SOLVING INITIAL-VALUE PROBLEM BY COMPARING LAPLACE TRANSFORM AND AN NUMERICAL PREDICTOR-CORRECTOR METHOD

Abstract

Different analytic and numerical methods have been proposed to solve differential equations. The goal of this paper is to expose the reader to modern computational tools for solving differential initial-value real problems of the form . The flow of electromagnetic forces, the temperature of a cup of coffee, the population of species as well as numerous other applications can be expressed as first order differential equations.  Process of modeling or formulating a mathematical model of a real-world problem through either: intuitive reasoning about the phenomenon or a physical law based on evidence from experiments and often takes the form of a differential initial-value problem. It is noticed that changes occur, and we want to predict future behavior on the basis of how current values changes, and the model of growth population is based on the assumption that the population grows at the rate proportional to the size of the population. Many differential initial-value problems cannot be solved using analysis and because of that a numeric approximation to the solution is often sufficient. The algorithm of FABM method can be used to compute such an approximation of the problem. The idea is to compare the solutions of the taken problem, analytically and numerically using Adams-Bashforth-Moulton (FABM) like a predictor-corrector numerical method with solutions obtained by Laplace Transform (LT) like an analytical method.  Laplace Transform like a powerful tool in applied mathematics and engineering, can be especially useful when dealing with piecewise-defined forcing functions. A predictor-corrector method is a set of two equations for . The first equation, called the predictor, is used to predict (obtain a first approximation) to , the second equation, called the corrector, is then used to obtain a corrected value (second approximation) to , it depends on the predicted value. Approaches from FABM are based on the analytical property that the initial value problem is equivalent to the Volterra integral equation, in the sense that a continuous function is a solution of the initial value problem if and only if it is a solution of it. For calculations we chose a population problem with fixed initial condition and growth rate with changing the rate at which people are being added to or subtracted from the population during the immigration and emigration. The results are shown by time series solution, tables and phase portraits of the model taken by algorithms coded on Mathematica Package. Phase portraits serve as a useful tool in understanding the behavior of solutions of the problem, depending on variable rate depended on immigration and emigration of the population. Illustrative example can be included to demonstrate the validity and applicability of the present techniques.