Phase-fitted and amplification-fitted Runge-Kutta type methods for solving linear differential equations with oscillatory solutions - Salih, Mohammed Mahmood (2016)
New phase-fitted and amplification-fitted Runge-Kutta methods (RK) based on the existing Runge-Kutta methods of order four and five were derived to solve second order ordinary differential equations with oscillatory solutions. The new method has the property of zero phase-lag and zero dissipation. The effects of phase-fitted and amplification-fitted relations are tested over a large interval on homogeneous and nonhomogeneous problems which have oscillatory solutions and the numerical results proved that the new methods are more accurate compared to the existing methods. Then, the first order Fuzzy Differential Equations (FDEs) are solved using the RK methods with phase-fitted and amplification-fitted and the numerical results show that the methods are more accurate than the existing methods. The technique of phase-fitted and amplification-fitted is then extended to diagonally implicit Runge-Kutta methods (DIRK) for solving second order ordinary differential equations (ODEs) with oscillatory solution. We derived the phase-fitted and amplification-fitted fourth DIRK method based on the fourth order existing DIRK methods. Numerical results show that the DIRK with phase-fitted and amplificationfitted is more accurate and efficient for solving oscillatory problems. In the next part of the thesis, we derived the order conditions of Runge-Kutta Nystrom method purposely for solving linear second order ordinary differential equations. Based on the order conditions we derived the new fifth order four-stage and sixth order five-stage explicit Runge-Kutta-Nyström methods for linear ordinary differential equations (LODEs). Then the methods are phase-fitted and amplification-fitted so that they will have zero-dispersion and zero-dissipation. The fifth order four-stage RKN method for LODEs has the property of First Same As Last (FSAL). Numerical results proved that the methods with phase-fitted and amplification-fitted are much more efficient than the existing methods with the same algebraic order Next we used the RKN methods for solving Hyperbolic partial differential equations (PDEs) that is the second order wave equations. We very well know that the second order PDEs can be converted to second order linear ODEs using the methods of lines. Thus we applied the RKN methods for LODEs to solve the resulting second order linear ODEs. Numerical results show that the RKN methods for LODEs are accurate and reliable for solving the second order wave equation. As a conclusion, in this thesis, we have derived phase-fitted and amplification-fitted RK and RKN methods for solving first and second order oscillatory problems. The phase-fitted and amplification-fitted RK method is also applied to first order fuzzy differential equations (FDEs). The non fitted RKN method for LODEs is also used for solving hyperbolic partial differential equations. Numerical results show that all the methods are more accurate then the existing methods in the secientific literature.