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California State University, Northridge Origin CALIFORNIA STATE UNIVERSITY, NORTHRIDGE ORIGIN, DEVELOPMENT OF SOLUTION METHODS, Ql AND APPLICATION OF DI.l!,JPERENCE EQUATIONS. A thesis submitted in partial satisfaction of the requirements for the degree of Master of Science in Mathematics. by Ronald P. Thomas June, 1977 p ' The Thesis of Ronald P. Thomas is approved: Tung---------~------------ Po ~in, Chairman California State University, Northridge ACKNOWLEDGEMENTS I would like to express my appreciation to the following who were instrumental in making this achievement possible: To the CSUN Department of Mathematics for allowing me to serve as a Graduate Assistant and as a Part Time Instructor and to their faculty for their expert instruction in mathematics. To Mr. Neil H. Jacoby, Jr., an Engineer and neighbo~ who helped me over the years master mathematical skills. And most j_mportantly To Dr. T.P. Lin, who was the professor for Math Models when I was enrolled in the Spring 1975 semester, for his inspiring teaching and for bringing to my atteLtion the topic of this thesis, for his time given to me in guiding the preparation of th~s thesis. ------------TABLE OF CONTENTS Title Page i Approval Page ii Acknowledgements iii Table of Contents iv Abstract v Part I - Introduction 1 - 3 Part II - Origin of Di.f.ference Eq'uations 4 - 6 Part III - Development o.f Methods o.f Solution 7 - 27 Part IV - The Application o.f Di.f.ference Equations 28 - 58 Bibliography 59 ABSTHACT ORIGIN, DEVELOPfJ!ENT Olt' SUJJUTION METHODS, AND APPLICATION OF DIFFEREnCE EQUATIONS. ·by Ronald P~ Thomas Master of Science - Mathematics This paper presents an elemetry study of difference equations. Difference equations are defined and their origin in the works of Brook Taylor and James Stirling, circa -- 1715, is viewed in the perspective of broad developments ... _:, . ~~:~.~~),' of mathematics to that time. Methods of solving various types of difference equations are presented. These include linear difference equations with constant coefficients an·d with variable coefficients. Difference equations are applied to problems in areas where work is done in discrete intervals. Discussion centers on examples of their application in sequences of numbers, systems of difference equations (pythagorean triples), finding the zeros of certain polynomials, biological-population dynamics and in numerical approximation of difference equations. PART I - INTRODUCTION Difference equations are equations of the form F (X' y' !J. y' JS..y' NY' ... }),"" j ) = 0 (1) where x, .Y are descrete variables and (y, /).y, •. ) are diff~rences of various orders. Equati~n (1) can be written in the form of G(x, f(x), f(x+h), f(x+2h), ... f(x+nh)) = 0 (2). where y = f(x) is an unknown function and h is a constant. The forms of difference equations shown in (1) and (2) are not the usually encountered forms. Subscript notation is frequently used. Let- x =a+ kh, a is a constant and yk = f(a+kh). Now equation (2) can be written as H k, yk' yk+l' •••• , yk+n = 0 (3) It is this form that difference equations will take in this thesis. Y = f(x) is a solution to equation (2) or Yk =f(a+kh) is a solution to equation.(3) if the -functions will reduce the given equation, after proper substitution, to an identity. There are many different types of difference ·equations. The various types depend on the configuration of subscripts, coefficients, constants or functions that are imbedded in the equation. Of particular, though not exclusive, interest in this thesis are Linear Difference Equations. 2 The reader who is familiar with differential equations will see many similarities between them and difference equations. This is as it should be. In defin­ ing a difference a finite increme~t h was added to the independent variable. Ih difference equations this h is kept finite. In differential equations this h is allowed to approach 0. Introduction to Applications The application of difference equations are many. The author's first encounter with them was in relation to sequences of numbers. They are used as recursion formulas. By applying the methods of solution given in Part III of this thesis. it is possible to find the value of a number in a sequence given the position of the number in the sequence (3rd term, lOth term, etc.). Difference equations are also used in finding the zeros of polynomial functions, numerical approximations to differential equations, pop­ ulation studies-at finite time intervals, and other pro­ blems where the interval of difference does not have to approach zero. Notations Used in Thesis The following are examples of notations used in this thesis: 3 Yk ~s used to indicate the kth value of the variable Y. Subscripts used on the other variables have a similar meaning. y(h) is used to indicate the solution to the k homogeneous difference equation or to the homogeneous part of the equation. y~P) is used to indicate the particular solution to a non-hornogenedus difference equation. yCl) is used to indicate the first solution to a k difference equation • . (2) yk the second solution to an equation. y(n) t·he Nth solution to an equation. k p ' PART II - THE ORIGIN. Oli' DIFFERENCE EQUATIONS A description of the various branches of mathematics existing to 1715 is being provided so that the reader will have knowledge of the mathematical methods that were available. 