Numerical solution of special second order initial value problems by hybrid type methods

We derived in this thesis new highly dispersive and highly dissipative two-step explicit hybrid methods for solving oscillatory problems. Dispersion conditions up to order ten and dissipation conditions up to order eleven for five stage hybrid methods are presented. The derivation of the methods was largely based on maximization of order of dispersion and dissipation while minimizing the principal error norm. Stability of the methods was investigated and their intervals of stability presented. The methods, which can be applied using constant step size,were tested on model problems. Numerical results revealed the superiority of the methods over several existing methods in the literature. In order to achieve higher accuracy and efficiency, trigonometrically fitted hybrid methods based on existing zero-dissipative methods were derived. Their ability to approximate the solution of problems with large fitted frequency using large step size proved their accuracy and efficiency for solving highly oscillatory problems compared to phase-fitted methods and trigonometrically fitted methods which are based on dissipative hybrid methods. Finally, semi-implicit hybrid methods based on explicit hybrid methods were derived.Dispersion and dissipation conditions for the methods were presented. Stability analysis along side stability or periodicity intervals of the methods was presented.Results obtained from numerical experiment showed the accuracy and efficiency of the new methods compared to the existing methods.

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