Numerical Stability Of Two-step Runge-kutta Methods For Pantograph Equations

Pantograph equation is a branch of delay differential equations. It has beenwidely applied to various fields such as electrodynamics，nonlinear dynamicalsystems, automatic control systems, ecology, financial field. It has promoted thesociety’s development. Generally speaking, the analysis solution of pantographequation is difficult to determine, so using numerical methods to solve thepantograph equation has important significance.Two-step Runge-Kutta methods were first studied by Byren and Lambert andextended to general form by Jackiewicz. In the two-step Runge-Kutta methods, theapproximate value of function at current step depends on stage value at twoconsecutive steps. We can gain extra degrees of freedom associated with a two-stepscheme without the need for extra function evaluations. Compared to one-stepRunge-Kutta methods, the numerical methods do not increase the amount ofcomputations. When the program runs, the efficiency of two-step Runge-Kuttamethods is higher than that of the one-step Runge-Kutta methods. Two-stepRunge-Kutta methods require fewer stages to achieve the same order as the one-stepRunge-Kutta methods and its precision is relatively high.In this thesis, we mainly study the numerical stability of two-stepRunge-Kutta methods for two kinds of pantograph equations. On the one hand, westudy the stability of two-step Runge-Kutta methods for linear pantograph equation.The two-step Runge-Kutta methods are applied on a constrained mesh to theasymptotically stable equation. We analyzed that whether the numerical methods canpreserve the asymptotic stability or not. Further, we discuss the asymptotic stabilityof pantograph equation whose coefficients are in the form of matrix. On this basis,we consider the conditions that the special numerical methods should satisfy, whichthe special methods can preserve asymptotic stability of equations. On the otherhand, we use the two-step Runge-Kutta methods to solve neutral pantographequation. Under the condition that the analytical solution of equation isasymptotically stable, we discuss the asymptotic stability of numerical solution.Further, the stability of numerical methods for neutral pantograph equation of matrixcoefficient is analyzed. At last, we use the numerical examples to verify theconclusion of this thesis.