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In this paper, function fitting Runge-Kutta method (FRK method ) on the basis of the general theory , by selecting a new set of basis functions , construct a triangle is called exponential fitting Runge-Kutta method (ETFRK method ) . This paper describes the structure and characteristics of FRK way to start , leads to the concept of basis functions and requirements , on this basis, given a particular set of basis functions , thereby obtaining ETFRK method . Next we give ETFRK coefficient method for solving methods , and the characteristics of the FRK method extensions come in order . In the method described ETFRK tectonic theory , we are given several types of explicit and implicit ETFRK method . In the explicit method, since the requirement for precision FRK method makes explicit ordinary Runge-Kutta method can not be established , then we introduce the extended Runge-Kutta method for optimization. However, as the explicit method itself will limit the number of basis functions of the two , to a certain extent also reduces the complexity of the explicit method . On the performance of the explicit method , we introduce the corresponding algebraic ETFRK method to estimate the order it , and in the numerical experiments verified . In the implicit method, we constructed entirely in accordance with the method ETFRK theory , not only on the number of basis functions with greater flexibility and can be configured to point to a better method of controlling the order , which is in the numerical experiments have been confirmed. Finally the method further discussion FRK development directions, one constructed new approach , the second is to improve the way the order of the three methods is to examine the stability .
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