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 discovery of extrasolar planets outside our solar system and around other stars is now well underway. In the presented paper, calculations of some physical properties for confirmed exoplanets have been done. We have estimated physical properties such as the semi-major axis for potentially habitable exoplanets, the mass of planets by applying Kepler's third law around the mass of solar, Jupiter, and Earth-mass, stellar luminosity, habitability zone for the inner center and outer regions, radial velocity amplitude, planetary equilibrium temperature (PET) or effective radiation emission temperature and planet density. The mass-radius (MR) relationship of planets was investigated for potentially habitable exoplanets of three different groups of extrasolar planets: Subterran (Mars-size), Terran (Earth-size), and Superterran (Super-Earths or Mini-Neptunes); introduced in PHL, and found well coefficient values for each group. The minimum and maximum values for the mass and radius of exoplanets have been selected mass between 0.1 to10 Earth mass, radius between 0.4 to 2.5 Earth radius. The same MR relationship has also estimated the same properties for a larger number of confirmed exoplanets with a mass and radius of mass between 0.1 to100 Earth mass and radius between 0.4 to 15 Earth radius, respectively, resulting their classification within 7 groups of mass and radius, with good coefficient values for each group. This is a new, possible cataloging that may need more effort for concluding a better understanding of the properties and varieties of the exoplanets.

The relationships between solar activities (sunspots, solar 10.7cm radio flux, solar irradiance, and solar proton events) and local earthquakes investigated in this paper. The geographical location of study is New Zealand area. All earthquakes data have been chosen for , from first of 1970 to Jun 2012. The study reveals the following conclusions: 1) The total numbers of earthquakes strongly show annually an increasing in number of earthquakes in New Zealand from 42 years ago. 2) The maximum earthquakes occur frequently around the minimum years of solar activities, 3) The maximum earthquakes occurs in minimum years of sunspots number with a good correlation coefficient. 4) The maximum earthquakes occur in the minimum solar cm radio flux with strong correlation coefficient.