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Influence of denominator functions on thesolutions of finite-difference schemesArthur L. SmithAtlanta University

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INFLUENCE OF DENOMINATOR FUNCTIONS ON THE

SOLUTIONS OF FINITE-DIFFERENCE SCHEMES

A THESIS

SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR

THE DEGREE OF MASTER OF SCIENCE

BY

ARTHUR L. SMITH

DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCES

ATLANTA, GEORGIA

APRIL 1989

ABSTRACT

MATHEMATICAL AND COMPUTER SCIENCES

SMITH, ARTHUR L. B.A. FISK UNIVERSITY, 1985

INFLUENCE OF DENOMINATOR FUNCTIONS ON THE SOLUTIONS OF

FINITE-DIFFERENCE SCHEMES

Advisor: Professor Ronald E. Mickens

Thesis dated April 1989

We investigate the influence on the solutions of

finite-difference schemes of using unconventional

denominator functions in the discrete modeling of the

derivatives for ordinary differential equations. The

derived results are a consequence of using a generalized

definition of the first derivative of a function. Two

explicit examples, the linear decay equation and the

Logistic differential equation, are used to illustrate in

detail the various solution possibilities that can occur.

ACKNOWLEDGMENTS

The work of this thesis reflects the cooperation of

several other persons who made the completion of this

research possible. I wish to thank Professor N. Warsi,

Chairperson, Department of Mathematics and Computer

Sciences, for providing me with the opportunity to do

research. I am deeply grateful to Professor Ronald E.

Mickens, advisor and professor of physics, for his

constructive criticism, suggestions, encouragement and

patience. Many thanks also go to my instructors at Atlanta

University who helped me develop a greater understanding

and appreciation of mathematics.

My research efforts were supported in part by NASA

Grant NAG 1-410.

11

TABLE OF CONTENTS

Page

Acknowledgments i i

Table of Contents iii

CHAPTER ONE: INTRODUCTION 1

1.1. Statement of the Problem 1

1.2. Summary of Thesis 3

1.3. Generalized Defin it ion of Derivative 3

1.4. Denominator Funct ions 5

1.5. Li near Stabi1ity Analysis 6

1.6. Outline of Thesis 11

CHAPTER TWO: THE LINEAR DECAY EQUATION 12

2.1. The Equation and Solution 12

2.2. Finite-Difference Models 12

2.3. Discussion 18

CHAPTER THREE: THE LOGISTIC EQUATION 20

3.1. The Equation and Solution 20

3.2. Finite-Difference Models 25

3.3. Discussion 29

CHAPTER FOUR: FUTURE INVESTIGATIONS 30

4.1. Summary 30

4.2. Partial Differential Equations 31

4.3. Extensions of Research 31

iii

APPENDIX: LINEAR DIFFERENCE EQUATIONS 33

REFERENCES 35

IV

CHAPTER ONE

INTRODUCTION

1.1. Statement of the Problem

Ordinary differential equations arise in the modeling

of many dynamic systems. However, very few of these

equations can be solved exactly and expressed in terms of a

finite sum of elementary functions. Aside from analytic

approximations to the solutions, numerical integration

techniques can prove to be extremely useful. ~ In parti-

cular, the method of finite-differences is one of the

most widely used procedures. It consists of modeling a con

tinuous system, the differential equations, by a discrete

12 13set of equations, the finite-difference equations. '

For example, the single ordinary differential equation

(ODE)

(1.1) £$ = f(x,t),

can be modeled by the scheme

(1.2) K+Xh S = f(xk,hk), k = integer,

where h = At is the step-size, the discrete time variable

is ti = hk, and x. is the discrete analogue of the

continuous function x(t) evaluated at t = t, . Note that

2

in obtaining Eq. (1.2) from Eq. (1.1), the following

replacements were made

(1.3a) t — hk, x(t) — xk,

r, nu\ dx(*) xk+l ~ xk(1.3b) H

The h, in the fraction on the right-side of Eq. (1.3b),

will be called the denominator function. More will be

presented on denominator functions in the section 1.4.

