No abstract

In this paper, a three-stage fifth-order Runge-Kutta method for the integration of a special third-order ordinary differential equation (ODE) is constructed. The zero stability of the method is proven. The numerical study of a third-order ODE arising in thin film flow of viscous fluid in physics is discussed. The mathematical model of thin film flow has been solved using a new method and numerical comparisons are made when the same problem is reduced to a first-order system of equations which are solved using the existing Runge-Kutta methods. Numerical results have clearly shown the advantage and the efficiency of the new method.

The analysis of the behaviors of physical phenomena is important to discover significant features of the character and the structure of mathematical models. Frequently the unknown parameters involve in the models are assumed to be unvarying over time. In reality, some of them are uncertain and implicitly depend on several factors. In this study, to consider such uncertainty in variables of the models, they are characterized based on the fuzzy notion. We propose here a new model based on fractional calculus to deal with the Kelvin-Voigt (KV) equation and non-Newtonian fluid behavior model with fuzzy parameters. A new and accurate numerical algorithm using a spectral tau technique based on the generalized fractional Legendre polynomials (GFLPs) is developed to solve those problems under uncertainty. Numerical simulations are carried out and the analysis of the results highlights the significant features of the new technique in comparison with the previous findings. A detailed error analysis is also carried out and discussed.

A new pair of embedded explicit Runge-Kutta-Nyström (RKN) methods is developed to integrate second-order differential equations of the form q = f (t, q) where the solution is oscillatory. The embedded formula has dispersion order eight and dissipation order seven for the fifth-order formula. The cost for this pair is four function evaluations at each step of integration. Numerical comparisons with several codes in the scientific literature such as RKN5(4)D, RKN5(4)B, RKN4(3)G and DOPRI5 show the efficiency of the new method developed.

Articles you may be interested inOn the derivation of second order variable step variable order block backward differentiation formulae for solving stiff ODEs AIP Conf. Abstract.A new block method that generates two values simultaneously is developed for the integration of stiff initial value problems. The method is proven to be A -stable and is a super class of the 2 -point block backward differentiation formula (BBDF). A comparison is made between the method, 1 point backward differentiation formula (BDF) and the 2 point BBDF methods. The numerical results indicate that the new method outperformed the 1 point BDF and the 2 point BBDF methods in terms of accuracy and stability. The total number of steps to complete the integration by the 1 point BDF method is reduced to half. Computation time for the method is also competitive.