A new method for solving initial value problems

• A more accurate method for solving ordinary differential equations is presented.
• In the proposed method, the coefficients in the numerical approximation are optimally determined to reduce the truncation error.
• The advantages of the proposed method are illustrated through several examples.

A more accurate method (comparing to the Euler, Runge–Kutta, and implicit Runge–Kutta methods) for the numerical solutions of ordinary differential equations (ODEs) is presented in this paper. The coefficients in the approximate solution for the ODE using the proposed method are divided into two groups: the fixed coefficients and the free coefficients. The fixed coefficients are determined by using the same way as in the traditional Taylor series method. The free coefficients are obtained optimally by minimizing the error of the approximate solution in each time interval. Examples are presented to compare the numerical solutions of the Rahmanzadeh, Cai, and White's method (RCW) to those of other popular ODEs methods.