Abstract – Ordinary Differential equation with Initial Value Problems (IVP) frequently arise in m... more Abstract – Ordinary Differential equation with Initial Value Problems (IVP) frequently arise in many physical problems. Numerical methods are widely used for solving the problems especially in case of numerical simulation. Several numerical methods are available in the literature for solving IVP. Runge-Kutta (which is actually Arithmetic Mean (AM) based method) is one of the best commonly used numerical approaches for solving the IVP. Recently Evans[1] proposed Geometric Mean (GM) based Runge-Kutta third order method and Wazwaz [2] proposed Harmonic Mean (HM) based Runge-Kutta third order method for solving IVP. Also Yanti et al. [3] proposed the linear combination of AM, HM and GM based Runge-Kutta third order method. We extensively perform several experiments on those approaches to find robustness of the approaches. Theoretically as well as experimentally we observe that GM based and Linear combination of AM, GM and HM based approaches are not applicable for all kinds of problems. To overcome some of these drawbacks we propose modified formulas correspond to those AM and linear combination of AM, GM, HM based methods. Experimentally it is shown that the proposed modified methods are most robust and able to solve the IVP efficiently.
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Papers by Rafiqul Sumon