BOARD AND

A

Program

of

Monographs, Textbooks, and Lecture Notes

EXECUTIVE EDITORS Earl

J. Taft

Rutgers University New Brunswick, New Jersey

Zuhair

Nashed University of Delaware

Newark, Delaware

EDITOFUAL BOARD

M. S. Baouendi Ani1 Nerode University of California, Cornell University

San Diego

Donald Passman Jane Cronin University of Wisconsin, Rutgers University Madison

Jack K. Hale Fred S. Roberts Georgia Institute of Technology Rutgers University

S. Kobayashi David L. Russell

University of California, Virginia Polytechnic Institute Berkeley and State University Marvin Marcus Walter Schernpp University of California, Universitat Siegen

Santa Barbara

Mark Teply Yale University Milwaukee

W. S. Massey University of Wisconsin,

DIFFERENCE EQUATIONS AND INEQUALITIES

Theory, Methods, and Applications

Second Edition, Revised and Eqpanded

Ravi P. Agarwal

National University of Singapore Kent Ridge, Singapore

M A R C E L

MARCEL DEKKER, _INC. NEW YORK - ^BASEL

D E K K E R

Aganval, Ravi

P.

Difference equations and inequalities : theory, methods, and applications l p. cm. - (Monographs and textbooks in pure and applied mathematics;

228)

Ravi P. Aganval

-

2nd ed., rev. and expanded

Includes bibliographical references and indexes.

ISBN 0-8247-9007-3 (acid-kee paper)

1. Difference equations. 2. Inequalities (Mathematics) I. Title. 11. Series.

QA431 .A43 2000

515'.6254c21 99-058166

"his book is printed on acid-free paper.

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0

2000 by Marcel Dekker, Inc.

A l l

Rights Reserved.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any mformation storage and retrieval system, without permission in writing from the publisher.

Current printing (last digit) l 0 9 8 7 6 5 4 3 2 1

PRINTED IN THE UNITED STATES

OF

AMERICA

Preface to the Second Edition

Since its publication in 1992,

Difference Equations and Inequalities

has been received very positively by the international scientific community. Its success prompted a request from the publisher for an updated edition.

In this edition, besides a new chapter on the

qualitative properties

of

solutions of neutral difference equations,

new material has been added in all the existing chapters of the first edition. This includes a variety of interesting examples from real world applications, new theorems, over 200 additional problems and 400 further references.

The theory of difference equations has grown at an accelerated pace in the past decade. It now occupies a central position in applicable analysis and will no doubt continue to play an important role in mathematics

as a

whole. It is hoped that this new edition will be a timely and welcomed reference.

It is a pleasure to thank all those who have helped in the preparation of this edition. I would especially like to thank Ms. Maria Allegra of Marcel Dekker, Inc., whose help was instrumental in the successful completion of this project.

Ravi P. Agarwal

Preface to the First Edition

Examples of discrete phenomena in nature abound and yet somehow the continuous version has commandeered all our attention - perhaps owing to that special mechanism in human nature that permits us to notice only what we have been conditioned to. Although difference equations manifest themselves

as

mathematical models describing real life situations in probability theory, queuing problems, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, quanta in radiation, genetics in biology, economics, psychology, sociology, etc., unfortunately, these are only considered as the discrete analogs of differential equations. It is an indisputable fact that difference equations appeared much earlier than

V

differential equations and were instrumental in paving the way for the development of the latter. It is only recently that difference equations have started receiving the attention they deserve. Perhaps this is largely due to the advent of computers, where differential equations are solved by using their approximate difference equation formulations. This self-contained monograph is an in-depth and up-to-date coverage of more than 400 recent publications and may be

_of

interest to practically every user of mathematics in almost every discipline.

It is impossible to acknowledge individually colleagues and friends to whom I am indebted for assistance, inspiration and criticism in writing this monograph. I must, however, express my appreciation and thanks to

_Ms.

Rubiah Tukimin for her excellent and careful typing of the manuscript.

Ravi P. Agarwal

Contents

Preface to the Second Edition Preface to the First Edition

Chapter 1 Preliminaries

1.1.

Notations

1.2.

Difference Equations

1.3.

Initial Value Problems

1.4.

Some Examples: Initial Value Problems

1.5.

Boundary Value Problems

1.6.

Some Examples: Boundary Value Problems

1.7.

Some Examples: Real World Phenomena

1.8.

Finite Difference Calculus

1.9.

Problems 1

.lo.

Notes

1 .l l.

References

Chapter 2 Linear Initial Value Problems

2.1.

2.2.

2.3.

2.4.

2.5.

2.6.

2.7.

2.8.

2.9.

2.10.

2.11.

2.12.

2.13.

2.14.

2.15.

2.16.

2.17.

2.18.

Introduction

Preliminary Results from Algebra Linear Dependence and Independence Matrix Linear Systems

Variation of Constants Formula Green’s Matrix

Adjoint Systems

Systems with Constant Coefficients Periodic Linear Systems

Almost Periodic Linear Systems Higher Order Linear Equations Method of Generating Functions Bernoulli’s Method

Poincark’s and Perron’s Theorems Regular and Singular Perturbations Problems

Notes References

V V

1 2 4 6 11 13 22 26 34 44 45

49 49 54 55 58 59 60 62 69 72 74 80 85 87 91 97 113 114

vii

Chapter 3 Miscellaneous Difference Equations

3.1.

3.2.

3.3.

3.4.

3.5.

3.6.

3.7.

3.8.

3.9.

3.10.

3.11.

3.12.

3.13.

3.14.

3.15.

3.16.

3.17.

3.18.

3.19.

3.20.

3.21.

Clairaut’s Equation Euler’s Equation Riccati’s Equation Bernoulli’s Equation Verhulst’s Equation

Best Discrete Approximations:

Harmonic Oscillator Equation Duffing’s Equation

van der Pol’s Equation Hill’s Equation

Mathieu’s Equation

Weierstrass’ Elliptic Equations Volterra’s Equations

Elementary Partial Difference Equations:

Riccati’s Extended Form Wave Equation

FitzHugh-Nagumo’s Equation Korteweg-de Vries’ Equation Modified KdV Equation Lagrange’s Equations Problems

Notes References

Chapter 4 Difference Inequalities

4.1.

Gronwall Inequalities

4.2.

Nonlinear Inequalities

4.3.

Inequalities Involving Differences

4.4.

Finite Systems

of

Inequalities

4.5.

Opial Type Inequalities

4.6.

Wirtinger Type Inequalities

4.7.

Problems

4.8.

Notes

4.9.