A major difficulty with the use of finite-difference

(FD) techniques is the occurrence of instabilities in

the solutions to the difference equations. A way to

characterize instabilities is to say that they are solu

tions to the FD equations that do not correspond to any

solution of the ODE.18'1

Numerical instabilities arise in one (or more) of

several ways:

a) the inappropriate modeling of nonlinear terms;

b) the inappropriate modeling of derivative terms;

c) the order of the FD equation is larger than the

order of the ODE.

In particular, chaotic solutions can occur if care is not

taken in the construction of a given FD scheme for an

ordinary differential equation. (A chaotic solution, for

3

our purposes, is defined to be a numerical unstable

solution that is bounded and does not approach any constant

15 19solution of the ODE.

The purpose of this thesis is to investigate the

influence of the denominator function on the behavior of

the solutions of FD models of ODE's.

1.2. Summary of Thesis

The influence on the solution behaviors of finite-

difference models of ordinary differential equations is

investigated by using complicated functional forms for the

denominator function in the replacement of the ordinary

derivative by a discrete derivative. The basis of our work

is a generalized definition of the first derivative of a

function.20 The particular examples of the linear decay

equation and the nonlinear Logistic equations are used to

illustrate in detail the various solution possibilities and

their stability properties. Our major conclusion is that

the functional form of the denominator plays a critical

role in determining the range of step-size values for which

stable numerical solutions exist.

1.3. Generalized Definition of Derivative

Conventionally, the first derivative of a function x,

at the point t, is defined as one of the following

13equivalent limits

x(t + h)

x(t) -

x(t + h)

- x(t)

h

x(t-h)

h

- x(t-h)

2h

The corresponding FD approximation for the first derivative

is, consequently, taken to be one of the three forms

d*

xk+l

x _

xk+l

h

h

2h

X

X

k

k- 1

which are called, respectively, the forward Euler, backward

Euler and central difference schemes.

However, it is possible to construct a more general

definition of the first derivative of x(t). One possibi

lity for example, is

dx ,• x(t+h) - x(t)(1.6) t2^ = Lim — rrC\^•^ dt h —0 (h)

where V>(h) has the property

(1.7) ^(h) = h + 0(h2) .

5

Note that the particular function V"(h) = h

13corresponds to the usual definition of the derivative.

1.4. Denominator Functions

The function V'(h), in Eq. (1.6), will be called the

denominator function.

Examples of functions V'(h), that satisfy the condition

given by Eq. (1.7), are

(1.8)

- e

In taking the Lim h —► 0, to obtain the derivative in Eq.

(1.6), the use of any of the ip(h) , as given by Eq. (1.8),

will lead to the usual result as expressed in Eq. (1.4).

However, for h finite, the discrete derivatives, con

structed from Eqs. (1.6) and (1.8), namely

xk+l ~ xk\ 5

:k-l

ck+l ~ xk-l

will differ from those conventionally given in the

6

literature 9-11 such as Eq. (1.5). This fact allows the

construction of a larger class of FD models of ODE's.

1.5. Linear Stability Analysis

As additional background information, the concept of

linear stability analysis will be introduced and discussed

for first-order ordinary differential and difference equa-

15,19tions.

Consider the, in general, nonlinear differential equa-

t ion

(1.10) $* = f(x).

Let the function, f(x), have N simple zeroes, i.e.,

(1.11) f(x*) = 0,

and denote them by

(1.12) {x}}, i = 1,...,N.

(N may be unbounded.) The zeroes of f(x) correspond to

constant solutions of Eq. (1.10).

Now select a particular constant solution, say x*., and

consider neighboring solutions that can be represented as

"fol 1 ows

(1.13) x(t) = x*. + e(t),

where the "perturbation function" has the property

(1.14) | c CO) | < |x*.

