Hybrid method for solving special fourth order ordinary differential equations

In recent time, Runge-Kutta methods that integrate special fourth order ordinary differential equations (ODEs) directly are proposed to address efficiency issues associated with classical Runge-Kutta methods. Although, the methods require approximation of y', y'' and y''' of the solution at every step. In this paper, a hybrid type method is proposed, which can directly integrate special fourth order ODEs. The method does not require the approximation of any derivatives of the solution. Algebraic order conditions of the methods are derived via Taylor series technique. Using the order conditions, eight algebraic order method is presented. Absolute stability of the method is analyzed and the stability region presented. Numerical experiment is conducted on some test problems. Results from the experiment show that the new method is more efficient and accurate than the existing Runge-Kutta and hybrid methods with similar number of function evaluation.

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