References

Chapter 5 Qualitative Properties of Solutions

of

Difference Systems

117 118 120 125 126 128 130 135 142 147 148 150 152 158 159 160 161 162 167 180 181

184 193 200 205 209 214 219 229 230

5.1.

Dependence on Initial Conditions and Parameters

234 5.2.

Asymptotic Behavior

of

Linear Systems

238 5.3.

Asymptotic Behavior of Nonlinear Systems

247

Contents ix

5.4.

5.5.

5.6.

5.7.

5.8.

5.9.

5.10.

5.11.

5.12.

5.13.

5.14.

5.15.

5.16.

5.17.

5.18.

Concepts of Stability Stability of Linear Systems Stability of Nonlinear Systems Nonlinear Variation of Constants Dichotomies

Lyapunov’s Direct Method for Autonomous Systems Lyapunov’s Direct Method for Non-Autonomous Systems Stability of Discrete Models in Population Dynamics Converse Theorems

Total Stability Practical Stability Mutual Stability Problems Notes References

250 255 262 269 272 281 289 292 298 303 305 306 308 326 328

Chapter 6 Qualitative Properties of Solutions of Higher Order Difference Equations

6.1.

General Properties of Solutions of

6.2.

Boundedness of Solutions of

(6.1.1)

6.3.

Recessive and Dominant Solutions of

(6.1.1) 6.4.

Oscillation and Nonoscillation for

(6.1.1) 6.5.

Riccati Type Transformations for

(6.1.1) 6.6.

Riccati Type Transformations for

6.7.

Olver Type Comparison Results

6.8.

Sturm Type Comparison Results

6.9.

Variety of Properties of Solutions of

(6.1.1)

p ( k ) ~ ( k +

¹⁾

+ ^{p ( k}

^- l ) u ( k - 1) =

q ( k ) u ( k )

(6.6.1) A ( p ( k ) A u ( k ) )

+ ~ ( k ) u ( k +

¹⁾

⁼

⁰

(6.9.1)

p ( k ) z ( k +

¹⁾

+

^{p(k -} l ) z ( k - l)

=

q ( k ) z ( k ) + ^{4 k )}

6.10.

Variety of Properties of Solutions

of (6.10.1)

A2u(k

-

1) + p ( k ) u r ( k ) = 0

6.11.

Oscilla,tion and Nonoscillation for

(6.11.1) A ( r ( k ) A u ( k ) )

+ f ( k ) F ( u ( k ) ) =

0 (6.12.1) A ( r ( k ) A u ( k ) )

+ f ( k ) F ( u ( k ) ) = g(k)

(6.13.1)

A2u(k) + ^f(k, u ( k ) )

= 0 (6.14.1)

A i u ( k ) = f ( k , u ( k ) ,

A p ( k ) ) 6.12.

Asymptotic Behavior of Solutions of

6.13.

l2 and

c0

Solutions of

6.14.

Oscillation and Nonoscillation for

335 337 342 348 349 355 362 365

367 370 378 384 387 391

6.15.

6.16.

6.17.

6.18.

6.19.

6.20.

6.21.

6.22.

6.23.

6.24.

6.25.

6.26.

Oscillation and Nonoscillation for

(6.15.1) A ( ~ ( k ) A u ( k ) ) + f ( k ) F ( k , u ( k ) ,

Au(k))

Variety of Properties of Solutions

of

(6.16.1) A4u(k

-

2)

= p(k)u(k)

Asymptotic Behavior of Solutions of Asymptotic Behavior, Oscillation and Nonoscillation for

= g ( k u ( k ) ,

W k ) )

(6.17.1)

Anu(k)

+ f ( k , ~ ( k ) ,

A u ( k ) , . ^{. .}

,

An-'u(k)) =

0 (6.18.1) Anu(k) +

h ( k ) F ( k ,

~ ( k ) ,

A u ( k ) , . . .

,

^An-lu(k))

= g ( k , u ( k ) , A u ( k ) , . . .

,

An-'u(k))

Oscillation and Nonoscillation

for

(6.19.1) An.(k) + CLl f i ( k ) F i ( ~ ( k ) ,

A u ( k ) , . . .

,

Oscillation and Nonoscillation for

An-' u(lc))

⁼

0 (6.20.1) ~ ( k + ¹⁾

^-

^{~ ( k )} + p ( k ) u ( k

^-

m )

=

0 Oscillation and Nonoscillation for

(6.21.1)6 A,u(k) + bC,"=, fz(k)Fz(u(gz(k)))

=

0 Oscillation and Nonoscillation for

(6.22.1) A ( p ( k ) ( A ~ ( k ) ) ~ ) + ^q(k + l ) f ( u ( k +

^{I)) =}

⁰

Oscillation and Nonoscillation for

(6.23.1) ~ ( t )

^{- ~ ( t - T )}

+ p ( t ) u ( t

- 0 )

= 0 Problems

Notes References

Chapter 7 Qualitative Properties of Solutions of Neutral Difference Equations

7.1. Oscillation and Nonoscillation for

(7.1.1)

A(u(k)

+

^{p ( k - T ) )}

+ q ( k ) u ( k

^{- o)}

= 0 of Nonoscillatory Solutions of (7.1.1)

7.2. Existence and Asymptotic Behavior

7.3. Oscillation and Comparison Theorems for (7.1

.l)

7.4. Global Asymptotic Stability Criterion for

(7.1.1)

7.5. Oscillation and Nonoscillation for

7.6. Oscillation and Nonoscillation for

(7.5.1) A (U(k)+pu(k-T))+q(k)zl(k-ol)"h(k).1L(lc-aa)

=

0 (7.6.1) A ^(u(k) +

^{p ( k - T ) )}

+ q ( k ) u ( k

- o) =

F ( k )

7.7. Oscillation and Nonoscillation for

393 398 403

408

414 420 425 427 436 443 473 474

485

493

509

512

517

519

Contents xi

7.8.

Oscillation and Nonoscillation for

(7.8.1) A

( ~ ( k ) ~ ( k )

-

p ( k ) ~ ( k

- ^{T ) )}

+ q ( k ) u ( k

-

~ ( k ) )

=

0

7.9.

Oscillatory and Asymptotic Behavior for

7.10.

Oscillation and Nonoscillation for

(7.9.1) A

( ~ ( k ) + p ( k ) u ( k

- T ) )

+ q ( k ) f ( u ( k

^{- 0 ) )} =

0

(7.10.116 A ( 4 k ) +P(k)'lL(k

+

ST)) ^-

q ( k ) f ( u ( g ( k ) ) ) = F ( k )

7.11.

Oscillation and Asymptotic Behavior for

7.12.

