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

2022 ◽  
Vol 2022 ◽  
pp. 1-6
Author(s):  
Ming-Jing Du
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Specific Method ◽  
Fractional Taylor’S Series ◽  
Fractional Delay ◽  
Delay Differential ◽  
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.

scholarly journals Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-28 ◽  
Author(s):  
Zeqing Liu ◽  
Ling Guan ◽  
Sunhong Lee ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Delay Differential Equation ◽  
Iterative Algorithms ◽  
Approximate Solutions ◽  
Higher Order ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper is concerned with the higher order nonlinear neutral delay differential equation[a(t)(x(t)+b(t)x(t-τ))(m)](n-m)+[h(t,x(h1(t)),…,x(hl(t)))](i)+f(t,x(f1(t)),…,x(fl(t)))=g(t),for allt≥t0. Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

scholarly journals Positive Solutions and Mann Iterations of a Fourth Order Nonlinear Neutral Delay Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-22
Author(s):  
Zeqing Liu ◽  
Jingjing Zhu ◽  
Jeong Sheok Ume ◽  
Shin Min Kang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Positive Solutions ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Approximate Solutions ◽  
Banach Fixed Point Theorem ◽  
Neutral Delay ◽  
Neutral Delay Differential Equation ◽  
Delay Differential

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given.

scholarly journals Hyers-Ulam Stability of the Delay Equationy'(t)=λy(t-τ)

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Soon-Mo Jung ◽  
Janusz Brzdęk
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Approximate Solutions ◽  
Initial Condition ◽  
Ulam Stability ◽  
Delay Differential

We investigate the approximate solutionsy:[−τ,∞)→Rof the delay differential equationy'(t)=λy(t-τ)(t∈[0,∞))with an initial condition, whereλ>0andτ>0are real constants. We show that they can be “approximated” by solutions of the equation that are constant on the interval[-τ,0]and, therefore, have quite simple forms. Our results correspond to the notion of stability introduced by Ulam and Hyers.

scholarly journals Exact and approximate solutions of delay differential equations with nonlocal history conditions

2005 ◽  
Vol 2005 (2) ◽  
pp. 181-194 ◽  
Author(s):  
S. Agarwal ◽  
D. Bahuguna
Keyword(s):  
Differential Equation ◽  
Global Existence ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Approximate Solutions ◽  
Classical Solutions ◽  
Global Existence Of Solutions ◽  
Delay Differential

We study the exact and approximate solutions of a delay differential equation with various types of nonlocal history conditions. We establish the existence and uniqueness of mild, strong, and classical solutions for a class of such problems using the method of semidiscretization in time. We also establish a result concerning the global existence of solutions. Finally, we consider some examples and discuss their exact and approximate solutions.

scholarly journals Solving Delay Differential Equations by an Accurate Method with Interpolation

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Ali Akgül ◽  
Adem Kiliçman
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Approximation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Accurate Method ◽  
Reproducing Kernel Method ◽  
Comparison Of The Results ◽  
Delay Differential

We use the reproducing kernel method (RKM) with interpolation for finding approximate solutions of delay differential equations. Interpolation for delay differential equations has not been used by this method till now. The numerical approximation to the exact solution is computed. The comparison of the results with exact ones is made to confirm the validity and efficiency.

scholarly journals An Approximation Technique for Fractional Order Delay Differential Equations

2019 ◽  
pp. 1539-1545
Author(s):  
Olutunde Samuel Odetunde ◽  
Abass Ishola Taiwo ◽  
Olusola Adebanwo Dehinsilu
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Decomposition Method ◽  
Convergent Series ◽  
Approximation Technique ◽  
Test Problems ◽  
Research Article ◽  
Iterative Decomposition ◽  
Fractional Delay ◽  
Delay Differential

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.

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