Properties of analytic solution and numerical solution of multi-pantograph equation

This paper deals with the properties of the analytic solution and the numerical solution of the multi-pantograph equation u'(t) = lambdau(t) + Sigma'(i=1) mu(i)mu(q(i)t), 0 < q(l) < q(l-1) < ... < q(1) < 1 u(0) = u(0). Here lambda, mu(1), mu(2),..., mu(1), mu(0) is an element of C. The existence and the uniqueness of the analytic solution of the multi-pantograph equation are proved, the Dirichlet series solution is constructed, and the sufficient condition of the asymptotic stability for the analytic solution is obtained. It is proved that the theta-methods with a variable stepsize are asymptotically stable if 1/2 < theta less than or equal to 1. Some numerical examples are given to show the properties of the theta-methods. (C) 2003 Elsevier Inc. All rights reserved.