If

(1.15) Lim |t(t) | = 0,t —>oo

then x(t) = x*. is said to be a stable constant solutionJ

Otherwise, it is an unstable constant solution.

Substitution of Eq. (1.13) into Eq. (1.10) gives

(1.16) d£ = f(x. + f)dt

= f(x*.) + Ae + 0(e2)•J

= Ae + 0(e2).

The second line of Eq. (1.16) is the two-term Taylor series

expansion of the function f (x*. + e) . The result of the

third line uses the fact that x*. is a zero of f(x). Also,J J

the definition

8

is made. Linear stability analysis retains only the first

(linear) term on the right-side of Eq. (1.16); this gives

the differential equation

(1.18) ^ = Ac,

whose solution is

(1.19) e(t) = f(0)eAt

Comparison of Eq. (1.19) with the condition given by Eq.

(1.15) shows that x(t) = x*. is linearly stable if the

following condition is satisfied

(1.20) A = df(x)dx x =x

. < 0.

A similar result can be obtained for first-order

difference equations. Consider the equation

(1.21) xk+1 = F(xk),

and denote the constant solutions of Eq. (1.21), i.e.,

(1.22) x = F(x),

by

(1.23) {x£}, I = 1,2,...,M,

where M may be unbounded. (It is also assumed that all the

zeros o:f

(1.24) Fa(x) = F(x) - x = 0,

are simple.) Select a particular constant solution, say

x-, and consider neighboring solutions*J

(1.25) xk = Xj + ek,

where

(1.26) |co

(1.27) Lim |c,| = 0,}K»-oo K

then Xi = x. is said to be a stable constant solution.K J

Otherwise, it is an unstable constant solution. Substitu-

10

tion of Eq. (1.25) into Eq. (1.21) gives

(1.28) Xj + ck+1 = F(X j

where

x=Xj-

Using the fact that x- = F(x.) and retaining only the

second term on the right-side of Eq. (1.28), the following

equation is obtained

(1.30) ek+1 = Bek.

This first-order, linear difference equation has the

solution (see the Appendix)

(1.31) ek = eo(B)k.

Comparison of Eq. (1.31) with the condition of Eq. (1.27)

gives the following linear analysis result for stability

(1.32) |B| < 1.

In other words , Xi = x- is linearly stable if theK J

11

following inequality is satisfied

(1.33)dF(x)

dx x=x

< 1

1.6. Outline of Thesis

In the earlier portions of Chapter One, preliminary

concepts and definitions were given. The problem of this

thesis was stated and the general results obtained by this

investigation were summarized. In Chapter Two, a detailed

analysis will be made of the influence of the denominator

function on the numerical solutions of the linear decay

differential equation. Chapter Three presents a similar

analysis for the nonlinear Logistic differential equation.

The major results of the thesis as well as how the research

can be extended to other related questions are given in

Chapter Four. The Appendix gives a brief summary of the

basic properties of first- and second-order linear

difference equations with constant coefficients.

CHAPTER TWO

THE LINEAR DECAY EQUATION

2.1. The Equation and Solution

Q

The linear decay equation is

(2.1) ^ = -x, x(0) = xQ (given).

This first-order, linear differential equation can be

solved by elementary means to give the solution

(2.2) x(t) = XQe-*.

Inspection of Eq. (2.2) shows that for any xQ

(2.3) Lim x(t) = 0.

Thus, x(t) = 0, the only constant solution of Eq. (2.1) is

stable.

2.2. Finite-Difference Models

A general class of FD models can be constructed by

using the result of Eq. (1.6) for finite h:

^ = X x0(2>4) ^(h) = ~Xk' x

12

13

where V'(h) has the property given by Eq. (1.7). This equa

tion can be rewritten to the form

(2.5) xk+1 = [1 - !Kh)]xk.

