37?

NONLINEAR BOUNDARY CONDITIONS

IN SOBOLEV SPACES

THESIS

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

DOCTOR OF PHILOSOPHY

By

Walter Brown Richardson Jr., B.S., M.S

Denton. Texas

December. 1984

Richardson. Walter B.. Nonlinear Boundaru Condit ions in

Sobolev Spaces. Doctor of Philosophy (Mathematics),

December. 1984. 65 pp.. 2 tables, bibliography, 28 titles.

The method of dual steepest descent is used to solve

ordinary differential equations with nonlinear boundary

conditions. A general boundary condition is B(u) = 0 where

where B is a continuous functional on the nth order Sobolev

space H n[0.1J. If F:H nCO,l] — • L 2[0,1] represents a

2 differential equation, define *(u) = 1/2 IIF < u) li and £(u) =

1/2 l!B(u)ll2. Steepest descent is applied to the functional

2 £ a * + £. Two special cases are considered. If f:lR — • R

C 2 )

is C ... a Type I boundary condition is defined by B(u) =

f(u(0>.u(1)). Given K : [ 0,1]xR —• R and g:[0,l] — • R of

bounded variation, a Type II boundary condition is B(u) =

J* K(x.u(x)) dg(x).

The following results are proved. A close relationship

is established between the two best known requirements for

convergence of general steepest descent - Rosenbloom's

strict convexity condition and Neuberger's gradient

inequality. If (5 is strictly convex, then Neuberger's 2

inequality holds for i = 1/2 llv/Jll . In dual steepest

descent, if the gradient inequality holds for 4> and $, and

the cosine of the angle between v4> and v£ is bounded away

from -1. then the gradient inequality holds for f. For A

Type I boundary condition, the gradient inequality is

equivalent to IIvfII being bounded away from 0. An explicit

form for v$(u) = B'(u) B(u) where B is a Type II boundary

condition is given. As a corollary, the least constant for

Sobolev's imbedding inequality. M f II S C • II f II „ , is shown 00 n n # 2

j- 1 n 2 sin Ckfl) 1 1 / 2

t o b e C n = { STT **-1 tanh(sln(ke)) ) w r , e r B ' n+l

Finally, dual steepest descent is implemented in a FORTRAN

program, and used to solve the test problem y* = y.

y ( 0 ) = y ( l ) 2 .

TABLE OF CONTENTS Page

LIST OF TABLES iv

Chapter

I. INTRODUCTION S

II. DUAL STEEPEST DESCENT 16

III. TYPE II NONLINEAR BOUNDARY CONDITIONS 23

IV. NUMERICAL SOLUTIONS OF SECOND ORDER PROBLEMS WITH

NONLINEAR BOUNDARY CONDITIONS 3?

APPENDIX S7

BIBLIOGRAPHY . . . . . 63

i i i

LIST OF TABLES

T a b l e Page

I. Best constant for the Sofaolev Imbedding

Inequality on H n C 0 . 1 3 50

II. Output from FH2.CGR . . . . . . . . 51

i v

CHAPTER I

INTRODUCTION

Differential equations, both ordinary and partial, in

which the solution is required to satisfy certain boundary

conditions occur in such diverse areas as mechanics,

accoustics, and electrical engineering. Taking second order

ordinary boundary-value problems as an example, the linear

case has been exhaustively studied over the past 200 years.

Much work has been done over the past forty years in

developing general methods of attacking the case in which

the differential equation is nonlinear. However, almost

without exception, the work has involved linear boundary

conditions of the form

A y (a) + A y ' ( a ) = 0 X M

Bl y(b) + B 2 y'(b) = 0 .

Very little is known about such problems when the boundary

conditions are nonlinear.

Mawhin (6. 3) has proposed the following problem to

2 describe nonlinear vibrations: find a y in H [0.1] such

that

2 y•(x) y(x) y ' 1 ( x ) = - for x in [0,1]

l + y < x )

y'(0) 2 - y(0) 2 = 0 and y(:l)*|y(l) - y'(l)| = 0

An example of a boundary-value problem which arises in the

study of electrical network theory is given by Morosanu (7).

