A Study of Single and Multi Step Methods To Solve Differential Equations

Abstract

Here the task was to study single and multi step methods to solve differential equations. As Rungc-Kutta method. a single step method has some property to represent a larnily so attention has been given to that method. It was found that to establish the method of any order the number of unknowns are more than the number of available equations. Thus there are options to choose certain values as someone wants (keeping in mind about the conditions). The opportunity has been taken and FIVE formulas of different orders, two fifth, two sixth and one seventh, has been proposed. Problems are solved to verify their capability and strength and it is found that the percentage errors with respect to the exact values are comparable in magnitude with respect to some available same order methods. In case of multi step methods, extensions in both explicit and implicit methods in terms of order are done. It is done keeping in mind that the extension in order will reduce the error. Extensions in Adams- Bash forth and Adams-Moulton formulas up to TENth order are done and with eighth order of both of them a problem is solved to demonstrate the strength of the extension of these predictor-corrector methods.