The exact solution of Eq. (2.5) is12

(2.6) xk = xQ[l - ^(h)]k.

Let us examine the behavior of the function, given by

Eq. (2.6), for the four denominator functions of Eq. (1.8).

Concern will not be placed on the closeness of the

"numerical values" of the FD scheme, as given by Eq. (2.6),

to the exact solution of the ODE of Eq. (2.1). The major

consideration will be whether the FD scheme reproduces the

general solution behavior of Eq. (2.1). Therefore, it is

to be noted again that all of the solutions to Eq. (2.1)

decrease monotonical ly to zero with increase of the

variable t. Cond it ions on the step-size h. to ensure that

the solut ion of Eq. (2.5) have th i s property will be deter-

Now, examination of Eq. (2.6) shows that all its

solutions will decrease monotonically to zero, with fixed

h > 0 and k —> oo, if and only if the following condition

is satisfied

14

(2.7) 0 < 1 - i/-(h) < I-

These inequalities may be rewritten to the following form

(2.8) 0 < ^(h) < 1.

If the inequalities of Eq. (2.8) are not satisfied,

then xk, as given by Eq. (2.6), will generally become

unbounded as k - oo. For example, the following cases

illustrate what can occur:

(a) Let V(h) = 1; then xR = 0 for all k = 1,2,3,....

(b) Let

(2.9) 1 < i/>(h) < 2,

and define (5 to be

(2.10) /? = 1 - V>(h) .

Note that

(2.11) -1 < /? < 0.

Therefore, Eq. (2.6) becomes

15

(2.12) xk - xo(^ i.) \p\ •

This corresponds to a numerical solution that oscillates

(from positive to negative values), but, overall decreases

exponentially to zero.

(c) For ip(h) = 2, Eq. (2.6) takes the form

(2.13) xk = xQ(-l)k.

The solution oscillates with constant amplitude.

(d) For ^(h) > 2, xk can be expressed as follows

(2.14) xk = xo(-l)k|7|k,

where

(2.15) 7 = 1 - tf(h) . 7 < -1-

This case gives a solution x^ oscillating, but, increasing

its magnitude exponentially as k —* oo.

Cases (a), (b) , (c) and (d) all correspond to situa

tions of numerical instability. For this example, they

occur because the step-size, h, can become too large.

We now consider in turn the four denominator functions

of Eq. (1.8). One of our interests will be the interval of

16

positive h values for which xk = 0 is a stable solution.

To begin, let V(h) = h. This is the choice for the

denominator function of conventional FD schemes. ~ From

Eq. (2.8), it follows that if

(2.16a) 0 < h < 1 ,

then the solution

(2.16b) xk = xo(l-h)k

will have the same behavior as Eq. (2.2), the exact

solution to the differential equation (2.1).

For ip(h) = sinh, the step-size must satisfy, from Eq.

(2.8), the restriction

(2.17a) 0 < h < v.

The solution, from Eq. (2.6), is now

(2.17b) xk = xQ[l - sinh]k

Note that the allowable step-size range, for h, is more

than three times that for V'(h) = 1.

Now consider

17

(2.18a) V(h) = eh - 1 .

For this case, the solution x. becomes

(2.18b) xk = xQ[2 - eh]k.

This situation corresponds to the following restriction on

h:

(2.18c) 0 < h < Ln 2.

Finally, "for the denominator function

(2.19a) V(h) = 1 - e~h

the solution x, can be written

(2.19b) xk = xoe"hk

Of importance is the fact that for this case the inequa

lities of Eq. (2.8) can be satisfied by any positive step-

size, h. Comparison of the solutions given by Eqs. (2.2)

and (2.19b) show that the following relationship holds

between them

18

(2.20) x(tk) = xk, tk = hk.

Consequently, for any fixed, finite value of the step-size,

h, the solution to the FD scheme, corresponding to Eq.

(2.19a), gives the value of the exact solution at the point

t = tk, for k an integer. Such FD schemes are called

j i 14,19exact FD models.