Given physical constants L > 0, R £ o. c > 0, G £ o,

R > 0, and C > 0; a nonlinear resistance f„ which is 0 0

maximal monotone in IR x IR ; and a voltage e per unit length

impressed along the line in series with it; find the current

is-u flowing in the line and th€» voltage v across the line,

so that

0\i <5v

"it"" <3v <3u

L g t - g x + R * u + e(t.x) = 0

i at <3x + G•v = o. for 0 < x < 1 and t > 0

u ( 0 . x ) = u Q ( x ) ; v ( 0 , x ) = v ( x ) : 0 5 x 5 1

dv(t . 1) -u(t.l) = C — + f' ( v ( t . D ) . t > 0 .

Ot 0

Historically, in deriving governing differential equations

from physical principles, nonlinearities were neglected

because they were difficult to handle. One wonders if many

boundary conditions should actually contain nonlinear terms

in order to give a more accurate! picture of the physics.

One seldom encounters a definition of boundary

conditions - rather a group of example boundary-value

problems is presented'to give a feel for what such

conditions are. Taking as our setting 2nd order problems ,

a linear boundary condtion for the differential equation

2 2 F(u) = o. where F:H [0,1] —» L . is defined to be one of the

form B(u) = 0, where B is a bounded linear functional on

2 H [0.1]. This includes standard textbook examples, interior

boundary conditions, and still more general ones of the form

B(u) = /Q u dg. A nonlinear boundary condition is B(u) = 0

where B is a continuous functional on H2[0.13. often assumed

(2) to be C . For Mawhin's problem mentioned earlier.

B(U) • l/2-|u'(0)2 - u(0) 2j + 1/2*ju(l)«|u<1) - U'(l)|| .

Perhaps the essence of the definition lies in the fact that

the range of B is a subset of the reals, rather than that

B(u) involves only the values of u at the topological

boundary of the domain of u.

There are many methods for handling nonlinear problems:

fixed-point theory, generalized inverse function theorems,

homotopy and degree theory, the Newton-Kantorovich method,

and the method of steepest descent. The last two are

particularly attractive because they yield not only valuable

theoretical information, but also numerical results when

implemented on a computer. Steepest descent dates back to

Cauchy (2), and although not used extensively on

minimization problems because of its slow convergence, the

method is very robust compared to Newton's method, which may

break down completely. Variations of steepest descent are

called general gradient methods and have found increasing

favor in numerical analysis. Recently. Neuberger (8. 9. 10.

11. 12) has used steepest descent in a Sobolev space setting

to solve differential equations with linear initial and

boundary conditions.

O n e of the chief a p p e a l s of steepest descent is its

s i m p l i c i t y . S u p p o s e each of X and Y is a Hilbert s p a c e and

(2 > F:X — » Y is C ; to use steepest descent to s o l v e the

e q u a t i o n F ( x ) = 0. o n e d e f i n e s *:X — » IR by * ( x ) =

2 1/2 IIF (x) I I . A zero of • is a s o l u t i o n of the o r i g i n a l

Y

e q u a t i o n and a m i n i m u m of * w o u l d be a "least s q u a r e s "

s o l u t i o n to the e q u a t i o n . In a d i f f e r e n t i a l e q u a t i o n

s e t t i n g 4> is a general v a r i a t i o n a l p r i n c i p l e a p p l i c a b l e to

m a n y p r o b l e m s w h e r e there is no c l a s s i c a l v a r i a t i o n a l

p r i n c i p l e . At the point x Q in X. the n e g a t i v e gradient of *

p o i n t s in the d i r e c t i o n to m o v e for the fastest

i n s t a n t a n e o u s d e c r e a s e of the function * . The fundamental

e x i s t e n c e and u n i q u e n e s s theorem for o r d i n a r y d i f f e r e n t i a l

e q u a t i o n s g u a r a n t e e s the e x i s t e n c e of z:[0,<»> — • X such that

z(0) ss XQ and z'(t) = - v # < z(t ) ) . O n e tries to find

c o n d i t i o n s that insure that 1im z(t> = u e x i s t s and that u t-K»

is a local m i n i m u m of * . Two such c o n d i t i o n s a r e the

c o n v e x i t y c o n d i t i o n of P. C. R o s e n b l o o m and the gradient

i n e q u a l i t y of N e u b e r g e r . A theorem in C h a p t e r I shows that

if a functional 4> s a t i s f i e s R o s e n b l o o m ' s c o n d i t i o n then 7 ( x )

2

= 1/2 llv*(x)ll s a t i s f i e s Neubergeir's gradient i n e q u a l i t y .

