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ABSTRACT
This project describes a differential equation of radioactive decay is numerically solved using the fourth order RungeKutta method. In order to properly estimate the quality of such methods, knowledge of the accuracy of the decay simulation is required. Here we consider the use of an EXCEL spreadsheet to tackle these drawbacks. In doing so, we employ the concept of relative row, relative column and fixed column in the spreadsheet to obtain the solution of systems of ODEs by the RK4 method. it is found that the way suggested here is faster than applying a scientific calculator and the solution obtained is significantly more accurate. I have numerically obtained the number of undecayed nuclei as a function of time. I haved displayed the numerical values graphically as well as in the form of data tables.
CHAPTER ONE
BACKGROUND STUDY
1.0 INTRODUCTION
Numerical solutions to engineering or science problems have historically been carried out using procedural programming languages. This is not efficient from a pedagogical perspective because students typically must put more effort into learning the language itself than they put into solving problems (Qureshi et al., 2013). For example, the numerical solution of a boundary value problem in one dimension using finite difference techniques generally involves the creation of a system of linear equations and the conversion of that system into an equivalent matrix equation that then can be solved. Many students find this process confusing, so, for instance, a simple change such as modifying the boundary conditions often takes substantial effort to incorporate into a working solution.
The difficulty here is that students become bogged down in forcing the algorithm to fit a structure required by the procedural language, rather than implementing the change in a more natural way. Modern computational tools can alleviate this difficulty, easing the programming effort required and allowing students to spend more time focusing on the performance of the algorithms and on the behavior of the resulting solutions (Chandio and Memon, 2010). Students are able to implement algorithms in a more convenient format, removing some of the steps typically required in reaching a solution and thus allowing more effort to be spent comparing various algorithms and studying the behaviors of the equations themselves (Abraham, 2007).
One example of a tool with which equations are easily solved is the spreadsheet,which is particularly wellsuited to the numerical solution of both differential and integral equations.
There are many physics problems that involve first order Odes. For example resistance, inductances, electrical circuits and radioactive decays (Boyce and DiPrima, 2001). Ordinary differential equations also appear in numerous problems in population biology and engineering giving mathematical descriptions of some phenomena. The numerical analysis of differential equations describes the mathematical background for understanding numerical methods giving information on what to expect when using them.
For studying numerical methods as a part of a more general course on differential equations, many of the basic ideas of the numerical analysis of differential equations are tied closely to theoretical behavior associated with the problem being solved (Glaser and Rokhlin, 2009). Differential equations can describe nearly all systems undergoing change and are essential parts of many areas of mathematics, from fluid dynamics to celestial mechanics. They are used by mathematicians, physicists and engineers to help in the designing of everything from bridges to ballistic missiles.
Ordinary Differential Equations (ODEs) are one of the most important and widely used techniques in mathematical modeling. However, not many ODEs have an analytic solution and even if there is one, usually it is extremely difficult to obtain and it is not very practical ( Amen et. al., 2004).
1.2 AIM AND OBJECTIVES OF STUDY
1.2.1 AIM
To solve the radioactive decay equation numerically using the rungekutta algorithm and compare same with its analytical solution.
1.2.2 OBJECTIVES
The specific learning objectives are:
(a) To develop a microsoft excel code to implement the RK4 algorithm for the radioactive decay equation.
(b) To plot the numerical solution sidebyside the analytical solution.
(c) To compare graphically relationship between the analytical and the numerical solutions.
1.3 LIMITATION
The limitation of analytical techniques to solve the nonlinear differential equations has impeded the use of numerical methods for obtaining an approximate solution of the problem (Garewal, 2000).
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