2.3. Pi scussion

Critical analysis of the work of the last section

leads to the following results and comments:

(a) The generalization of the definition of the first

derivative of a function suggests new ways of constructing

discrete approximations to the derivative for purposes of

modeling ODE's by difference equations.

(b) The functional form of the denominator function

can have a major influence on the possible solution

behaviors of FD models of ODE's.

(c) The above investigation of the decay equation

shows that use of more complicated denominator functions

can greatly extend the interval of step-size values for

which the solutions of the FD model shows no numerical

instabi1 ities.

(d) Thus, based on (a), (b) and (c) , it is strongly

19

suggested that in the replacement of the derivative, dx/dt,

by its discrete FD representation, more complicated func

tional forms be used for the denominator function.

CHAPTER THREE

THE LOGISTIC EQUATION

3.1. The Equation and Solution

Consider the following elementary, but, nontrivial

15,16first-order nonlinear differential equation

(3.1) JjS = x(l-x).

This equation is called the Logistic equation and provides

a simple model of a population interacting with itself.

A great deal of work has been done on this equation to

investigate mechanisms of how numerical instabilities

17 15occur. (See, for example, Ushiki and Mickens. )

Equation (3-1) has two constant solutions obtained

from solving the expression

(3.2) f(x) = x(l-x) = 0.

They are

(3.3) x^ = 0, x^ = 1.

Applying linear stability analysis gives

A - dfO)(3.4) = 1,x= 0

20

21

and

(3.5)

Therefore, based on the analysis of section 1.5, the

following conclusions hold:

(a) x* = 0 is linearly unstable;

(b) x* = 1 is linearly stable.

These results can also be obtained from a global

perspective. First, note that from Eq. (3.1), it follows

immediately that the derivative has the property

< 0, for x > 1;

(3.6) g* <> 0, for 0 < x < 1;

< 0, for x < 0.

Second, two different solutions of the differential equa

tions cannot intersect. (This is a consequence of the

uniqueness of the solutions. ) It therefore follows that

every solution that starts with an initial condition,

x(0) = Xq > 1, will decrease monoton ical ly to constant

solution x(t) = x* = 1. Likewise, every solution with 0 <

Xq < 1, will increase monotonical ly to the constant solu

tion x(t) = x* = 1. However, for Xq < 0, all solutions

move away from the constant solution x(t) = x* =0. Thus,

22

for t > 0, "the constant solution x(t) = x* = 0 is

(globally) unstable, while x(t) = x^ = 1 is (globally)

stable.

Equation (3.1) can also be solved exactly. First,

note that it is a separable equation and can be written

(3.7) ,?x . = dt.v J x(l—x)

Applying the partial fraction procedure to the left-side of

Eq. (3.7) gives

1 cl C2(3-8) x(i-x) = -x- + r=^

c1(l—x) +

= x(l-x)

Comparison of the left- and right-sides of Eq. (3.8) gives

(3.9) Cj = 1, c1 - c2 = 0

or

(3.10) c1 = c2 = 1.

Therefore,

23

= Ln x - Ln(l—x) = t +

= Lnte) = * + C3'

where Co is an arbitrary constant of integration. Solving

the last line for x(t) gives

where c = exp(c3). If x(0) = xQ is given, then c can be

easily determined, i.e., from Eq. (3.12)

(3.13) = c.q

Substituting this value of c into Eq. (3.12) and solving

for x(t) gives

y

This explicit solution of Eq. (3.1) clearly has the

properties discussed above. In particular, note that

(3.15) Lim x(t) = 1, for xQ > 0;

(3.16) for xQ = 0, x(t) = 0;

24

(3.17) for xQ = 1, x(t) = 1.

Also, of interest is the situation for xQ > 0. For

this case, the solution x(t) becomes unbounded for a finite

value of t = t. This can be shown as follows. Let Xq =

-|xo| < 0. Placing this representation of xQ into Eq.

