Research Article Novel Exponentially Fitted Two-Derivative Runge-Kutta Methods with Equation-Dependent Coefficients for First-Order Differential Equations Yanping Yang, 1 Yonglei Fang, 1 Xiong You, 2 and Bin Wang 3 1 School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, China 2 Department of Applied Mathematics, Nanjing Agricultural University, Nanjing 210095, China 3 School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China Correspondence should be addressed to Yonglei Fang; ylfangmath@163.com Received 5 February 2016; Accepted 27 March 2016 Academic Editor: Allan C. Peterson Copyright © 2016 Yanping Yang et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te construction of exponentially ftted two-derivative Runge-Kutta (EFTDRK) methods for the numerical solution of frst-order diferential equations is investigated. Te revised EFTDRK methods proposed, with equation-dependent coefcients, take into consideration the errors produced in the internal stages to the update. Te local truncation errors and stability of the new methods are analyzed. Te numerical results are reported to show the accuracy of the new methods. 1. Introduction In this paper, we are interested in the numerical integration of initial-value problems (IVPs) of frst-order diferential equations in the form   () =  (,), ( 0 )= 0 , (1) whose solutions exhibit a pronounced exponential behaviour [1, 2]. For the numerical solution of problem (1), classi- cal general-purpose methods such as Runge-Kutta (RK) methods and linear multistep methods (LMMs) can not produce satisfactory results due to the special structure of the problems. Te possible ways to construct the numerical methods adapted to the character of the solutions can be obtained by using exponential ftting (EF) technique [3–9]. Tere have been a large number of results concerning this topic in [10–15]. In standard approach to deriving exponentially ftted Runge-Kutta(-Nystr¨ om) methods, the efect of the error in the internal stages to the error of the fnal stage is completely neglected. D’Ambrosio et al. revisited the EFRK(N) method with two stages by considering the error in the internal stages in [1, 2]. Ixaru [16] introduced A-stable explicit fourth-order Runge-Kutta methods with three stages by evaluating the errors in each internal stage. For more accurate request, inspired by the work of Chan and Tsai [17], we improve the EFRK methods of D’Ambrosio et al. [1] by introducing the second derivative into the scheme. Te paper is organized as follows. In Section 2, we construct two versions of exponentially ftted TDRK methods. Te second version, revised version, has coefcients depending on the equation to be solved. In Section 3, we present the local truncations errors and analyze the linear stability of the new EFTDRK methods. Numerical experiments are reported in Section 4. Finally, in Section 5, we give some conclusive remarks. 2. Construction of the New Methods For the numerical integration of (1), we consider the explicit TDRK methods of the form  1 =  ,   =  +  ℎ(  ,  )+ℎ 2 −1 ∑ =1   (  +  ℎ,  ), =2,...,, Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2016, Article ID 9827952, 6 pages http://dx.doi.org/10.1155/2016/9827952