Continuous Runge–Kutta schemes for pantograph type delay differential equations

Pantograph differential equations are important types of delay differential equations. Using continuous mono-implicit RK schemes, we propose a numerical method for numerically approximating pantograph delay differential equations that are reliable and unconditionally stable. The method combines the accuracy of implicit methods with the efficiency of implementation. The method works for stiff and non-stiff initial value problems, reducing the computational cost of fully implicit methods. Some examples are provided to demonstrate the effectiveness of the numerical method.