(3.14) gives

(3.18) x(t) = — ■ -f- —t--|xQ| + (l+|xQ|e

Now x(t) will become unbounded if the denominator becomes

zero. Inspection of Eq. (3.18) shows that this is

possible. To determine the value of t where this occurs,

set the denominator zero and solve for this value of t = t.

Doing this gives

(3.19) -|xo| + (l+lxQDe-* = 0,

and

(3.20) t = Ln( Ix.l0)'

Finally, it should be stated that in practical appli

cations (population dynamics, etc.) only xQ > 0 is needed.

Consequently, the interesting behavior of the solutions to

25

Eq. (3.1) for Xq > 0 never arise.

3.2. Finite-Difference Models

Two finite-difference models will be constructed for

the Logistic differential equation.

For the first model the denominator function V"(h) = h

will be selected. This corresponds to the conventional

choice and gives

(3.21) Xk+1h~ Xk = xk(l-xk).

The constant solutions, xk = x, to Eq. (3.21) are

(3.22) x1 = 0, x2 = 1.

Since the nonlinear difference equation (3.21) has no exact

solution that can be easily written in terms of elementary

functions of the discrete variable k, the properties of its

solutions must be determined indirectly by means of a

linear stability analysis. Now Eq. (3.21) is a discrete

model of the ordinary differential equation given by Eq.

(3.1). For Eq. (3.1), it is known that x(t) = 1 is

linearly stable and x(t) = 0 is linearly unstable. The

discrete model should have these same properties.

At xk = x-, = 0, linear stability analysis gives

26

(3.23) xk = 5^ + ck = 0 +

and from Eq. (3.21) the result

(3.24) x. ,-. = F(x.) = (l+h)xk -

and for instability the condition

dF(x,= 1 + h > 1.

x, =0

(See Eq. (1.33) for this condition.) Examination of the

result given by Eq. (3.25) shows that the constant solution

at xk = x-. = 0 is (linearly) unstable for all h > 0.

This agrees with the corresponding property of Eq. (3.1).

Now consider xk = xk = 1. For this case, the con

stant solution must be (linearly) stable and from Eq.

(1.33) the requirement

(3.26)

gives

dF(*k>dx,

< 1,

(3.27) 0 < 1-h < 1,

or

(3.28) 0 < h < 1.

27

(Oscillations in xk are not allowed; thus, in Eq. (3.27),

the full range of h values, -1 < 1-h < 1, does not

occur.)

From Eqs. (3.25) and (3.28) the following conclusions

may be reached: The FD model of Eq. (3.1), with the

denominator function xp(h) = h, has the same (linearly)

stability properties for the constant solutions, x* = x^ =

0 and x* = x2 = 1 , for values of the step-size in the

range 0 < h < 1. For h > 1, the FD model of Eq. (3.21)

can have oscillating stability or oscillating instability

for the constant solution x* = x2 = 1 • The particular

possibility depends on the value of the step-size h.

Hence, for h > 1, numerical instabilities can occur for

this FD model of Eq. (3.1).

The second FD model for Eq. (3.1) will be one for

which the denominator function is taken to be

(3.29) V(h) = 1 - e~h

This gives the following FD model for Eq. (3.1)

(3.30) xk+1 = xk + [l-e-h]xk(l-xk)

or

(3.31) xk+1 = F(xk),

where

28

(3.32) F(xk) = [2-e"h]xk - [1-e""] x2, ,

and

(3.33) 4E- = [2-e-h] - 2[l-e"h]xk.

Consider the constant solution x^ = x^ = 0. From Eq.

(1.33) it follows that the stability of xk = x1 = 0 is

determined by the value of the derivative evaluated at

X,, = 0. Its value is

(3.34) = £-£ = 2 - e~h > 1x, =0

Since B-, is always greater than one for any positive h, our

conclusion is that xk = 5^ = 0 is (linearly) unstable for

all h > 0.