Classification of Solutions

_for

(7.11.1)

A2(u(k) + p ( k ) u ( k

- ^{7 ) )}

+ q ( k ) f ( u ( k + ¹

^{- 0 ) ) =}

⁰

(7.12.1) A(a(k)A(u(k)

+ p ( k ) u ( k

- T ) ) )

+ ^{q ( k} + l ) f ( u ( k +

^{1 - U ) )}

^{= 0}

7.13.

Existence of Solutions for

(7.12.1)

7.14.

Oscillation of Mixed Difference Equations

7.15.

Oscillation and Nonoscillation for

(7.15.1) An(u(k)

+ p ( k ) u ( k

- ^{T ) )}

+ q ( k ) f ( u ( k

^{- g ) )} =

0

7.16.

Oscillation and Nonoscillation for

(7.16.1) A (a(k)An-'(u(k) -

p ( k ) u ( k

^{- T ) ) )}

+ k ( k ) f ( u ( g ( k ) ) )

=

0

7.17.

Problems

7.18.

Notes

7.19.

References

Chapter 8 Boundary Value Problems for Linear Systems

8.1.

8.2.

8.3.

8.4.

8.5.

8.6.

8.7.

8.8.

8.9.

8.10.

8.11.

8.12.

Existence and Uniqueness

Method of Complementary h n c t i o n s Method of Particular Solutions Method of Adjoints

Method of Chasing

Method of Imbedding: First Formulation Method of Imbedding: Second Formulation Method of Sweep

Miller's and Olver's Algorithms Problems

Notes References

Chapter 9 Boundary Value Problems for Nonlinear Systems

9.1.

Preliminary Results from Analysis

9.2.

Existence and Uniqueness

527 531 544 548

556 565 572 583

590 593 620 621

629 634 643 643 651 656 662 666 671 674 676 677

681 684

9.3.

Approximate Picard’s Iterates

9.4.

Oscillatory State

9.5.

Stopping Criterion

9.6.

Application t o the Perturbation Method

9.7.

Monotone Convergence

9.8.

Periodic Boundary Value Problems

9.9.

Newton’s Method

9.10.

Approximate Newton’s Method

9.11.

Initial-Value Methods

9.12.

Invariant Imbedding Method

9.13.

Problems

9.14.

Notes

9.15.

References

Chapter 10 Miscellaneous Properties of Solutions of Higher Order Linear Difference Equations

10.1.

Disconjugacy

10.2.

Right and Left Disconjugacy

10.3.

Adjoint Equations

10.4.

Right and Left Disconjugacy for the Adjoint Equation

10.5.

Right Disfocality

10.6.

Eventual Disconjugacy and Right Disfocality

10.7.

A Classification of Solutions

10.8.

Interpolating Polynomials

10.9.

Green’s Functions

10.10. Inequalities and Equalities for Green’s Functions

10.11.

Maximum Principles

10.12.

Error Estimates in Polynomial Interpolation

10.13.

Problems

10.14.

Notes

10.15.

References

Chapter 11 Boundary Value Problems for Higher Order Difference Equations

691 695 698 700 703 707 714 719 724 731 734 739 739

745 746 750 751 752 754 755 757 762 772 774 775 778 789 790

1 1.1. Existence and Uniqueness

795

11.2.

Picard’s and Approximate Picard’s Methods

803 11.3.

Quasilinearization and Approximate Quasilinearization

809

11.4.

Monotone Convergence

819

11.5.

Initial-Value Methods

824

11.6.

Uniqueness Implies Existence

833

Contents

...

x111

1 l .7.

Problems

839

11.8.

Notes

841

1 1.9.

References

842

Chapter 12 Sturm-Liouville Problems and Related Inequalities

12.1.

Sturm-Liouville Problems

844

12.2.

Eigenvalue Problems for Symmetric Matrices

848 12.3.

Matrix Formulation of Sturm-Liouville Problems

850 12.4.

Symmetric, Antisymmetric and Periodic Boundary Conditions

852 12.5.

Discrete Fourier Series

12.6.

Wirtinger Type Inequalities

12.7.

Generalized Wirtinger Type Inequalities

2.8.

Generalized Opial Type Inequalities

2.9.

Comparison Theorems for Eigenvalues

2.10.

Positive Solutions of

(12.10.1)

A3u(IC) +

X a ( k ) f ( u ( k ) )

=

0, IC E IN(2,K

+

²⁾

(12.10.2)

u(0)

= u(1) =

u ( K +

^{3) = 0}

2.1

1. Problems

1

12.12.

Notes

12.13.

References

Chapter 13 Difference Inequalities in Several Independent Variables

13.1.

Discrete Riemann’s Function

13.2.

Linear Inequalities

13.3.

Wendroff Type Inequalities

13.4.

Nonlinear Inequalities

13.5.

Inequalities Involving Partial Differences

13.6.

Multidimensional Linear Inequalities

13.7.

Multidimensional Nonlinear Inequalities

13.8.

Convolution Type Inequalities

13.9.

Opial and Wirtinger Type Inequalities in Two Variables

13.10.

Problems

13.11.

Notes

13.12.

References Author Index Subject Index

856 859 865 867 869

873 882 896 897

901 905 908 91 1 914 921 925 930 933 941 944 945 949 963

Chapter 1

Preliminaries

W e begin this chapter with some notations which are used throughout this monograph. This is followed by some classifications namely: linear and nonlinear higher order difference equations, linear and nonlinear first order difference systems, and initial and boundary value problems. W e also include several examples of initial and boundary value problems, as well as real world phenomena from diverse fields which are sufficient to convey the importance of the serious qualitative as well as quantitative study of difference equations. The discrete Rolle's theorem, the discrete Mean value theorem, the discrete Taylor's formula, the discrete I'Hospital's rule, the discrete Kneser's theorem are stated and proved by using some simple inequalities.

1.1. Notations

Throughout, we shall use some of the following notations: W = {0,1,. .

.}

the set of natural numbers including zero,

W ( a )

=

{ a , a +

^{1, . .}

^.}

^where

a ^E

^W, W ( a ,

b

^- 1) =

{ a , a +

^{1,. ..}

^{, b}

^{- l} where a.}

^{< b}

^{- 1}

^<

⁰⁰

and

a , b E N.

Any one of these three sets will be denoted by

m.

