Abstract
In this paper, a computational procedure for solving singularly perturbed nonlinear delay differentiation equations (SPNDDEs) is proposed. Initially, the SPNDDE is reduced into a series of singularly perturbed linear delay differential equations (SPLDDEs) using the quasilinearization technique. A trigonometric spline approach is suggested to solve the sequence of SPLDDEs. Convergence of the method is addressed. The efficiency and applicability of the proposed method are demonstrated by the numerical examples.
Highlights
Consider a nonlinear singularly perturbed delay differential equation in the form εθ′′ = F s, θ, θ′ðs − δÞ on ð0, 1Þ, ð1Þ under the interval and boundary conditions θðsÞ = μðsÞ on − δ ≤ s ≤ 0, θð1Þ = γ, ð2Þ where 0 < ε ≪ 1 is a perturbation parameter and δ is a delay parameter of oðεÞ
Motivation for the research and solution of the singularly perturbed nonlinear delay differentiation equations (SPNDDEs) has been increasing in the last few years
Classical methods for solving such types of problems are ineffective since a boundary layer structure is present when the perturbation parameter goes to zero
Summary
Consider a nonlinear singularly perturbed delay differential equation in the form. εθ′′ = F s, θ, θ′ðs − δÞ on ð0, 1Þ, ð1Þ under the interval and boundary conditions θðsÞ = μðsÞ on − δ ≤ s ≤ 0, θð1Þ = γ, ð2Þ where 0 < ε ≪ 1 is a perturbation parameter and δ is a delay parameter of oðεÞ. Classical methods for solving such types of problems are ineffective since a boundary layer structure is present when the perturbation parameter goes to zero. For these equations, effective numerical methods should be established, the accuracy of which does not depend on ε. For generating numerical spectrum solutions to linear and nonlinear second-order boundary value problems, a new operational matrix approach based on shifted Legendre polynomials is introduced and studied in [13]. In [16], the authors used shifted Legendre polynomials for studying the spectral collocation approach to solve neutral functional-differential equations with proportional delays.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
