The long time behavior of a spectral collocation method for delay differential equations of pantograph type (Q1948829)

The authors present a spectral collocation method with domain decomposition (with a uniform mesh) for the pantograph-type delay differential equation of the form \[ y'(x)=a(x)y(x)+b(x)y(qx)+f(x),\;x \in [0,T],\;y(0)=y_0. \] An overview of existing numerical methods for solving this equation, along with comments relating to related problems and issues with their application, is given in Section 1. The authors comment on the increasing popularity of using a spectral method with partial differential equations and justify their decision to develop a ``novel `time stepping' scheme based on uniform meshes for large \(T\)'', stating that it has some ``remarkable advantages'' over Runge-Kutta methods and that it is more appropriate for predicting the long time behaviour of a dynamical system. In Section 2, the spectral collocation method is introduced and the existence and uniqueness of the spectral collocation solution is discussed. The numerical approach is derived using Legendre-Gauss interpolation and the numerical implementation of the collocation scheme is described. The discrete scheme is rewritten in matrix form as \(DU_m^N\), \(1\leq m \leq M\), where \(U_m^N(t)\) is the global solution to the pantograph-type equation. The structure of the resulting \(M\) systems depends strongly on the delay term in the spectral collocation equation, and changes for each value of \(m\) as the three phases, I, II and III (\(m=1, 1<m \leq q^{II}:= \lfloor{\frac{1}{1-q}}\rfloor\) and \(m \geq q^{III}:=\lceil{\frac{1}{1-q}}\rceil\)). In the third phase, the pure delay phase, it is necessary to deal with two cases separately. A brief discussion of the existence and uniqueness of the spectral collocation solution, which is defined by a linear algebraic system, is included. A theorem is presented and proved. Convergence analysis is the focus of Section 3. Exponential convergence of the global errors is proved, both theoretically and numerically, and error estimates are proved. The main result of the paper is presented in Theorem 3.3. Illustrative numerical results are included in Section 4 to demonstrate the efficiency of the new method. In the first and third examples, the errors are compared with those from the multi-domain Legendre-Gauss collocation method, developed by Wang and Wang (2010). Very similar results are observed. The third example demonstrates an improvement in the numerical results over the Legendre collocation method developed by Brunner and Tang (2009), and state that their new method is still valid for a large time interval. Exponential decay of the errors in demonstrated in Example 2.