The scalar valued functions on

m

will be denoted by the lower case letters

u.(k), ~ ( k ) ,

. . . whereas the vector valued functions by the bold face let- ters

u(k),

v ( k ) , . . . and the matrix valued functions by the calligraphic letters

U ( k ) , V ( k ) ,

^{. . . .} Let

f ( k )

be a function defined on

m,

then for all

k l ,

k-2 E and

kl > k z , E : & f ( e )

^{= 0 and}

n:lk1 _{f ( e )} ⁼

^{1, i.e.}

enlpty sums and products are t,aken to be 0 and 1 respectively. If

k

and

k +

^{1 are in}

m,

then for this function

f ( k )

we define the

shift operator E

as

E f ( k )

=

f(k +

^1). In general, for a positive integer

m

if

k

and

k +'m

are in

m,

^then

E"f ( k )

=

E [ E " - l f ( k ) ]

⁼

f ( k + ^m,).

Similarly, the

forwa7.d

and

backward d i f f e r e n c e operators

A and

V

are defined as

A f ( k )

=

f(k +

^{1) -}

^{f ( k )}

^and

V f ( k ) = f ( k )

-

f(k

- 1) respectively. The higher order differences for a positive integer

m

are de- fined as

A " f ( k ) =

A [Arn,-'f(k)] . Let I be the

iden,tity operator,

i.e.

If(k) = f ( k ) ,

then obviously A

= E

- 1 and for a positive integer

m

2 Chapter 1 we may deduce the relations

and

m

(1.1.2)

E " f ( k ) =

( I

+ A)"f(k)

⁼

c ( I I ' ) A i f ( k ) , A o

= I.

i = O l,

As usual R denotes the real line and R.+ the set of nonnegative reals.

For

t E

R and m a nonnegative integer the

factorial expression, (t)'")

is defined as

(t)'")

=

nzil(t

^{- i ) .} Thus, in particular for each k E

IN, =

k ! .

1.2. Difference Equations

A d i f f e r e n c e equation

in one independent variable k E

m

and one unknown

u.(k)

is a functional equation of the form

(1.2.1)

f(k,?L(k),?L(k +

^{l), . . .}

^{, u ( k} +

^{n ) ) =}

^0,

where

f

is a given function of IC and the values of

u.(k)

at IC E

m.

If (1.1.2) are substituted in (1.2.1) the latter takes the form

(1.2.2)

g(k,u(k),Au(k),...,AnZl,(k)) =

0.

It was this notation which led (1.2.1) to the nanle difference equation.

The

order

of (1.2.1) is defined to be the difference between the largest and smallest arguments explicitly involved, e.g. the equation u ( k

+

^{3) -}

3 u ( k +

²⁾

+ ^{7 u ( k} +

^{1) = 0} is of order two, whereas w,(k

+

¹⁰⁾ = k ( k - 1) is of order zero.

The difference equation (1.2.1) is

linear

if it is of the form (1.2.3)

C

n,

a i ( k ) u ( k + ^{2 )}

^{= b(k).}

.i=O

If b(k) is different from zero for at least one k

E m,

then (1.2.3) is a

nonhom,ogen.eous

linear difference equation. Corresponding to (1.2.3) the equation

n

(1.2.4) C a i ( k ) u ( k + i ) = 0

i=O

is callcd a homogeneous linear difference equation.

Equation (1.2.1) is said to be nor.nral if it is of the form (1.2.5)

u,(k + ⁿ⁾

^{= f ( k ,}

u,(k),

^7/4k

+

^{l), ' .}

^{, u ( k} +

^{n, - 1))}

or

or

(1.2.7)

A " ' I / . ( ~ )

= f ( k ,

~ ( k ) , ~ ( k +

^{l), . . . , ~ ( k}

+

^{TZ - 1)).}

We shall also consider system of difference equations

(1.2.8) u ( k

+

^{1) =} f ( k , u ( k ) ) ,

k

E

m

where U and f are 1 x n vectors with c:orrlponerlts

w,i

and f i , 1

5 i 5 n,

respectively.

The nth order equation (1.2.5) is equivalent to the system

Ui(k +

^{1) =} U . i + I ( k ) , 1

5

1:

5

R - 1

?/,"(k + ¹⁾

⁼ f ( k , l / , l ( k ) , 7 / , 2 ( k ) , ' . ' , U n ( k ) ) ,

k

E

W

(1.2.9)

in the scnse that

~ ( k )

is a solution of (1.2.5) if and only if (1.2.10) 7/>i(k) =

v ( k +

^{i - l), 1}

5

1:

5

R .

A

system of linear difference equations has the form (1.2.11) U ( k + 1) = A ( k ) u ( k )

+ b ( k ) ,

^k

where A ( k ) is a given norlsingular

n x n

matrix with elcrrlcnts

a , i j ( k ) ,

1

5

i , j

5

n,,

b ( k )

is a givcn

n,

X 1 vector with corrlporlents b i ( k ) , 1

5 i 5

n,, u(k) is an unknown R X 1 vector with components

u,i(k), 1 5 i 5 n,.

If

b ( k )

is different from zero for at least one

k

E

m,

then the system (1.2.11) is called rlorlhorrlogcrlcolls. Corresponding to (1.2.1 1) the system (1.2.12)

u ( k +

1)

=

A ( k ) u ( k ) ,

k

E

'R

is said to be homogeneous.

4

Chapter

1 If

a o ( k ) q , ( k ) #

0 for all

k

E

m,

then the nth order equation (1.2.3) is equivalent to the system (1.2.11) where

(1.2.13)

A(k)

⁼

and

(1.2.14)

It1 the above difference equations (systems) t,he funckions arc assumed to be defined in all of their arguments. Therefore, not all the systems can be written as higher order difference equations, e.g.

7I.l(Ic

+ ^{1) =}

^211(k)

+

^k.z(k)

~ z ( k

+

^{1) =}

^{( k}

^- 1 ) ~ 1 ( k )

+ ~ z ( k ) ,

k E IN.

1.3. Initial Value ^Problems

is said to be a solu.tion, of the given nth order difference equation on if the values of

u ( k )

reduce the difference equation to an identity over W. Similarly, a function

u(k)

defined

on wl

is a solution of the given difference system 011 provided the values of

u(k)

reduce the difference system to an equality over

N.

-

The

gen,eral

solution of an nth order difference equation is a solution w(k) which depends on n, arbitrary constants, i.e. ~ ( k , cl,. . . ,c,) where ci E

R,

1

5

i

5 R.

W e observe that these constants ci can be taken as

periodic f u m c t i o n ~ ~

ci(k) of period one, i.e.

ci(k + ^{1) =} c i ( k ) , k

E mn-l.

Similarly, for the systems the general solution depends on an arbitrary vector.