A d e t a i l e d d e s c r i p t i o n of using steepest descent w i t h

the S o b o l e v inner product to s o l v e the test p r o b l e m y* = y,

y(0) = l on [0.1] is given by N e u b e r g e r in (11). B r i e f l y ,

letting X = jy in H 1 f 0 . 1 ] : y(0> = oj. Y = L 2 [ 0 , 1 ] . and

defining F:X — • Y by F(x) = <x+_L)' - (x+i.), one has that if

z is a zero of F. then y = z + i is a solution to the

original problem. A discrete steepest descent is

implemented to find a numerical solution and. most

importantly, the rate of convergence of the finite

difference scheme is shown to be far superior to that of the

same process, using the Euclidean inner product.

suppose the problem is complicated by introducing the

simplest nonlinear boundary conditions: y' = y.

2

y(0) = y(l) . There are several ways one might attack the

new problem. Define F:H 1[0,1] — • L 2[0.1] and B:H i[0,ll —» IR

by F(x) = x' - x and B(x> = x(0) - x(l) 2. Neuberger in (8)

uses a scheme which is based on steepest descent for the 2

functional #<x) = 1/2 HF(x)ll and employs the family of

projection operators <P > 1 where P is the X x i n H L O t l ] X

orthogonal projection onto the null space of B'(x). This

approach is closely related to the gradient-projection

method used in nonlinear optimization (14, is, 19). It has

the property that, both in the continuous and discrete

steepest descent processes, one always stays in the set of

all functions which satisfy the boundary conditions.

We shall use another tack and attempt to solve the

problem by applying steepest descent simultaneously to the

differential equation and to the boundary condition part of

the problem. The name penalty function is given to this

method when applied to the constrained optimization of a

10

real-valued function. However, the idea does not seem to

have found its way into the area of differential equations

with nonlinear boundary conditions. Here convergence

criteria would be harder to come by. because of the infinite

dimensional setting and resulting loss of local compactness.

(2)

Given F: X — • Y and B:X — • IR. both C . we seek an x in X

such that F (x) = 0 and B(x) = 0. Define *:X — • IR by *(x) =

1/2 it F < x) li2 and £:X — • IR by $<x) = 1/2 llB(x)ll2. These are

measures of how close an element of X comes to satisfying

the differential equation and the boundary condition,

respectively. If G:X —» XxIR is given by G(x) = ( ' a n d

0 V X /

e< x) = 1/2 1,G{x)l,xX|R - 1 / 2 IIF<X>"2 + 1/2 IIB (X > 112 = *(x) +

^(x). then steepest descent can be used. x is a solution to

the pair of equations only in case it is a zero of the

funct ional $.

A few comments about this proposed method are in order.

First, although it will work on a problem with linear

boundary conditions, it clearly will not be as efficient as

steepest descent applied to the subspace of all functions

satisfying the boundary conditions or as classical

algorithms designed to solve a specific class of problems.

The method wi'.l 1 come into its own when the boundary

conditions are nonlinear. While not depreciating the

importance of, say, fast Poisson-so1vers, there is a great

11

need for a widely applicable, type-iindependent method which

handles general boundary conditions and in which there is a

close relationship between theoretical and computational

aspects.

Second, when the boundary conditions of the above

2

example are written as y(0) = y(l) it is easy to find an

initial guess that satisfies them and then begin the

projection method - never leaving the set K of all functions 2

f such that f(0) = f(l) . However, if the problem were 2

posed in an H [0.1] setting, the boundary conditions might

2

have been written equivalently as y<0) = y'(i) and it is no

longer as easy to find an initial guess. Also, one stays in

a different set K of functions. Lastly, in partial

differential equations one oftens encounters a situation in

which it is not known a priori from physical or theoretical

considerations what a "natural" set of boundary conditions

is. In this case, repeated trials of the dual steepest

descent using different choices for the functional B should

yield valuable information about what boundary conditions

are feasible.

