Abstract—In this paper we constructed the sets of explicit third-order Improved Runge-Kutta (IRK) methods. The method used in two and three stage which indicated as the required number of function evaluations per step. The third-order IRK method in two-stage has a lower number of function evaluations than the classical third-order RK method while maintaining the same order of local accuracy. In three-stages, the new method is more accurate compared to the classical third-order RK method. The stability region of methods are given and numerical examples are presented to illustrate the efficiency and accuracy of the new methods.

Index Terms—Improved Runge-Kutta Methods, two-step runge-kutta methods, order conditions, stability region, ordinary differential equations.

Faranak Rabiei is with the Department of Mathematics, Faculty of Science, University Putra Malaysia (email: rabiei@math.upm.edu.my)

Cite: Faranak Rabiei and Fudziah Ismail, "Third-Order Improved Runge-Kutta Method for Solving Ordinary Differential Equation," International Journal of Applied Physics and Mathematics  vol. 1, no. 3, pp. 191-194, 2011.