For a given n.th order difference equation 011

m

we are usually inter- ested in a

particu.lar solution,

on

m m ,

i.e. thc one for which the first n, consecutive values termed as

initial condition,s

(13.1) u ( a + i - l ) = W,i,

l < i < n

or

(1.3.2)

A" " u (~ ) =

I I , ~ ,

1 5 i 5 n, ( a = 0

if

ET

=

IN)

are prescribed. Each of the differcnc:e equations (1.2.1), ..., (1.2.7) togcther with (1.3.1) or (13.2) is called an initial

value pgroblem,.

Similarly, the system (1.2.8) together with

(1

3.3) U(.) = ^{U 0}

is called an initial value problem. For the linear systems

(1.2.11)

and (1.2.12) we shall also consider more general initial condition

(1.3.4) u(k0) = uo,

where k.0 E m 1 is fixed.

For

k

=

a ,

equation (1.2.5) becomes

~ ( a .

+ ^{n ) = f ( a ,} ~ ( a ) , u.(a +

^{I), . . .}

^,

^u(a,

+

^{R - 1)).}

Using the initial corldit,iorls (1.3.1), we find

Hence the value of

~ ( a +

ⁿ⁾ is uniquely determined in terms of known quantities. Next, setting

k

=

a +

^{1 in (1.2.5)} and using the values of v,(a

+

^{l), . . . , u ( n}

+

ⁿ⁾ we find that

u ( a +

¹

+ ⁿ⁾

is uniquely deter- mined. Now using irlductive arguments it is easy to see that the initial value problem (1.2.5), (13.1) has a unique solution

u ( k ) ,

k E

W ,

and it can be constructed recursively. Because of this reason difference equations are also called

recumiue relatl,on.s.

The

ezisten,ce

and

uniquen,ess

of each of the initial value problems (1.2.5), (1.3.2); (1.2.6), (1.3.1) or (1.3.2);

(1.2.7), (1.3.1) or (1.3.2); (1.2.8), (1.3.3) follow similarly. For the initial value problem (1.2.11), (1.3.4) the existence and uniqueness of the solution u ( k ) ,

ko 5 k

E is now obvious, whcreas for

ko 2 k

E

m

we need to write (1.2.11) as

(1.3.5)

~ ( k )

=

d-l(k)u(k +

^{1) -}

d-'(k)b(k)

ant1 from this

u(k0

- 1) and then

u(k0

^{- 2)} ant1 so forth, can be obtained miquely.

Finally, WC note that the initial value problem (1.2.3), (1.3.1) need not

have

a solution or a unique solution, e.g. the problcrrl

k u ( k +

^{2) -}

u,(k)

= 0,

k

E I N ,

~ ( 0 )

= 1, u , ( l ) = 0 has no sollltion.

In

fact, for

k

=

0

the difference equation gives

~ ( 0 )

= 0, which violates the initial conditions.

Also, the initial value problem

ICu(k+

2)

-7/,(k)

0 has infinitely many solutions

. ,

odd

for

k

even,

if ao(k)a,

( k ) #

0 for all

1.4. Some Examples: Initial Value Problems

The following cxamples provide a variety of situations of oc:mrrcrlc'e of initial value problems.

Example 1.4.1. Let

k 2

1 given points i n a plane be s ~ ~ h that any thrcc of them arc noncollinear.

WC

shall find the 1111rr1bcr of straight lines that can be forrncd

by

joining together cvcry pair of points. For this, let ~ ( k ) represents the nurrltm of suc.11 lincs. Let a new point t x added to the set of

k

points, which is also rlorlcdlirlcar with any other pair.

T he

rn1mt)cr of lines can now be written as

~ ( k + ^1).

^This

^{~ ( k} +

^{1) ('an be} found from

~ ( k ) by

adding the

k

IKW possible lines from the new

(IC +

1)th point to each of thc previous

k

poirltk Thnls, it follows that

(1.4.1)

?/,(k + ¹⁾

⁼

^{U ( k )} + ^{k ,}

^{IC E}

^IN(1).

Since when

k

= 1 there is no pair of points, it is obvions that

(1.4.2) ?/,(l) = 0.

The first,

order

initial value problem (1.4.1), (1.4.2) has a unique solution

~ ( k )

=

(1/2)k(k

- l),

k E W ( 1 ) .

Example 1.4.2. In nllmber theory the following result is fundanlental:

Theorem 1.4.1.

Every

positive integer greater than one can be expressed as the product of only a single set of prime nu~rhers.

The classical method of proving that

thcre

is no greatest prime mnnber is as follows: Suppose the contrary he true and the finite system of primes is ?!(1),

?)(a),

. .

, u ( k )

where ~ ( 1 )

<

~ ( 2 )

<

. . .

< o ( k ) .

Then, the rlurrlber m = ~ J ( ~ ) T J ( ~ ) . . - I I ( ~ )

+

¹ is prime to

~ ( 1 ) ~ ?1(2),...,v(k)

_. Hence, from Theorem 1.4.1,

m

is a prime which is greater than

~ ( k ) .

Let u s write t,his process of dcrivation of 'greater primes'

from

'lesser primes' thus

Then, we have

Thus, the problem gives rise to a nonlincar difference equatiou, which by writing u ( k ) = ~ ( k ) - (1/2) takes the compact form

(1.4.3) u ( k + 1) =

u 2 ( k ) + 4, ^{1 k E nv(1).}

Further, since ~ ( 1 ) = 2, for the difference equation (1.4.3) we find the initial condition

(1.4.4) W . ( l ) = 3/2.

Example 1.4.3. Consider the definite integral

(1.4.5)

It can easily be

seen

that 0

< u ( k ) < u ( k

- 1) and u ( k ) + 0 as

k

+ m, also

(1.4.6)

~ ( k +

^{1) = 1 -}

^{( k} + 1 ) ? ~ ( k )

(1.4.7) ~ ( 1 ) = -. ¹

e

8

Chapter

1 With 1/e correct to any number of places, the difference equation (1.4.6) provides unrealistic values. Indeed, rounding all the calculations to six decimal places, we obtain

~ ~ ( 1 )

= 0.367879

~ ( 2 ) = 0.264242

~ ( 3 ) = 0.207274

~ ( 4 ) = 0.170904

~ ( 5 ) = 0.145480

~ ( 6 )

=

0.127120

~ ( 7 ) = 0.110160 w.(8) = 0.118720

~ ( 1 0 ) = 1.684800

~ ( 1 2 ) = 211.393600.

~ ( 9 ) = -0.068480

~ ( 1 1 ) = -17.532800

Example 1.4.4. Let P K ( ~ ) =

x k : = O a ( k ) t k

be a given polynomial of degree

K .

Consider the problem of finding a polynomial

Q K ( ~ ) = x A : = o ~ ( k ) t k

of degree

K

such that Q K ( t ) - Q L ( t )

=

P K ( ~ ) , t E

R.