Now consider the other constant solution at xk = yt^ =

1. For this case

(3.35) B2 = £-x, =1

and Bq satisfies the condition

(3.36) 0 < B2 < 1 , h>0.

29

Hence, x, = x2 = 1 is linearly stable for all h > 0.

Therefore, in this second FD model for Eq. (3.1), the

linear stability properties of the constant solutions are

independent of the step-size and are of the same nature as

the original differential equation.

3.3. Discuss ion

The analysis of section 3.2 clearly shows the

advantages of using more complicated, nonstandard denomina

tor functions in constructing FD models of ODE's. In parti

cular, for the differential equation considered in this

Chapter, the Logistic equation, the use of conventional

techniques lead to Eq. (3.21) as a discrete model of Eq.

(3.1). Correct (linear) stability properties of the con

stant solutions was obtainable only if the step-size, h,

was restricted to the range 0 < h < 1. However, by using

the expression of Eq. (3.29), for the denominator function,

a FD model was constructed having the correct (linear)

stability properties for all h > 0.

CHAPTER FOUR

FUTURE INVESTIGATIONS

4.1. Summary

The following are the two main results of this investi-

gat ion:

(a) A generalization of the conventional definitions

of the derivative leads to a larger class of finite-

difference (FD) models for ordinary differential equations

(ODE).

(b) The use of more complex functional forms for

denominator functions allow the construction of FD schemes

that have the correct linear stability properties over

larger ranges of step-size values than conventional

techn iques.

The findings of this thesis, which are based on the

work of Mickens ' ' ' have important consequences for

the field of numerical integration of differential equa

tions. A major problem in this area is the numerical inte

gration of an ODE for large intervals of the independent

variable (which is taken to be the time). Conventional

numerical integration techniques require small time-steps

18and this can lead to difficulties with round-off errors.

The small time-step requirement is a reflection of the fact

that conventional schemes will have (numerical)

30

31

15,18 T__instabilities if the time-step becomes too large. It

should be clear that if the procedures of this thesis are

applied to this problem, many of the difficulties involving

numerical instabilities will be resolved.

4.2. Partial Differential Equations

The work of this thesis was centered on FD schemes for

ODE's using a generalization of the usual definition of the

derivative of a function. It should be indicated that

Mickens has applied these concepts and procedures also to

14,20-22certain classes of partial differential equations.

A number of important results were obtained. A good sum

mary is presented in reference 19.

4.3. Extensions of Research

The procedures of this thesis have general applicabi

lity. Three problems where their application might lead to

interesting results are:

(a) The numerical integration of first-order ODE's

(4.1) ^ = f(x),

where f(x) has certain predetermined properties such as the

number and type of zeroes, etc.

(b) The numerical integration of systems of first-

32

order ODE's for which a special case is that of two first-

order equations.

(4.2a) ^| = f(x,y),

(4.2b) ^ = g(x,y).

Such types of equations arise in the analysis of many

3,6systems in the sciences and engineering.

(c) The numerical integration of partial differential

equations, in particular, the diffusion/heat equation in

,. . 1,3,9,18one space dimension

(4 ^\ du — 9 u

and Laplace's equation in two space dimensions ' '

(4.4) & + & = 0.d2 dJ

APPENDIX

LINEAR DIFFERENCE EQUATIONS

The purpose of this Appendix is to present a brief

review of how to solve first- and second-order difference

equations with constant coefficients. The book by

Mickens12 provides the full proofs and details for the

following material.

Let (a1,b1,a2,b2,c2) be constants. A first-order

linear difference equation with constant coefficients can

be written in the form

(A.I) a1xk+1 + blXk = 0,

where the independent discrete variable k is an integer.

Rewriting Eq. (A.I) gives

(A.2) x k+1

If x0 is given, then it is clear that the solution to Eq.

