensuring local stability and the development of a Hopf bifurcation of the model. Finally, numerical simulations are performed to support the analytical results.</p></abstract>

Existence of local and global solutions to fractional order fuzzy delay differential equation with non-instantaneous impulses

<abstract><p>The main concern of this manuscript is to examine some sufficient conditions under which the fractional order fuzzy delay differential system with the non-instantaneous impulsive condition has a unique solution. We also study the existence of a global solution for the considered system. Fuzzy set theory, Banach fixed point theorem and Non-linear functional analysis are the major tools to demonstrate our results. In last, an example is given to illustrate these analytical results.</p></abstract>

Existence and uniqueness of mild solutions to neutral impulsive fractional stochastic delay differential equations driven by both Brownian motion and fractional Brownian motion

Existence and uniqueness of mild solutions to neutral impulsive fractional stochastic delay differential equations driven by both Brownian motion and fractional Brownian motion

Stability and Bifurcation Analysis of a Fractional Order Delay Differential Equation Involving Cubic Nonlinearity

Stability and Bifurcation Analysis of a Fractional Order Delay Differential Equation Involving Cubic Nonlinearity

Numerical Approach for Solving the Fractional Pantograph Delay Differential Equations

A new class of polynomials investigates the numerical solution of the fractional pantograph delay ordinary differential equations. These polynomials are equipped with an auxiliary unknown parameter a, which is obtained using the collocation and least‐squares methods. In this study, the numerical solution of the fractional pantograph delay differential equation is displayed in the truncated series form. The upper bound of the solution as well as the error analysis and the rate of convergence theorem are also investigated in this study. In five examples, the numerical results of the present method have been compared with other methods. For the first time, a‐polynomials are used in this study to numerically solve delay equations, and accurate approximations have been displayed.

On degree theory for non-monotone type fractional order delay differential equations

<abstract><p>In this paper, we establish a qualitative theory for implicit fractional order differential equations (IFODEs) with nonlocal initial condition (NIC) with delay term. Because area related to investigate existence and uniqueness of solution is important field in recent times. Also researchers are using existence theory to derive some prior results about a dynamical problem weather it exists or not in reality. In literature, we have different tools to study qualitative nature of a problem. On the same line the exact solution of every problem is difficult to determined. Therefore, we use technique of numerical analysis to approximate the solutions, where stability analysis is an important aspect. Therefore, we use a tool from non-linear analysis known as topological degree theory to develop sufficient conditions for existence and uniqueness of solution to the considered problem. Further, we also develop sufficient conditions for Hyers- Ulam type stability for the considered problem. To justify our results, we also give an illustrative example.</p></abstract>

Fractional view analysis of delay differential equations via numerical method

<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>

The Analysis of Fractional‐Order System Delay Differential Equations Using a Numerical Method

To solve fractional delay differential equation systems, the Laguerre Wavelets Method (LWM) is presented and coupled with the steps method in this article. Caputo fractional derivative is used in the proposed technique. The results show that the current procedure is accurate and reliable. Different nonlinear systems have been solved, and the results have been compared to the exact solution and different methods. Furthermore, it is clear from the figures that the LWM error converges quickly when compared to other approaches. When compared with the exact solution to other approaches, it is clear that LWM is more accurate and gets closer to the exact solution faster. Moreover, on the basis of the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional‐order delay differential equations.

Attractivity and Ulam-Hyers stability results for fractional delay differential equations

In this paper, we look into some results for the global attractivity and Ulam stability of solutions for fractional delay differential equations via Hilfer-Hadamard fractional derivative. The results are obtained by using Krasnoselskii?s fixed point theorem and Banach contraction principle.

Numerical solution for multi-term fractional delay differential equations

In this paper, a multi-term nonlinear delay differential equation (DDE) of arbitrary order is studied.Adomian decomposition method (ADM) is used to solve these types of equations. Then the existence andstability of a unique solution will be proved. Convergence analysis of ADM is discussed. Moreover, themaximum absolute truncated error of Adomian’s series solution is estimated. The stability of the solutionis also discussed.

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

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.

Existence and stability analysis of [formula omitted]th order multi term fractional delay differential equation

Study of fractional differential equations equipped with multi term operator have been carried out in the recent years. Several novel models need to be formulated. Therefore, keeping in view the importance of multi term differential operators in the study of differential equations, we propose a model of nth order multi term fractional delay differential equation (MTFDDE). We will provide results related to solution’s existence, and its uniqueness as well as will explore results related to four different kinds of functional stability. To get the results under consideration, standard fixed point theorems will be used. Moreover, functional stability results are obtained successfully and at last data dependence results are elaborated with the help of different results.

