Browsing by Author "Wimaladharma, N. A. S. N."

Now showing 1 - 3 of 3

An approximate solution to Lane-Emden equation of the polytrophic index three by using Differential Transform Method
(Faculty of Science, University of Kelaniya, Sri Lanka, 2021) Aththanayaka, A. M. S. K.; Wimaladharma, N. A. S. N.
Lane-Emden equation is a second order dimensionless non-linear ordinary differential equation which can be used to describe internal structure of a star, the thermal behaviour of a spherical cloud of gas, isothermal gas spheres etc. Self-gravitating spheres of plasma, such as stars, can also be described approximately by using these equations. Lane-Emden equation was solved by using Adomian Decomposition Method (ADM), Homotopy Analysis Method for some values of polytrophic index n. There are exact, analytical solutions for Lane-Emden equation in particular values n = 0, 1, 5. Since its non-linearity, the exact solutions cannot be found easily. Differential Transform Method (DTM) is an iterative method with a Taylor series solution gives good approximation in very small region. DTM can be applied for both linear and nonlinear nth derivative functions. In this research, a numerical solution to Lane-Emden equation with n = 3 has been found by using Differential Transform Method. To increase the range of convergence of the solution, the Pade approximation has been applied. Pade approximation is a ratio of two McLaurin’s expansion of the polynomials. The obtained solution for Lane-Emden equation has been compared with the solutions obtained by using the Fourth Order Runga-Kutta (RK4) method, ODE45 and Forward Euler method, which are effective and accurate methods for solving differential equations. The Einstein-Maxwell equations for a static spherical distribution of matter which is called Electrically Counterpoised Dust (ECD) under gravitational attraction and electrical repulsion can be simplified to the Lane-Emden equation when n = 3. It has been shown that the mass of a sphere of electrically counterpoised dust is an increasing function of its radius and it has a maximum value. Since the solution obtained gives us a physically acceptable result, it can be justified that the obtained solution using DTM is acceptable and gives better approximate solution with the form of a polynomial for linear and nonlinear differential equations.
Differential transform method for an immunology model of HIV
(4th International Research Symposium on Pure and Applied Sciences, Faculty of Science, University of Kelaniya, Sri Lanka, 2019) Silva, M. B. G. M. K.; Peiris, C. M. B. I. N.; Wimaladharma, N. A. S. N.
Human Immunodeficiency Virus (HIV) mainly attacks a person’s immune system. The virus destroys 𝐶𝐷4+𝑇 cells, which mainly fight against the infection. As a result, the probability of facing the risk of various deadly infections increases and sometimes it leads to a cancer due to the weakening of the immune system. The main objective of the research is to solve a system of ordinary differential equations for a dynamic model of HIV using semi numerical analytical method, namely differential transform method (DTM). The solutions, which were obtained from DTM were compared with the solutions of modified Euler method and forth order Runge Kutta (RK4) method. Moreover, Pade approximation was applied for DTM. Pade approximated solutions were obtained by using a limited number of coefficients of solutions of power series given by DTM. The results of the research show that DTM is an efficient method to solve systems of nonlinear ordinary differential equation such as dynamic model of HIV. The solutions well behaved for small time intervals. Hence, the Pade approximation was applied with DTM in order to obtain accurate solutions for large time intervals.
Thin shell model for Majumdar Papapetrou spacetimes.
(International Research Symposium on Pure and Applied Sciences, 2017 Faculty of Science, University of Kelaniya, Sri Lanka., 2017) Dilini, N. I.; Wimaladharma, N. A. S. N.
Einstein – Maxwell field equations are nonlinear partial differential equations which are difficult to solve. Therefore different assumptions are needed to solve them. Also it is very hard to find properties of the matter distributions with black holes. An exact solution to Einstein-Maxwell filed equations describing gravitational fields of the extremely charged thin spherical shells has been found in a Majumdar Papapetrou spacetime. The boundary conditions are applied considering the facts that the metric must be continuous across the shell and the absence of matter in the thin shell and outside of the shell. The metric for the exterior vacuum region of the thin shell is in the same form of conventional Extreme Reissner Nordstorm (ERN) metric which describes the exterior region of a black hole. Therefore, by replacing the black hole by a thin shell so that the centre of the thin shell is on the point of existence of black hole, the singularities of ERN metric can be removed in a Majumdar Papapetrou spacetime. This process has been generalized for N-ERN black holes with any finite number of black holes in Majumdar-Papapetrou spacetimes. In the case of two ERN black holes, the matter densities of each shell which were located on the points of singularities have been calculated. Two spherical shells with different radii and center locations are considered. Calculating the redshift of a light pulse emitted at a point on the interior flat region of the thin shell as observed by an observer at infinity, it is shown that the solution is physically acceptable.