This leads to the following initial value problem

K

K

(1.4.8) ~ ( k ) = ( k

+

^l)?I,(k

+

¹⁾

+

~ ( k ) , k

E W(0, K

- 1) (1.4.9)

u ( K )

=

a ( K ) .

Example 1.4.5. Often we need to compute the value of

P K ( ~ )

=

C f = ( = , a ( k ) t k

at some to E

R.

The cornputation of

a ( k ) t b

=

a(k)

x t o x . ^{. .} x to needs k multiplications, and hence to find PK(t0) we require in total

K ( K +

^1)/2 multiplications and

K

summations. Homer's m,eth,od is an

algorith,m

^(a list of instructions specifying a sequence of operations to be uscd i n solving a certain problem) which reduces these rrmltiplications to only K and the same number of sumrnations.

At t

=

to, we begin with the representation

Thus, if the numbers

~ ( k )

are obtained from the scheme (1.4.10) ~ ( k ) =

a ( k ) + t o ~ ( k + ^l),

^{k E}

N ( O , K

- 1) (1.4.11) w.(K)

= a ( K ) ,

then v.(O)

=

PK(t(=,).

It is easy to see that the initial value problem (1.4.10), (1.4.11) is equiv- alent to

(1.4.12) ~ ( k

+

^{1) 1}

^{a ( K}

^{- k - 1)}

+ t o ~ ( k ) ,

k

E W(0, K

- 1) (1.4.13)

~ ( 0 )

=

a ( K )

and

u ( K ) =

P ~ ( t 0 ) .

Example 1.4.6. Consider the initial value problem (1.4.14) (t

+

1 ) ~ "

+

^{y'+ ty}

⁼

⁰

(1.4.15)

~ ( 0 )

= 1,

~ ' ( 0 )

= 0.

Evidently t = 0 is an ordinary point of the differential equation (1.4.14).

Insertion of y ( t )

= CEO ^7r(k)tk

into (1.4.14) yields

cc cc cc cc

c k ( k - l ) ? r ( k ) t " " + x k ( k - l ) l r ( k ) % " " + ~ k u ( k ) t k " + x

v,(k)t"+l

= 0,

1.=0 A:=O b:=O k.=O

which is the same as

cc

2 ~ ( 2 ) + ^v.(l) + c ^{[ ( k +}

^l)ku,(k

⁺

¹⁾

^{+ ( k + 2)(k +}

l ) v . ( k

+ ^{2 )}

k=1

+ ( k + ^1)11.(k + ¹⁾ + ^{~ . ( k}

^{- l)]}

^tk

⁼

^0.

Thus,

on

equating thc coefficients of tk to zero, we obtain (1.4.16)

2 ~ . ( 2 )

+u(1)

= 0

(1.4.17)

~ ( k + 2 )

=

- -

( k ⁺ % ( k + l ) - 1

( k + 2 ) ( k +

^{l ) ( k}

+

²⁾

u ( k - l ) , k

E N(1).

From

the initial conditions (1.4.15) it

is

obvious that

u ( 0 ) =

1, u(1)

= 0

and from (1.4.16) WC find

v.(2) =

0. Thus, i n turn we have a third

order

difference equation (1.4.17) together with the initial conditions

(1.4.18) U,(()) = 1,

u(1)

=

0, ~ ( 2 )

=

0.

Example 1.4.7.

A

system of polynomials

{Pk:(t)}, k

E

N

is called an orthonorrrlal system with respect to rlormegative wcight function

w ( t )

over the interval [cy, [I]

,

^if

1. Pk:

(t) is a polynomial of degree

k

W e will writc

(1.4.19)

P k ( t ) = a ( k ) t k + b(k)t"" +

^{'. '}

10

Chapter l

(:,q,k-l =

a ( k

-

l ) / u ( k ) .

To obtain c k , k we cornpare the coefficients of tk, this gives ck,A; =

- b ( k )

-

h(k +

¹⁾

a(k)

a ( k +

^{1) '}

Thus, (1.4.20) takes the form

(1.4.21)

t P k ( t )

=

a ( k ) ^{h ( k )} b ( k +

¹⁾

a(k

+

¹⁾

^{a ( k} +

¹⁾

Hence,

on

identifying

Pk(t)

^as v.(k) we observe that any three succes- sive orthonormal polynomials satisfy the second order difference equation (1.4.21).

In particular, if w ( t ) =

(1 -t2)"I2

and /? = --cy = 1, then (1.4.21) rc- duces to known recurrence formula for the

Chebysh,ev

polynmm,ials denoted by

TL:

( t )

(1.4.22) T k + l ( t ) = 2tT,:(t) - a(k)Tk-l(t),

k

E W(1) where a(1) =

fi

and a ( k )

=

1 for all

k

E N ( 2 ) .

The initial functions for (1.4.22) are defined to be (1.4.23)

Example 1.4.8. Let g(u) = 0 be a system of n nonlinear equations i n n unknowns I L ~ , . . . , U , .

Newton's method

for solving this system is in fact an initial value problem of the type (1.2.8), (1.3.3) where

1.5. Boundary Value Problems

Another choice to pin out the solution

u(k)

of a given difference system

011

N(a,

h -

1)

can

be

described as follows: Let B ( a , h) be the space of all real R vector functions defined

on

W(a, h) and 3

be

an operator mapping

B ( a ,

h) into

R",

then our concern is that

u(k)

must satisfy tllc boundary condition

(1.5.1) F [ U ]

= 0.

System (1.2.8) or (1.2.11) or (1.2.12)

together

with

(1.5.1)

is called a

b o m d - ary oalue

pr.oblern,. Obviously, initial c:ondition (1.3.3) as well as (1.3.4) is a special case of (1.5.1). The tcrrn bolmdary (:onclition c'orrlcs from the fact that

. F

allows the possibility of tlcfiuing cwlditiorls at the points a.

and b of W(a,

h ) . For

example, let kl

<

. . .

<

k,. ^(T

2

2)

be

sorne fixed points in N(a, h), t,hen we seek a solution u(k) of the diffcrcncc system on N(k.1, k,. - 1) satisfying

(1.5.2) $i(U(k-l), .'.

,u(k,,.))

=

0,

1

5

i

5

n

In the case when

. F

is lirlcar we shall prefer to write the boundary condition (1.5.1) as

(1

5.3)

L[u]

= 1,

where

thc vector

1

is known. Similarly, if

4i,

1

5

i

5

R are linear, then (1.5.2) will

be

written as

(1.5.4)

(1.5.5)

i=l q=l

It is of interest to note that (1.5.4), or equivalently (1.5.5), include i n particxlar the

12 Chapter

1

(i) Periodic Conditions:

T =

2 and for simplicity we let

k.1 =

0,

=

K

(1.5.6) u(0) = u ( K ) .

