scholarly journals Exponential Collocation Method for Solutions of Singularly Perturbed Delay Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Şuayip Yüzbaşı ◽  
Mehmet Sezer
Keyword(s):  
Differential Equations ◽  
Matrix Equation ◽  
Delay Differential Equations ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Algebraic Equations ◽  
Analysis Technique ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Delay Differential

This paper deals with the singularly perturbed delay differential equations under boundary conditions. A numerical approximation based on the exponential functions is proposed to solve the singularly perturbed delay differential equations. By aid of the collocation points and the matrix operations, the suggested scheme converts singularly perturbed problem into a matrix equation, and this matrix equation corresponds to a system of linear algebraic equations. Also, an error analysis technique based on the residual function is introduced for the method. Four examples are considered to demonstrate the performance of the proposed scheme, and the results are discussed.

A shifted Legendre method for solving a population model and delay linear Volterra integro-differential equations

2017 ◽  
Vol 10 (07) ◽  
pp. 1750091 ◽  
Author(s):  
Şuayip Yüzbaşi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Matrix Equation ◽  
Population Model ◽  
Estimation Method ◽  
Approximate Solutions ◽  
Algebraic Equations ◽  
Integro Differential Equation ◽  
Linear Algebraic Equations ◽  
The Matrix

In this paper, we propose a collocation method to obtain the approximate solutions of a population model and the delay linear Volterra integro-differential equations. The method is based on the shifted Legendre polynomials. By using the required matrix operations and collocation points, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. The matrix equation corresponds to a system of linear algebraic equations. Also, an error estimation method for method and improvement of solutions is presented by using the residual function. Applications of population model and general delay integro-differential equation are given. The obtained results are compared with the known results.

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 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.

scholarly journals An Asymptotic-Numerical Hybrid Method for Solving Singularly Perturbed Linear Delay Differential Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Süleyman Cengizci
Keyword(s):  
Differential Equations ◽  
Hybrid Method ◽  
Delay Differential Equations ◽  
Numerical Procedure ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Singularly Perturbed ◽  
Convection Term ◽  
Linear Delay ◽  
Delay Differential

In this work, approximations to the solutions of singularly perturbed second-order linear delay differential equations are studied. We firstly use two-term Taylor series expansion for the delayed convection term and obtain a singularly perturbed ordinary differential equation (ODE). Later, an efficient and simple asymptotic method so called Successive Complementary Expansion Method (SCEM) is employed to obtain a uniformly valid approximation to this corresponding singularly perturbed ODE. As the final step, we employ a numerical procedure to solve the resulting equations that come from SCEM procedure. In order to show efficiency of this numerical-asymptotic hybrid method, we compare the results with exact solutions if possible; if not we compare with the results that are obtained by other reported methods.

Sign in / Sign up

Export Citation Format

Share Document