Numerical solutions of single delay differential equations and special second order oscillatory initial value problems using Runge-Kutta and hybrid methods

The first part of the thesis focuses on adapting existing methods for solving first and second order delay differential equations (DOEs). The methods are Improved Runge-Kutta (IRK) and Runge-Kutta (RK) methods which are adapted for solving first order DDEs. The accuracy and stability of the methods when applied to linear first order DDEs are looked into. Next we adapt the existing hybrid methods for solving special second order DDEs. Numerical results are compared in terms of accuracy and computational time with the Runge-Kutta Nystrom (RKN) method. Stability of the methods when applied to linear second order DDEs are presented. The new Semi-Implicit Hybrid methods (SIHMs) are derived for solving system of oscillatory problems. The methods have highest possible order of dissipation and dispersion with small error coefficients. The periodicity intervals of the methods are also given. Numerical results indicate that SIHMs are more efficient compare to the existing methods. Then the zero-dissipative Phase-Fitted Hybrid methods (PFHMs) are constructed based on the existing explicit hybrid methods. The dispersion relations are developed in order to obtain methods with phase-lag of order infinity. Numerical illustrations indicate that PFHMs are much more efficient than the existing methods. Finally, we constructed Optimized Hybrid methods (OPHMs) based on the existing non-zero-dissipative hybrid methods. To develop OPHMs; dissipative, dispersive and first derivatives of dispersive relations are required. We found that the non-zero-dissipative hybrid methods are more suitable to be optimized than phase-fitted. Numerical results are also given to prove the claim. In conclusion, the IRK methods and hybrid methods are more efficient in solving first and second order DDEs respectively. The new methods constructed in this thesis are suitable for solving second-order ODEs and they are more efficient compared to the existing methods.

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