Project description

  • PROJECT TITLE: SOLUTION OF FIRST ORDER DIFFERENTIAL EQUATION USING NUMERICAL NEWTON’S INTERPOLATION AND LAGRANGE.
  • DEPARTMENT: MATHEMATICS
  • PRICE: 3000 | CHAPTERS: 5 | PAGES: 57 | FORMAT: Microsoft Word, PDF | | PROJECT DELIVERY: Instant Download »

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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 dissolve (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 well-posed 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.

1.3 objective of the study

    This research work will give a vivid look at differentiation and combined the newton’s interpolation and Lagrange method to solve first order differential equation.

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1.4 Significance of The Study

The significance of this study cannot be over emphasized especially in this modern era where everything in the entire world is changing with respect to time, because the rate of change is an integral part of operation in science and technology, hence there is need to ascertain the origin of calculus and its application.

Finally, the goal of this work is to review the application of differentiation in calculus.

1.5 LITERATURE REVIEW

Calculus, historically known as infinitesimal calculus, is a mathematical discipline focused on limits, functions, derivations, integrals and infinite series.

Ideas leading up to the notion of function, derivatives and integral were developed throughout the 17th century but the decisive step was made by Isaac Newton and Gottfried Leibniz.

Ancient Greek Precursors (Forerunners) of The Calculus

Greek mathematicians are credited with a significant use of infinitesimals.

Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-section with a cone’s smooth slope prevented him from accepting the idea, at approximately the same time. 

Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they create.

Antiphon and later Eudoxus are generally creadited with implementing the method of exhaustion which implementing the method of exhaustion which made it possible to compute the area and volume of regions and solids by breaking them up into an into an infinite number of recognizable shapes.

Archimedes of Syracuse developed this method further, while also inventing heuristic method which resemble modern day concept somewhat. It was not until the time of Newton that these methods were incorporated into a general framework of integral calculus.

          It should not be thought that infinitesimals were put on a rigorous footing during this time, however.

Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true.

Pioneers of modern calculus

          In the 17th century, European mathematicians Isaac barrow, Rene Descartes, Pierre deferment the idea of a deferment.

          Blaise Pascal, john Wallis and others discussed the idea of a derivative. In particular, in method sad disquirendam maximum et minima and in De tangetibus linearism Curvarum, Fermat developed an adequality method for determining maxima, minima and tangents to various curves that was equivalent to differentiation.

          Isaac Newton would latter write that his own early ideas about calculus came directly from formats way of drawing tangents

          On the integral side cavalierre developed his method of in divisible in the 1630s and 40s, providing a modern form of the ancient Greek method of exhaustion and computing cavalierr’s quadrate formula, the area under the curves Xn of higher degree, which had previously only been computed for the parabola by Archimedes.

          Torricelli extended this work to other curves such as cycloid and then the formula was generalized to fractional and negative powers by Wallis in 1656.

          In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly.

Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature.

          James Gregory influenced by Fermat’s contributions both to tangency and to quadrature, was then able to prove a restricted version on the second fundamental theorem of calculus in the mid -17th century. The first full proof of fundamental theorem of calculus was given by Isaac barrow.

          Newton and Leibniz building on this work independently developed the surrounding theory of infinitesimal calculus in the late 17 centuries.

Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts.

Newton provided some of the most important applications to physics, especially of integral calculus.

Before Newton and Leibniz, the word “calculus” was a general term used to refer to anybody of mathematics, but in the following years, “calculus”. Became a popular term for a field of mathematics based upon their insight.

The work of both Newton and Leibniz is reflected in the notation used today.

 Newton introduced he notation f for the derivative of function f.

Leibniz introduced the symbol  for the integral and wrote the derivative of a function y of the variable x as    both of which are still in use today         


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