(A.2) is

(A. 3) xk = xo(-e1) .

For arbitrary Xq, this provides the general solution to Eq

(A.I).

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34

A second-order linear difference equation with constant

coefficients can be written in the form

(A. 4) a2xk+1 + b2xk+1 + c2xk = 0.

The associated characteristic equation is

(A. 5) a2r2 + b2r + c2 = 0.

The latter equation has two roots, r1 and r2; they can be

expressed in terms of the constants a2, b2 and c2. The

general solution to Eq. (A.4) can be expressed as follows

(A.6) xk = D1(r1)k + D2(r2)k,

where D-, and D2 are arbitrary constants

REFERENCES

1. D. Potter, Computational Physics (Wiley, New York,

1973).

2. M. Gitterman and V. Halpern, qualitative Analysis of

Physical Problems (Academic Press, New York, 1981).

3. C. C. Lin and L. A. Segel , Mathematics AppI ied to

Deterministic Problems in the Natural Sciences

(Macmillan, New York, 1974).

4. M. R. Spiegel, Applied Differential Equations

(Prentice-Hall, Englewood Cliffs, NJ; 3rd edition).

5. A. H. Nayfeh, Perturbation Methods (Wiley, New York,

1973).

6. R. E. Mickens, Nonlinear Oscillations (Cambridge

University Press, New York, 1981).

7. D. Zwillinger, Handbook of Differential Equations

(Academic Press, New York, 1989).

8. D. W. Jordan and P. Smith, Nonlinear Ordinary

Differential Equations (Clarendon Press, Oxford,

1977).

9. L. Lapidus and G. F. Pinder, Numerical Solution of

Partial Differential Equations in Science and

Engineering (Wiley, New York, 1982).

10. J. Lambert, Computational Methods j_n Ordinary

Differential Equations (Wiley, New York, 1973).

11. R. D. Richtmyer and K. W. Morton, Difference Methods

for Initial-Value Problems (Wiley-Interscience, New

York, 1967, 2nd edition).

12. R. E. Mickens, Difference Equations (Van Nostrand

Reinhold, New York, 1987).

13. J. Marsden and A. Weinstein, Calculus 1 (Springer-

Verlag, New York, 1985, 2nd edition), section 1.3.

14. R. E. Mickens, "Difference Equation Models of

Differential Equations Having Zero Local Truncation

Errors," in Differential Equations. I. W. Knowles and

R. T. Lewis, editors (North-Holland, Amsterdam, 1984),

pps. 445-449.

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36

15. R. E. Mickens, "Runge-Kutta Schemes and Numerical

Instabilities: The Logistic Equation," in Differential

Equations and Mathematical Physics. I. W. Knowles and

Y. Saito, editors (Springer-Verlag, Berlin, 1987),

pps. 337-341.

16. R. M. May, "Simple Mathematical Models with very

Complicated Dynamics," Nature 261, 459-467 (1976).

17. S. Ushiki, "Central Difference Scheme and Chaos,"

Physica D4, 407-415 (1982).

18. F. B. Hildebrand, Finite-Difference Equations and

Simulations (Prentice-Hall, Englewood Cliffs, NJ ,

1968).

19. R. E. Mickens, "Pitfalls in the Numerical Integration

of Differential Equations," in Analytical Techniques

for Material Characterization. W. E. Collins, B. V. R.

Chowdari and S. Radhakrishna, editors (World

Scientific, Singapore, 1987), pps. 123-143.

20. R. E. Mickens, "Stable Explicit Schemes for Equations

of Schrodinger Type," Physical Review A (accepted for

publication, 1989).

21. R. E. Mickens, "Exact Solutions to Difference Equation

Models of Burgers' Equation," Numerical Methods for

Partial Differential Equations, 2, 123-129 (1986).

22. R. E. Mickens, "Difference Equation Models of

Differential Equations," Mathematics and Computer

Modelling, H, 528-533 (1988).