Solution of First Order Differential Equation Using Numerical Newton’s Interpolation and Lagrange
Chapter One
Objective of the study
This research work will give a vivid look at differentiation and combine newton’s interpolation and Lagrange methods to solve the first-order differential equations.
CHAPTER TWO
FUNDAMENTALS OF CALCULUS
FUNCTIONS OF SINGLE VARIABLE AND THEIR GRAPHS
Functions are essential ingredients in the study of calculus. Their absence means an incomplete study of this branch of mathematics which is more or less the care of many aspects of mathematics.
Calculus by its basis is the notion of correspondence for example, the surface area (A) of a sphere relates to its radius by the formula
A= 4πr2
The sphere volume (V) of a given mass of gas is related to the pressure (P) of the gas.
V=
These examples give us an idea what a function is. Conclusion from them could imply that if the value of one quantity, say y depends on the value of another quantity, say x, then for every value of x there corresponds one and only the value one of y.
On this basis we say that y is a function of x. thus, A is a function of r and V is a function of p as above.
Though, the same element of y may correspond to different element of x. for example, two different books may have the same number of pages.
Definition 2.0: A function f from a set x to a set y is a correspondence that assigns to each element x of X a unique element y of y.
The set of values of X is called the domain of the function. The element y is called the image X under f and is denoted by f (x). the range of the function consists of all images of elements of X.
We can represent these explanations by means of diagram.
Values of f at each of the points indicated (or the images under f) can be obtained by more evaluation
Example2.1 f (x)= x3 +4x-3
F (2)= 23 + 4 (2) -3
=8+8-3
=16-3
=13
F(-3) =(-3)3+4(-3)-3
=-27-12-3
=-42
F (c) = c3+4 (c) -3
=c3+ 4c-3
CHAPTER THREE
DIFFERENTIATION
Differentiation as a limit of rate of change of elementary function
In this section we shall discuss about the gradient of a line and a curve, gradient function also called the derived function and finally, the main topic differentiation as a limit of rate of change of elementary function.
Gradient: Straight line and curve
As we rightly said in the study of rate of change of a function, the gradient of a line is measured by taking the ratio of the increase in Y and the increase in x in moving from one point to another on the line. Such that if (x1, y1,) and (x2, y2,) are two point on a line, the gradient is then taken as.
CHAPTER FOUR
Newton’s interpolation and Lagrange method to solve first order differential equation
The study will combine both Newton’s interpolation method and Lagrange method to solve first order differential equation. Since the problem is an initial value problem (IVP), the first value for y is available. We will use newton’s interpolation to find the second two terms then use the three values for y to form a quadratic equation using Lagrange method as follows;
Newton’s interpolation method
CHAPTER FIVE
DISCUSSION
The method that has been used gives results very close to the exact value. This is noted by the percentage error that is very minor. The method is very accurate and easy to use after getting the quadratic equation. Thus, one can get the value of y at any value of x without necessary getting preceding values of y.
Conclusion
Numerical methods used such as Runge kutta, Euler, Taylor series methods etc. are cumbersome and have a bigger percentage error. I therefore recommend the use of combined Newton’s interpolation and Lagrange method to solve first order differential equation.
REFERENCES
- Aashikpelokhai U.S.U et al: An introductory course in higher Mathematics Calculus, Vector and Mechanics 2009 PON publisher Ltd, Edo State, Nigeria.
- Boyer Carl the history of calculus New York, Dover publication 1949.
- Frank Ayres Jr. et al: Schaum’s outlines fifth edition (2009) The McGraw-Hill publishing Companies USA
- George B Thomas Jr. et al: Calculus and Analytic Geometry 9th edition 1998 Addison-Wesley publishing company USA
- Harper et al: The calculus with Analytical Geometry fifth edition 1986 Harper and Row publishers. New York
- Mary W. Gray: Calculus with finite mathematics 1972 Addison-Wesley Publishing company London
- Murray R Epiegel: Schaum’s Outline 1971 Advance Mathematic for Engineers and Scientist. Mc Graw-Hill, New York
- Raymond A. Barnett et al: Applied Calculus 3rd edition 1987, Dellen Publishing company San Francisco California
- Stroud K.A. et al : Engineering Mathematics sixth Edition 2007 Published by: Palgrave Macmillan.
