M. Khalid

Numerical Solution of Sixth-Order Differential Equations Arising in Astrophysics by Neural Network

In the current paper, a neural network method to solve sixth-order differential equations and their boundary conditions has been presented. The idea this method incorporates is to integrate knowledge about the differential equation and its boundary conditions into neural networks and the training sets. Neural networks are being used incessantly to solve all kinds of problems hailing a wide range of disciplines. Several examples are given to illustrate the efficiency and implementation of the Neural Network method.

Solving Polluted Lakes System by using Perturbation-Iteration Method

Water pollution is a major global problem that requires ongoing evaluation and revision of water resource policies at all levels, in order to create a healthy living environment. Differential equations are an effective way to analyze such situations. In this paper a system of linear equations with interconnecting pipes is considered for analyzing the pollution of system of lakes through differential equations. Perturbation-iteration method is used to compute an approximate solution of three input models i.e. periodic, linear step model and exponentially decaying model. The fourth order RungeKutta method (RK4) numerical solution of the lakes system problem is used as a reference to compare with the analytical approximations showing the high accuracy of the results.

Delta : Polynomial Approximation of Time Period 1620–2013

The difference between the Uniform Dynamical Time and Universal Time is referred to as (delta ). Delta is used in numerous astronomical calculations, that is, eclipses,and length of day. It is additionally required to reduce quantified positions of minor planets to a uniform timescale for the purpose of orbital determination. Since Universal Time is established on the basis of the variable rotation of planet Earth, the quantity mirrors the unevenness of that rotation, and so it changes slowly, but rather irregularly, as time passes. We have worked on empirical formulae for estimating and have discovered a set of polynomials of the 4th order with nine intervals which is accurate within the range of ±0.6 seconds for the duration of years 1620–2013.