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CHAPTER ONE
1.0 INTRODUCTION
1.1 BACKGROUND OF STUDY
Differential equation is one of the major areas in mathematics with series of method and solutions. A differential equation as for example u(x) = Cos(x) for 0 <x< 3 is written as an equation involving some derivative of an unknown function u (E.W Weisstein, 2004). There is also a domain of the differential equation (for the example 0 <x< 3).
In reality, a differential equation is then an infinite number of equations, one for each x in the domain. The analytic or exact solution is the functional expression of u or for the example case u(x) = sin(x) + c where c is an arbitrary constant. This can be verified using Maple and the command dsolve(diff(u(x),x)=cos(x)); . Because of this non uniqueness which is inherent in differential equations we typically include some additional equations. For our example case, an appropriate additional equation would be u(1) = 2 which would allow us to determine c to be 2 − sin(1) and hence recover the unique analytical solution u(x) = sin(x)+2 − sin(1). Here the appropriate Maple command is dsolve(diff(u(x),x)=cos(x),u(1)=2);. The differential equation together with the additional equation(s) are denoted a differential equation problem.
Note that for our example, if the value of u(1) is changed slightly, for example from 2 to 1.95 then also the values of u are only changing slightly in the entire domain. This is an example of the continuous dependence on data that we shall require: A wellposed differential equation problem consists of at least one differential equation and at least one additional equation such that the system together have one and only one solution (existence and uniqueness) called the analytic or exact solution (Joshn Wiley, 1969); to distinguish it from the approximate numerical solutions that we shall consider later on. Further, this analytic solution must depend continuously on the data in the (vague) sense that if the equations are changed slightly then also the solution does not change too much. The study in this regard wishes to determine the solution of first order differential equation using numerical Newton’s interpolation and Lagrange.
1.2 STATEMENT OF RESEARCH PROBLEM
What really instigated the study was due to the need to solve first order differential equations using numerical approaches. Most of the researches on numerical approach to the solution of ordinary differential equation tend to adopt other methods such as Runge Kutta method, and Euler’s method; but none of the study has actually combined the newton’s interpolation and lagrange method to solve first order differential equation. However the study will try to use Newton’s interpolation and Lagrange to solve the problems below:
1. Find the polynomial interpolating the points
x
1
1.3
1.6
1.9
2.2
f(x)
0.1411
0.6878
0.9962
0.5507
0.3115
Where f(x) = sin (3x), and estimate f (1.5)
2. The following below will be solved using Newton’s interpolation method
Evaluate f (15), given the following table of values:
x
10
20
30
40
50
Y=f(x)
46
66
81
93
101
3. Find Newton’s forward difference, interpolating polynomial for the following data:
x
0.1
0.2
0.3
0.4
0.5
Y=f(x)
1.4
1.56
1.76
2
2.28
1.3 AIMS AND OBJECTIVES OF STUDY
The main aim of the study is to determine the solution of first order differential equation using numerical Newton’s interpolation and Lagrange interpolation. Other specific objectives of the study include:
1. to determine the difference between Newton’s interpolation and the Lagrange interpolation
2. to examine the accuracy of Newton’s and Lagrange interpolation methods in solving first order differential equations
3. to investigate on the factors affecting the use of Newton’s interpolation and Lagrange interpolation method
1.4 RESEARCH QUESTION
The study came up with research question so as to ascertain the above stated objectives of the study. The research questions for the study are stated below as follows:
1. What is the difference between Newton’s interpolation and the Lagrange interpolation?
2. What is the accuracy of Newton’s and Lagrange interpolation methods in solving first order differential equations?
3. What are the factors affecting the use of Newton’s interpolation and Lagrange interpolation method?
1.5 SIGNIFICANCE OF STUDY
The study on solution of first order differential equation using numerical Newton’s interpolation and Lagrange will be of immense benefit to the entire mathematics departments in Nigeria, lecturers and students that wishes to carry out similar research on the above topic as it will proffer numerical solution to first order differential equation using Newton’s interpolation and Lagrange. The study will also discuss on the limitations of the above methods. Finally the study after completion will contribute to the existing literature on numerical solution to first order differential equations using Newton’s interpolation and Lagrange methods
1.6 SCOPE OF STUDY
The study is limited to first order differential equation using numerical Newton’s interpolation and Lagrange. The study will cover numerical solution to first order differential equation using Newton’s interpolation and Lagrange
1.7 LIMITATION OF STUDY
Financial constraint Insufficient fund tends to impede the efficiency of the researcher in sourcing for the relevant materials, literature or information and in the process of data collection (internet).
Time constraint The researcher will simultaneously engage in this study with other academic work. This consequently will cut down on the time devoted for the research work.
1.8 DEFINITION OF TERMS
First order differential equation: A firstorder differential equation is an equation
= ƒ(x, y)
in which ƒ(x, y) is a function of two variables defined on a region in the xyplane. The equation is of first order because it involves only the first derivative dy dx (and not higherorder derivatives). We point out that the equations
Interpolation: the insertion of something of a different nature into something else
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