Hybrid and linear multistep methods for solving oscillatory second-order differential equation

This thesis is focused mainly on developing methods for solving special second order ordinary differential equations (ODEs) and delay differential equations (DDEs) with oscillatory solutions. The first part of this thesis is on the derivation of semi-implicit hybrid methods using the technique of trigonometrically-fitted for solving oscillatory ordinary as well as delay differential equations. The implementation of trigonometrically fitting technique is supposed to enhance the efficiency of the methods. Numerical results are illustrated using efficiency curve where the common logarithm of the maximum global error versus the CPU time is taken. Results indicated that the new method work efficiently for solving both ODEs and DDEs. The stability of the methods are presented. In the second part of the thesis, phase-fitting technique is applied to the existing hybrid methods for solving oscillatory ODEs. The modification causes the nullifying of phase-lag of the methods. Numerical results illustrated that the new phase-fitted method is efficient compared to the existing fitted and non-fitted methods. The derivation of vanishing phase-lag and amplification fitted semi-implicit hybrid method are shown in the third part of the thesis. The general formula of hybrid method is modified with additional coefficients which depend on the value of the fitted frequency. The theory of zero dissipation and zero dispersion techniques are investigated. Numerical solutions show that the new method is a promising tool for integrating oscillatory problems.

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