Abstract
In this paper, we propose an efficient numerical method for delay differential equations with vanishing proportional delay qt (0 < q < 1). The algorithm is a mixture of the Legendre-Gauss collocation method and domain decomposition. It has global convergence and spectral accuracy provided that the data in the given pantograph delay differential equation are sufficiently smooth. Numerical results demonstrate the spectral accuracy of this approach and coincide well with theoretical analysis.
Highlights
We consider the delay differential equation (DDE):y (x) = a(x)y(x) + b(x)y(qx) + f (x), x ∈ J := [0, T ], y(0) = y0, (1)where q ∈ (0, 1) is a given constant and a, b, f are smooth functions on [0, T ]
Cheng [17] proposed and analyzed a spectral collocation method for the Volterra integral equations of the second kind, and their idea was extended to pantograph DDEs with a single delay and multiple delays in [1] and [2], respectively
The maximum point-wise errors of long time behavior between the numerical solution obtained by the numerical scheme (18) and the exact solution are given in Table 1 for q = 0.01, q = 0.5 and q = 0.99, respectively
Summary
We consider the delay differential equation (DDE):. where q ∈ (0, 1) is a given constant and a, b, f are smooth functions on [0, T ]. Pantograph delay differential equations, spectral collocation method, exponential convergence, domain decomposition, vanishing proportional delay. Cheng [17] proposed and analyzed a spectral collocation method for the Volterra integral equations of the second kind, and their idea was extended to pantograph DDEs with a single delay and multiple delays in [1] and [2], respectively. It is desirable to partition the time interval [0, T ] and use moderate mode N to construct numerical solution on each subinterval, step by step This process has the spectral accuracy, and the global convergence. We will introduce the spectral collocation method for the above pantograph-type DDE with vanishing proportional delay and describe the corresponding computational (‘time-stepping’) scheme.
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