(ii) Implicit Separated Conditions:

(1.5.7) x ~ , i ( ~ , ) , ~ u , ~ ( k , ; )

= 1

5 i 5

T

(2 5

T

5 n , but fixed)

n,

q= 1

where

s1 =

1 , 2 , .

. .

,pl;

^{. . .} ; ^S,,. = 1,2, . . . ,[j,.

and C:=,

^[ji

^{= n.}

The

subscript

i ( . s i )

allows thc possibility that a t thc same point ki several boundary conditions arc prescribed.

(iii) Separated Conditions:

(1.5.8)

U+")(k.L) =

P

,,,, Z ( S & ) , 1

5 i 5

^T

(2 5

^T

5 n,, but fixed) whcre

.s,i,

1 5 i 5

T

arc the

same

as in (1.5.7).

I n (1.5.8) the subscript

i(si)

allows the possibility that the set of variables spccificd a t the boundary points

may

not be disjoint. For instance if

n =

7,

T = 4, ul(kl), 7Lg(kl), 7 ~ 2 ( k 2 ) , ug(kg), ~ 1 ( k 4 ) , u ~ ( k 4 )

and

7 ~ 7 ( k 4 ) ,

then

u,1

is fixed a t kl and

k4,

and

71,s

is fixed a t kl and

kg,

whcrcas no condition is prcscribcd for

764

and

71,s.

The indexing for the boundary conditions is specified by

I(1) = 1,

I ( 2 )

= 3,

2(1)

= 2, 3(1) = 3, 4(1) = 1,4(2) = 6

and

4(3) =

7.

For

a

given n t h ordcr diffcrcncc equation on

N ( a , b -

1)

^WC

shall consider sorrle of the following conditions.

(i) Niccoletti Conditions:

~ = I c ~ < I c ~ + 1 < ~ ~ < k ~ + 1 < ~ ~ ~ < k , ~ ~ < k , _ ~ + 1 < k , = b - 1 + ~ , where cach ki

E N ( u , b - 1

+

^v.)

(1 5.9)

~ ( k i ) = A i ,

1

5 i 5 n.

(iii) Abel-Gontscharoff Conditions:

(iv)

(n,p)

Conditions:

A”u(a.)

=

A ; , 0 5

i

5

^R - 2 (1.5.12)

A”v.(b -

1 + ^n.

^{- p )}

^{B , (0 5}

^p

^{5 n,}

^-

^1,

but fixed).

(v) (p,

n,)

Conditions:

A ” u ( ~ )

= I?,

(0 5

p

5

^R - 1, but fixed) (15.13)

A’~?r(b+

1)

= Ai,

O

5

i

5

n, - 2.

1.6. Some Examples: Boundary Value Problems

The following examples are sufficient to dernonstrate how discrete bound- ary value problems appear.

Example 1.6.1. Consider a string of length

K + ^1,

^{whose mass may}

be

neglected, which is stretched betwecn two fixed cnds

A

and B with a force

f

and is loaded at intervals 1 with

K

cqual masses

M

not under the influence of gravity, and which is slightly disturbed so that the tension in the string is constant along each segment and equal to

f.

Let

7 J ( k ) ,

1

5 k 5 K

(Figure 1.6.1) be the ordirlatcs at tirrle

t

of the

K

particles. Then, the restoring force in the negative direction is given by

F ( k ) = f [ ( u ( k

-

1)

- v ( k ) )

+ ^{( ~ ( k} + ¹⁾

^-

~ ( k ) ) ] .

Thus, by Newtonj’s second law the equation of motion of the kt11 particle is

14

Chapter 1

Figure 1.6.1.

Since each particle is vibrating, let

v ( k ) = w.(k) cos(wt

+ ^{4) in the}

above equation, t o obtain

- u h f u , ( k )

+ f ( - u . ( k

^{- 1)}

+ ^{2u,(k) -}

^U>(k

+

^{1)) =}

^0,

which is the same as

(1.6.1)

u,(k

+ ¹⁾

^-

^cu.(k) + ^{u ( k}

^{- 1)}

⁼

^0,

^k

^E

^IN(1,K)

where

c =

2

- ( w 2 ~ / f ) .

This second order homogeneous difference equation represents the am- plitude of the motion of every particle except the first and last. In order that it may represent these also, it is necessary t o suppose that

v ( 0 )

and

v ( K

+ ¹⁾ are both zero, although there are no particles corresponding to the values of k equal to 0 and K + ^1. With this understanding, we find that

(1.6.2) ~ ~ ( 0 ) =

u,(K

+ ^{1) =}

^0.

Equation (1.6.1) together with (1.6.2) is

a

second order boundary value problem.

Example 1.6.2.

Consider the electric circuit shown in Figure 1.6.2. As-

sume that

V0 =

A is a given voltage and V K + ~

=

0, and the shaded

region indicates the ground where the voltage is zero. Each resistance in

the horizontal branch is equal to R and in the vertical branches equal t o

4R.

W e want to find the voltage

Vk

for

1 2 k 5 K .

For this, according to

K i d o f f ' S curren,t

law, the sum of the currents flowing into a junction point is equal to the

s u m

of the currents flowing away from the junction point. Applying this law at the junction point corresponding to the voltage

Vj:+l,

^WC have

I k + l

=

h + 2

+

i k + l .

Using

Ohlm

^'S

law,

^{I =}

VIR,

the above equation can be replaced by

K:

- K:+1 -

K:+1

- K:+2

&.+l

-

0

R

⁺

4R '

R

^-

which is

on

identifying Vk. as w.(k) leads to the second order difference equation

(1.6.3)

_4~,(lC

+ ²⁾

^-

9 ~ ( k +

¹⁾

+ 4 ~ ( k . )

=

0, k

E

N(0, K

-

1)

and the bo'undary conditions arc

(1.6.4) v.(O)

=

A, ?],(K +

^{1) =}

^0.

7 ,,,,,,,,,,,,,,,,~, _,

I

I I _ I , I , _ .

t

Figure

1.6.2.

Example 1.6.3.

To

test whether a batch of articles is satisfactory, we introduce a scoring system. The score is initially set at

( K / ? ) +

^{( n - 1).}

If a randomly sampled item is found to be defective, we subtract

(TI

-

1).

