## Functional Continuous Runge-Kutta methods for Retarded Functional Differential Equations

### Alexey Eremin

Saint Petersburg State University

Continuous methods are the methods constructed for retarded functional differential equations (RFDEs) which include delay differential and many integral-differential equations. In them the delay can fall in the current time-step - a situation known as "overlapping," and explicit methods cannot be applied directly - they become fully implicit. Functional Continuous methods are specially constructed to remain fully explicit.

This talk starts with Functional Continuous Runge-Kutta methods, their main idea, order conditions and minimal stages number to achieve a certain order. Next, ways to reduce the number of stages necessary for a given order are considered. For instance, the idea, used by E. Nyström to construct more effective methods for second-order ordinary differential equations, is used for analogous RFDEs to construct FCRKs with fewer stages than for the first-order equations. Systems of RFDEs with certain types of right-hand side dependencies on the retarded solution also open the possibility to construct methods with reduced number of stages. This idea is based on partitioning the system and using two different (though interconnected) computational schemes. Order conditions and examples of the methods are presented.