11 University of Sindh Journal of Information and Communication Technology (USJICT) Volume 2, Issue 1, January 2018 ISSN-E: 2523-1235, ISSN-P: 2521-5582 © Published by University of Sindh, Jamshoro Website: http://sujo.usindh.edu.pk/index.php/USJICT/ Modification of Heun’s Iterative Method for the Population Growth Rate Problems Aliya Pirzada, Asif Ali Shaikh,Syed Feroz Shah Mehran University of Engineering and Technology Jamshoro, Pakistan aliyapirzada@hotmail.com, asif.shaikh@faculty.muet.edu.pk, feroz.shah@faculty.muet.edu.pk Abstract: In this paper Modified Heun’s algorithm of Heun’s algorithm is presented with different formulations which are applied on exponential Population growth rate problems. In Heun’s algorithm the average of two formulations is considered as A.M mean, where as in Modified Heun’s algorithm G.M and Modified Heun’s algorithm H.M are also considered as averages which are also applied on exponential population growth rate problems respectively. Comparison between numerical results of both Modified Heun’s algorithm and existing Heun’s algorithm shows that Modified Heun’s algorithm of Heun’s algorithm is more convergent then Heun’s algorithm. Both algorithms will be analyzed by different errors for the convergent purpose. Keywords: Exponential Population Growth Rate problems, Heun’s Algorithm, Modified Heun’s Algorithm, Convergence, Error. I. INTRODUCTION Differential equation arise from many problems in oscillations of mechanical and electrical system, bending of beam, conduction of heat, velocity of chemical reaction etc, and such as play a very important role in all modern and scientific and engineering studies. Differential equations whether ordinary, partial or algebraic; that evolves change of some variables with respect to other variables. Mathematical models are very useful to solve real word problems, most of differential equations are difficult to solve analytically, then it must rely some numerical method to solve them, there are number of numerical method which are used for differential equation to solve them numerically, like Euler’s method Runge Kutta method Adms Bash fourth method etc. In this paper, we solve exponential population growth rate problems using Heun’s algorithm and modified Heun’s algorithm. There are many excellent and exhaustive text on this subject that may be consulted such as, Euler’s method is presented from the point of view of Taylor’s algorithm and Runge Kutta method which are used on ordinary differential equation for stability, accuracy, consistency, and convergence [1]. There methods are present to solve initial value, problems, first order Euler’s, second order Heun’s and rational Block method. The numerical results shows the block method is more convergent then both methods [2]. A new nonlinear adaptive numeric solution for ordinary differential equation with initial conditions the main features is to implement nonlinear polynomial expansions in a nural network-like adaptive framework [3]. There are comparative studies of numerical methods for the numerical methods namely; Runge Kutta method, Euler’s method and an implici t linear multistep method of order six which are used for ordinary differential equation. II In these three methods the implicit linear multistep method of order six is more accurate and converges and faster than both methods [4]. Here is suggested Taylor’s series expansion algorithm of numerical solution for ordinary differential equation, which are competes strongly with other existing algorithm [5]. Adams- Bashfourth method, Runge Kutta method, Adams- Moultan method which are used for ordinary differential equation and stiff problems for consistency, stability, and convergence [6]. There exist a huge number of numerical methods that iteratively construct approximation to solution of ordinary differential equation [7]. A numerical method namely, Stochastic terms involves a numeric are of pseudo random number for solving ordinary differential equations. The present work is very useful for exponential population growth rate problems and shows convergence in numerical results. II. NUMERICAL METHOD Numerical techniques forms an important part of solving initial and boundary value problems in ordinary differential equation; most important in case where there no closed form solution. Here is present some numerical algorithm Existing Heun’s algorithm and modified Heun’s algorithm of Heun’s algorithm. Population Growth Rate Differential Equation Problems. Differential equation dp/dt=kp of the Growth rate problems, p(t) be the quantity that increase with time