If it is acceptable, we add 1. The procedure stops when the score reaches either

K + ⁽ⁿ

^{- 1) or} less than ( n - 1). If

K +

^{(n. - l),} the batch is accepted; if less than

(n,

- l), it is rejected. Snppose that the probability of selecting an acceptable item is p, and q = 1 - p . Let

Pj.

denote the

16 Chapter. 1 probability that the batch will be rejected when the score is a t k . Then after the next choice, the score will be increased by 1 with probability

p

or decreased by

(n, - 1)

with probability

q.

Thus

4

= P k + 1

+

qpk-(n,-l),

which on identifying

P k as

u ( k ) can be written as the n,th order difference equation

1

P P

(1.6.5) ~ ( k + ^n.)

^-

^{- u ( k} + ⁿ

^{- 1)}

+ - ~ ( k )

cl

= 0, k E N(0, K

-

1) with the boundary conditions

(1.6.6) u ( 0 )

= u(1)

=

^{' ' '} =

u(n.

^- 2) =

1, 7f4K

- 1

+

^{n,) =}

^0.

Example 1.6.4.

To overcome thc difficulty realized in Example 1.4.3, using the known behavior of u ( k ) , Dorn

and

RilcCracken [l61 used the famous recurrence algorithm proposed by Miller

[34]. They took u ( K ) =

0 for suffic:icrltly large K

and

recursed (1.4.6) backward. To c h x k the accuracy of results, they arbitrarily chose

K1

> ( K )

and

obtained another set of values of the integral (1.4.5). Thc search for K and

K1

continues urltil thc results agree to the desired degree of accuracy. However, this rrlet,hod does not appear to be practicable. To cvaluate the integral (1.4.5) we notice that u ( k ) also satisfies

(1.6.7)

~ ( k +

^{2) =}

^{( k} +

^{l ) ( k}

+ 2 ) ~ ( k )

-

( k +

^l),

^{k E N(0, K}

^{- 1)}

together with

(1.6.8) v,(l) =

1 -

u ( 0 )

and for sufficiently large K ,

w.(K+2)

=

u ( K + l ) and hence (1.4.6) implies

(1.6.9) 1

u ( K +

1)

= -

_{K + 3 '}

The boundary value problem (1.6.7)

^~

(1.6.9) will be solved satisfactorily later in Example 8.5.2.

Example 1.6.5.

For the continuous boundary value problem (1.6.10)

V'' =

f

^{( t ,}

!/l

(1.6.11)

_Y(Q)

= A , ?/(P)

=

B

the following result is well known

[SO]:

Theorem

1.6.1. Let

f ( t ,

^y) be continuous 011 [a,

a ]

x

R

and (1.6.12)

Then, the boundary value problem

(1.6.10), (1.6.11)

has a urlique solution.

However, even if

f ( t , W)

= f(t)y+g(t) the analytical solution of

(1.6.10),

(1.6.11) may not be determined. Faccd with this difficulty we find an approximate solution of (1.6.10)

,

^(1.6.11) by employing discrctc variable mcthods. One of such well known and widely used discrete rnetllods is due to Nom,e,row which is defined as follows: W e introduce the set { t k } , wherc

t k ;

= ^cy

+ ^{kh,, h}

⁼

^{( a}

^-

^cr)/(K + ^{l), k}

^E

W ( O , K +

^{1). Let}

u , ( k )

be the approximation to the

true

solution y ( t ) of

(1.6.10),

(1.6.11) at

t

= t k . W e assume that

7r(k)

satisfics the following second order diffcrence cqnatiorl (1.6.13)

u ( k + ¹⁾

^-

^27r(k) + ^{u ( k}

^-

^{1) = -h,2(} 1

^f

(O + ^{( k}

^- l ) h , u ( k -

1))

12

+lOf(cu+

kh,,Il,(k)) + ^{f ( 0} + ^{( k} + l ) h , , u , ( k +

^{l))), k}

^{E IN(1,K)}

together with the boundary conditions

(1.6.14) ~ ( 0 )

=

A , ?/,(K + ¹⁾

⁼

^B

The existerlcc and uniqueness of the boundary value problem

(1.6.13),

(1.6.14) and its uscfulness in coI1juction with initial valuc rncthods will be given in later chapters.

Example

1.6.6. Let [cy,@ be a given intcrval, and

P

: a = to

<

tl

<

. ..

<

t K + l = be a fixed partition. W e scck a function

S p ( t )

E C ( 2 ) [ a , p ] which coincides with a cubic polynomial in each subinterval

[tk-I,tk],

k

E

IN(1,K +

¹⁾ and satisfies

S,(tk)

= yk:,

k E N(0, K + ¹⁾

where the ordinates yk arc prescribed. The function S p ( t ) is called a cubic

spline

with respect to the partition

P.

Designating S g ( t ~ . ) by M k ,

k

E

N ( 0 , K +

¹⁾ the linearity of S g ( t ) in each subinterval

[tk-l,

^{t k ] ,}

k

E W(1,

K + ¹⁾

implics that

(1.6.15)

where hk

= tl;

- t k - 1 . If we integrate twice

(1.6.15)

and evaluate the constants of integration, we obtain the equations

18 Chapter 1

and

From (1.6.17), we have

In virtue of (1.6.15) and (1.6.16), the functions

S $ ( t )

and

S p ( t )

are continuous on [a,

01.

The continuity of

S b ( t )

at t

=

t~; yields by means of (1.6.18) and

(1.6.19)

the following secorld order difference equation

k

E W(1,

K ) .

0nc:e two appropriate boundary conditions, say,

MO

and M K + ~ are prescribed, the solution of (1.6.20) serves to determine

S p ( t )

in each subinterval [tI;-l,

tk].

Difference equation (1.6.20) can be used to

find

an approximate solution of the problem (1.6.10), (1.6.11).

For

this, we observe that

Mc;

are given to be

f(tk,yl;)

and the problem is to find VI;,

k

E

N(0, K +

^{1). Thus,}

if i n particular

hI;

= h,

k

E W ( 1,

K + ¹⁾

then once again we have t~;

=

N i- kh, h

= (p

-

a ) / ( K + ¹⁾

and on identifying yk as u ( k ) we need to solve the following difference equation

(1.6.21) u ( k

+ ¹⁾

^{- 2u,(k)}

+

^{u ( k - 1)}

⁼

- h 2 ( f ( o 1

+ ^{( k}

^-

l ) h , u ( k

- 1)) 6

+ 4 f ( a +

^kh,

^{4 k ) )} +

^f(o

+ ^{( k} +

l ) h , w.(k

+

^l))),

^k

^{E N(1,}

^{K )}

together with the boundary conditions (1.6.14).

Example 1.6.7. Let in a domain

D C RK+’

the function

q5(tl,.

. .

, t ~ + ~ )

be given for which third order derivatives exist. Then, a necessary condition