Convergence analysis of an Arnoldi order reduced Runge-Kutta method for integro-differential equations of pantograph type

In this paper, we consider a numerical method for linear functional integro-differential equations with pantograph delays. To deal with the pantograph delays, we introduce a geometrically increasing mesh, and propose a new kind of Runge-Kutta methods based on the Arnoldi order-reduced technique. The convergence analysis of the new method is presented, and the method is proved to be at least first order. The numerical experiments attached show that, the new method can achieve almost the same accuracy as the corresponding Runge-Kutta method for some properly chosen Krylov subspaces.