Based on the theory of functional fitting Runge-Kutta methods(FRK methods), we construct a class of methods called trigonometric-exponential fitting Runge-Kutta methods by selecting a group of new basic functions, and note them ETFRK methods.The paper begins with introducing the structure and characteristics of FRK methods, raising the concept and requirements of the basic functions. On this basis, we develop ETFRK methods which are based on a group of new basis functions. The coefficients of ETFRK methods are derived, and the characteristics of FRK methods are expanded in.After introducing the constructure theory of ETFRK methods, we give several kinds of explicit and implicit ETFRK methods. In explicit methods, we introduce extended Runge-Kutta methods for optimization, because general explicit Runge-Kutta methods cannot satisfy the requirements of the exact solution. But explicit methods can only have two basis functions, which reduce the complexity. We define the corresponding algebraic methods of the ETFRK methods to investigate the order of the methods, which is verified in the practical experiments. In the implicit methods, we develop the methods fully in accordance with the theory, so that the basis functions are rich, and we can also use the collocation techniques to control the order of the ETFRK methods, which is verified in the practical experiments.At last, we also suppose the development direction of FRK methods. Firstly, new methods can be developed. Secondly, better order is expected. In addition, we can discuss the stability of the methods and so on. |