scholarly journals Solving Delay Differential Equations of Small and Vanishing Lag Using Multistep Block Method

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Nurul Huda Abdul Aziz ◽  
Zanariah Abdul Majid ◽  
Fudziah Ismail
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Divided Difference ◽  
Interpolation Polynomial ◽  
Block Method ◽  
Step Size ◽  
Delay Term ◽  
C Program ◽  
Delay Differential ◽  
A Current

This paper considers the numerical solution of delay differential equations for solving the problem of small and vanishing lag using multistep block method. This problem arises when the size of a delay value is smaller than the step size,x-τ<h, and the delay time may even vanish whenτ→0in a current step. The proposed approach that is based on interpolation of Newton divided difference has been implemented by adapting this problem to the multistep block method. In order to achieve the required accuracy, this approach considered the appropriate degree of interpolation polynomial in approximating the solution of delay term. The developed code for solving small and vanishing lag is done using C program and we called it as DDEB5. TheP-stability andQ-stability of this method are also studied. Numerical results are presented and compared to the existing method in order to illustrate the efficiency of the proposed method.

scholarly journals Computation of First order Delay Differential Equations via Simpson Block Method

2020 ◽  
Vol 5 (2) ◽  
Author(s):  
Ridwanulahi I Abdulganiy ◽  
Olusheye A Akinfenwa ◽  
Osaretin E Enobabor ◽  
Blessing I Orji ◽  
Solomon A Okunuga
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Step Width ◽  
Block Method ◽  
Delay Term ◽  
First Order ◽  
Collocation Technique ◽  
Delay Differential

A family of Simpson Block Method (SBM) is proposed for the numerical integration of Delay Differential Equations (DDEs). The methods are developed through multistep collocation technique using constant step width. The convergence and accuracy of the methods are established through some standard DDEs in the reviewed literature. Keywords— Block Method, Collocation Technique, Delay Term, Delay Differential Equation, Self Starting.   

scholarly journals Numerical Approach for Solving Delay Differential Equations with Boundary Conditions

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1073
Author(s):  
Nur Tasnem Jaaffar ◽  
Zanariah Abdul Majid ◽  
Norazak Senu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Numerical Approach ◽  
Block Method ◽  
Data Simulation ◽  
Step Size ◽  
Function Calls ◽  
Delay Differential

In the present paper, a fifth-order direct multistep block method is proposed for solving the second-order Delay Differential Equations (DDEs) directly with boundary conditions using constant step size. In many life sciences applications, a delay plays an essential role in modelling natural phenomena with data simulation. Thus, an efficient numerical method is needed for the numerical treatment of time delay in the applications. The proposed direct block method computes the numerical solutions at two points concurrently at each computed step along the interval. The types of delays involved in this research are constant delay, pantograph delay, and time-dependent delay. The shooting technique is utilized to deal with the boundary conditions by applying a Newton-like method to guess the next initial values. The analysis of the proposed method based on the order, consistency, convergence, and stability of the method are discussed in detail. Four tested problems are presented to measure the efficiency of the developed direct multistep block method. The numerical simulation indicates that the proposed direct multistep block method performs better than existing methods in terms of accuracy, total function calls, and execution times.

scholarly journals Periodic Averaging Principle for Neutral Stochastic Delay Differential Equations with Impulses

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Peiguang Wang ◽  
Yan Xu
Keyword(s):  
Differential Equations ◽  
Operator Theory ◽  
Delay Differential Equations ◽  
Averaging Principle ◽  
Neutral System ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Delay Term ◽  
Delay Differential

In this paper, we study the periodic averaging principle for neutral stochastic delay differential equations with impulses under non-Lipschitz condition. By using the linear operator theory, we deal with the difficulty brought by delay term of the neutral system and obtain the conclusion that the solutions of neutral stochastic delay differential equations with impulses converge to the solutions of the corresponding averaged stochastic delay differential equations without impulses in the sense of mean square and in probability. At last, an example is presented to show the validity of the proposed theories.

Computational Treatment of First Order Delay Differential Equations Using Hybrid Extended Second Derivative Block Backward Differentiation Formulae

2020 ◽  
Vol 1 (1) ◽  
Author(s):  
C. Chibuisi ◽  
Bright Okore Osu ◽  
C. Olunkwa ◽  
S. A. Ihedioha ◽  
S. Amaraihu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Derivative ◽  
Delay Term ◽  
First Order ◽  
Block Form ◽  
Collocation Approach ◽  
Backward Differentiation Formulae ◽  
Delay Differential ◽  
Step Number

This paper considers the computational solution of first order delay differential equations (DDEs) using hybrid extended second derivative backward differentiation formulae method in block form without the implementation of interpolation techniques in estimating the delay term. By matrix inversion approach, the discrete schemes were obtained through the linear multistep collocation approach from the continuous form of each step number which after implementation strongly revealed the convergence and region of absolute stability of the proposed method. Computational results are presented and compared to the exact solutions and other existing method to demonstrate its efficiency and accuracy.

DECAY AND GROWTH RATES OF SOLUTIONS OF SCALAR STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY AND STATE DEPENDENT NOISE

2005 ◽  
Vol 05 (02) ◽  
pp. 133-147 ◽  
Author(s):  
JOHN A. D. APPLEBY
Keyword(s):  
Differential Equations ◽  
Growth Rates ◽  
Delay Differential Equations ◽  
Decay Rates ◽  
Stochastic Delay Differential Equations ◽  
Diffusion Term ◽  
Stochastic Delay ◽  
Delay Term ◽  
Unbounded Delay ◽  
Delay Differential

This paper studies the growth and decay rates of solutions of scalar stochastic delay differential equations of Itô type. The equations studied have a linear drift which contains an unbounded delay term, and a nonlinear diffusion term, which depends on the current state only. We show that when the nonlinearity at zero or infinity is sufficiently weak, the same non-exponential decay and growth rates found in the corresponding underlying linear deterministic equation are recovered, in an almost sure sense.

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