Numerical stability of Grünwald–Letnikov method for time fractional delay differential equations

This paper is concerned with the numerical stability of time fractional delay differential equations (F-DDEs) based on Grünwald–Letnikov (GL) approximation for the Caputo fractional derivative. In particular, we focus on the numerical stability region and the Mittag–Leffler stability. Using the boundary locus technique, we first derive the exact expression of the numerical stability region in the parameter plane, and show that the fractional backward Euler scheme is not \(\tau (0)\)-stable, which is different from the backward Euler scheme for integer DDE models. Secondly, we prove the numerical Mittag–Leffler stability for the numerical solutions provided that the parameters fall into the numerical stability region, by employing the singularity analysis of generating function. Our results show that the numerical solutions of F-DDEs are completely different from the classical integer order DDEs, both in terms of \(\tau (0)\)-stability and the long-time decay rate. Numerical examples are given to confirm the theoretical results.

Investigation of nonlinear fractional delay differential equation via singular fractional operator

Abstract The present research work is basically devoted to construction of a fractional order differential equation with time delay. Initially, integral representation is given to solution of the underline problem. Afterwards, operator form of solution is studied under some auxiliary hypothesis. Since uniqueness of solution is required, therefore we also provide results for exploring the uniqueness of solution for the underlying model. Using Lebesgue dominated convergence theorem and some other results from analysis, this work provides results devoted to existence of at least one solution. Also, for investigating the nature of solution for the proposed model, we study different kind of stability analysis. These stability related results show, how the solution behave with time. At the end of the article, we illustrate the obtained results via some examples.

Interpolating Stabilized Element Free Galerkin Method for Neutral Delay Fractional Damped Diffusion-Wave Equation

A numerical solution for neutral delay fractional order partial differential equations involving the Caputo fractional derivative is constructed. In line with this goal, the drift term and the time Caputo fractional derivative are discretized by a finite difference approximation. The energy method is used to investigate the rate of convergence and unconditional stability of the temporal discretization. The interpolation of moving Kriging technique is then used to approximate the space derivative, yielding a meshless numerical formulation. We conclude with some numerical experiments that validate the theoretical findings.

Variational Approximate Solutions of Fractional Delay Differential Equations with Integral Transform

The idea of the paper is to consolidate Mahgoub transform and variational iteration method (MTVIM) to solve fractional delay differential equations (FDDEs). The fractional derivative was in Caputo sense. The convergences of approximate solutions to exact solution were quick. The MTVIM is characterized by ease of application in various problems and is capable of simplifying the size of computational operations. Several non-linear (FDDEs) were analytically solved as illustrative examples and the results were compared numerically. The results for accentuating the efficiency, performance, and activity of suggested method were shown by comparisons with Adomian Decomposition Method (ADM), Laplace Adomian Decomposition Method (LADM), Modified Adomian Decomposition Method (MADM) and Homotopy Analysis Method (HAM).

A stable collocation approach to solve a neutral delay stochastic differential equation of fractional order

In this article, a step-by-step collocation technique based on the Jacobi polynomials is considered to solve a class of neutral delay fractional stochastic differential equations (NDFSDEs). First, we convert the NDFSDE into a non-delay equation by applying a step-by-step method. Then, by using a Jacobi collocation technique in each step, a non-delay nonlinear system is obtained. The convergence analysis of this numerical technique is discussed. Finally, several examples are implemented to confirm the efficiency and effectiveness of the proposed method.

Analytical solution of a fractional differential equation in the theory of viscoelastic fluids

The aim of this paper is to present analytical solutions of fractional delay differential equations (FDDEs) of an incompressible generalized Oldroyd-B fluid with fractional derivatives of Caputo type. Using a modification of the method of separation of variables the main equation with non-homogeneous boundary conditions is transformed into an equation with homogeneous boundary conditions, and the resulting solutions are then expressed in terms of Green functions via Laplace transforms. This results presented in two condition, in first step when 0 ≤ α, β ≤ 1/2 and in the second step we considered 1/2 ≤ α, β ≤ 1, for each step 1,2 for the unsteady flows of a generalized Oldroyd-B fluid, including a flow with a moving plate, are considered via examples.