In this paper, an improved trigonometrically fitted zero-dissipative explicit two-step hybrid method with fifth algebraic
order is derived. The method is applied to several problems where by the solutions are oscillatory in nature. Numerical
results obtained are compared with existing methods in the scientific literature. The comparison shows that the new
method is more effective and efficient than the existing methods of the same order.
In this paper, we develop algebraic order conditions for two-point block hybrid method up to order five using the approach
of B-series. Based on the order conditions, we derive fifth order two-point block explicit hybrid method for solving
special second order ordinary differential equations (ODEs), where the existing explicit hybrid method of order five is
used to be the method at the first point. The method is then trigonometrically fitted so that it can be suitable for solving
highly oscillatory problems arising from special second order ODEs. The new trigonometrically-fitted block method is
tested using a set of oscillatory problems over a very large interval. Numerical results clearly showed the superiority
of the method in terms of accuracy and execution time compared to other existing methods in the scientific literature.