Implementation of Chebyshev Pseudo-Spectral Method for Obtaining the Approximate Solutions of Delay Differential Equations in Fractional-Order

2016 ◽  
Vol 13 (10) ◽  
pp. 6563-6567
Author(s):  
Mohammed M Babatin
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Delay Differential Equations ◽  
Applied Mathematics ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Fractional Delay ◽  
Delay Differential ◽  
Pseudo Spectral Method ◽  
Bio Engineering

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The method is based upon Chebyshev approximations and introduce a new approximate formula for the fractional derivative. The fractional derivative is described in the Caputo sense. Special attention is given to study the convergence analysis and estimate the upper bound of the error of the proposed formula. The properties of Chebyshev polynomials are utilized to reduce FDDEs to linear or nonlinear system of algebraic equations. Numerical simulation and the exact solutions of FDDEs are presented.

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 A new stable collocation method for solving a class of nonlinear fractional delay differential equations

2020 ◽  
Vol 85 (4) ◽  
pp. 1123-1153
Author(s):  
Lei Shi ◽  
Zhong Chen ◽  
Xiaohua Ding ◽  
Qiang Ma
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Orthonormal Bases ◽  
Fractional Delay ◽  
Nonlinear Condition ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

AbstractIn this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal bases of $W^{1}_{2,0}$ W 2 , 0 1 . Error estimations of approximate solutions are given and the highest convergence order can reach four in the sense of the norm of $W_{2,0}^{1}$ W 2 , 0 1 . To overcome the nonlinear condition, we make use of Newton’s method to transform the nonlinear equation into a sequence of linear equations. For the linear equations, a rigorous theory is given for obtaining their ε-approximate solutions by solving a system of equations or searching the minimum value. Stability analysis is also obtained. Some examples are discussed to illustrate the efficiency of the proposed method.

scholarly journals Hypergeometric fractional derivatives formula of shifted Chebyshev polynomials: tau algorithm for a type of fractional delay differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Waleed M. Abd-Elhameed ◽  
José A. Tenreiro Machado ◽  
Youssri H. Youssri
Keyword(s):  
Differential Equations ◽  
Chebyshev Polynomials ◽  
Delay Differential Equations ◽  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Tau Method ◽  
Fractional Delay ◽  
Delay Differential ◽  
Derivatives Of ◽  
Fractional Delay Differential Equations

Abstract This paper presents an explicit formula that approximates the fractional derivatives of Chebyshev polynomials of the first-kind in the Caputo sense. The new expression is given in terms of a terminating hypergeometric function of the type 4 F 3(1). The integer derivatives of Chebyshev polynomials of the first-kind are deduced as a special case of the fractional ones. The formula will be applied for obtaining a spectral solution of a certain type of fractional delay differential equations with the aid of an explicit Chebyshev tau method. The shifted Chebyshev polynomials of the first-kind are selected as basis functions and the spectral tau method is employed for obtaining the desired approximate solutions. The convergence and error analysis are discussed. Numerical results are presented illustrating the efficiency and accuracy of the proposed algorithm.

scholarly journals Variational Approximate Solutions of Fractional Delay Differential Equations with Integral Transform

2021 ◽  
pp. 3679-3689
Author(s):  
Eman Mohmmed Namah
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Integral Transform ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Adomian Decomposition ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

     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).

scholarly journals The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
M. M. Khader
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Approximate Formula ◽  
Laguerre Polynomials ◽  
Algebraic Equations ◽  
Numerical Examples ◽  
Generalized Laguerre Polynomials ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

scholarly journals Second-Order Delay Differential Equations to Deal the Experimentation of Internet of Industrial Things via Haar Wavelet Approach

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Yongtao Xuan ◽  
Rohul Amin ◽  
Fakhar Zaman ◽  
Zohaib Khan ◽  
Imad Ullah ◽  
...  
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Approach ◽  
Haar Wavelet ◽  
Second Order ◽  
Algebraic Equations ◽  
Elimination Algorithm ◽  
Collocation Points ◽  
Delay Differential ◽  
Mean Square Errors

In this article, an efficient numerical approach for the solution of second-order delay differential equations to deal with the experimentation of the Internet of Industrial Things (IIoT) is presented. With the help of the Haar wavelet technique, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for various collocation points. The results show that the Haar wavelet method is an effective method for solving delay differential equations of second order. The convergence rate is also measured for various collocation points, which is almost equal to 2.

Sign in / Sign up

Export Citation Format

Share Document