1. Number systems were developed as a means of expressing "different quanti ties. Various bases w.ere used. 2. Arithmetic, which originated in Babylonia and Egypt, supplied a way of relating numbers by certain rules; 2 ie: +, -, x, -, I , ( ) , etc. 3. Algebra, which had its beginnings in Babylonia, is concerned with solving problems via equations (linear, second and some 3rd and 4th degree). Also of concern were problems of' applied geometry. ·4. Geometry (Babylonian, Pythagorean, Euclidean) had its beginnings as a concern for measure~ent of land. Many problems expressed in geometric terms were really algebraic in.nature. Various geometric figures were considered. It was the formalization by the Greeks (Euclid in particular) that created geometry as a mathematical system of its own. This ca~e about as many people were concerned as to why certain things were ~rue (a circie is bisected by its diameter). 5. TrigonQmetry had its beginnings with the early Baby- lonian astronomers who were concerned with various 4 5 astronomical phenomena. The period of th~ mbon's rev­ olution about the earth and the inclination of its orbit. were two of these.· The Greeks used trigonometry as a means of locating places on earth (longitude and latitude). t1odern trigonometry, complete with the usual trigonometric functions, was developed by Euro­ .pean mathematicians; arnong'tbem Pitcus, Rheticus and Fincke. A table of trigonometric values was published by Rheticus in 1583. 6 .. Analytic Geometry, sometimes called coordinate geometry, was developed primarily by Descartes in 1637. He applied algebra to the study of geometric theorems and problems. Uut of this came some of the coordinate systems we use today. 7. The Calculus was developed by Newton and Leibniz in the latter 1600's. The-work is based on the ideas and concepts of limits, convergence and ·infinitesimals of a function. Also the rates of change of a function at a point, slope of a tangent to a curve at a point were considered. With the development of mathematics up through the calculus of Newton and Leibniz certain problems were not ye·t. solvable. Problems such as interpolating for inter­ mediate values in a table or finding a function to fit observed data derived from sor.J.e experiment. The work of 6 Brook Taylor and James Stirling provided the means for solving these problems. Br0ok Taylor's work "Methodus Incrementorium" con­ cerned itself with functions and rates of changes of the function for a given change, or increment, of the inde­ pendent variable. Also considered is how these differences interrelate. This work led to'his development of the series expansion of a function about a point which now bears his name. James Stirling's work "Methodus Differentailis" concerned itself with infinite series, summation of those series and interpolation formulas. The work with infinite series dealt with ways of expressing series, differences between terms and dj._fferences between vari-ous partial .sums. Series of various types were considered. The work on interpolation formulas dealt with ~ays of finding values between those in a table of a mathematical relation Div~ded differences were the end result of this work. Since then Lagrange, Euler and others have made use of difference equations in developing techniques of num­ erical analysis. With the advent of computers, some tech­ niques in nucierical analysis involving difference equa­ tions can now be efficiently performed. This has.caused an increased interest in difference equations. PART III - DEVELOH'IEW.r OF f1IETHODS OF SOLUTION TO DIJ?l!,EHENCE EQUATIONS The method os solution to various types of difference equations will be presented and proved in this part. Of primary interest w~ll be Linear Difference equations with constant coefficients. Also considered will be Linear Difference equations with variable coefficients. Section 1. Linear Difference Equations with Constant Coefficients. Defn: A difference equation is linear over a set S .of integers if it can be written in the form where f , f , ••• fn' G are defined functions of k 0 1 : vk Es. Equation (4) is said to be a linear difference equation with constant coefficients if ·all f fn 0 ·are constant functions. Defn: A function Yk = f(k) is a solution of a .difference equation over a setS if the values of Yk reduce the difference equation to an identity over S.
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