This paper investigates two specific forms of nonlinear

boundary conditions. The natural setting for a first order

problem is H 1 ; suppose f:IR2 —• IR is a given function

and define B : H 1 [ 0 , 1 ] — • IR by B(u) = f (u(0 ) . u( 1 ) ) . This is

the most general nonlinear boundary condition which involves

1 2

only u(0) and u(l) and is still manageable. A sufficient

condition for the gradient inequality to hold for this

boundary condition is given in Chapter II. The second type

of condition uses the values of u everywhere in [0,1],

although the non 1 i near i t i es are milder. Given g:[0.l] • |R

and K : [ 0, 1 ] x(R — • IR. define B:H n[0,l] — • |R by B(u) =

K(x.u(x)) dg(x). This includes conditions such as u(0) =

2 u(l) and, in fact, any problem in which the nonlinearities

occur pointwise, but encounters difficulties with u(0)*u(l>

2 + uCl/2) = 0, because of the first term.

An interesting spinoff of the above work relates to one

of the large class of inequalities in analysis which go

under the name of Sobolev. These state that the norm of a

function in one space is dominated by a constant times the

norm of its derivative in a second space. The existence of

such constants dates back to Sobolev (17) and is critical to

the development of theoretical partial differential

equations (l). Within the last ten years several articles

have been written which address the question of what is the

best or least constant that can be used in a particular

Sobolev inequality. Talenti (18). Rosen (13). and Lieb (4)

seek the least c such that lluII < c-ilu'll where l < p,q < <»

and domain(u) = IRn. Using variational methods. Marti (5)

found the best c for the inequality Hull^ < c-llul^ 2 and gave

numerical results for the higher order Sobolev spaces.

13

H ntO,13, whore n • 2 6. As a Corollary to a Theorem in

Chapter III we show that the least c such that for all u in

H ntO,l], Hull S c Hull is given bu <» n, 2

2 sin 3(k©) >1/2 n , 1 • 2 sin (k©) v

'n ~ |n+l Xk=l tanh(sin(k9)) j where 0 = — — •

n+1

and require none of the machinery of the calculus of

variations.

Chapter IV describes how the dual steepest descent can

4 4 be implemented on a computer. Given F:IR —» IR and B: R — • IR

2

we seek a member u of H [0,13 such that for t in Ca.bl,

F(t.u(t).u ,(t).u , ,(t)> = 0 and B(u(a).u(b>.u'(a>.u•(b)) = 0.

A finite-difference scheme is used to reduce the problem to

solving a set of nonlinear equations by steepest descent.

Much improved convergence over traditional steepest descent

is obtained by employing the Sobolev rather than the

Euclidean inner product and by using a nonlinear version of

Hestenes* conjugate-gradient algorithm. Finally, the code

of a FORTRAN program, FH2.CGR, which uses the above ideas is

discussed, as is the output from the test problem y* = y,

y(0) a y(1) 2.

CHAPTER BIBLIOGRAPHY

1. Brezis, Haim. and Nirenberg,, L.. "Positive Solutions of Nonlinear Elliptic Equations Involving Critical Sobolev Exponents." Communi cat ions on Pure and Applied Mathematics. XXXVI (July. 1983). 437-477.

2. Cauchy. A., "Methods generals pour la resolution des systems d'equations simultanees." Comptes Rendus Hebdomada i res des Seances de 1'Academi e des Sciences. 25 (1847). 536-538.

3. D'Hondt. T.. "The Approximate Solutions of Differential Systems with Nonlinear Boundary Conditions," Zei tschri ft fur Anqewandte Mat hemat i k und Mechanik. (July. 1978), 423-42S.

4. Lieb, Elliott. "Sharp constants in the Hardy-Litt1ewood-Sobolev and related inequalities." Annals of Mathemat ics. 118 (September. 1983), 349-374.

5. Marti, J. T.. "Evaluation of the Least Constant in

Sobo lev's Inequality for H l(0.s)." SI AM Journal on Numerical Anal us i s. 20 (December, 1983), 1239-1242.

6. Mawhin, Jean, "Recent Trends in Nonlinear Boundary Value Problems." Proceedi nqs of t he Seventh Internat ional Conference on Nonlinear Osci1lat ions. (Berlin September. 1975). 51-70.

7. Morosanu. G., "On a Class of Nonlinear Differential Hyperbolic Systems with Non-local Boundary Conditions." Journal of D i f ferent ial Equat ions. 43 (March. 1982). 345-368.

8. Neuberger, J. W. "Gradient Inequalities and Steepest Descent for Systems of Partial Differential Equations." Proceedings of the Conference on Numer i ca1 Ana 1 us i s, Texas Tech. Un i vers i t y. 19 8 3.

9. , "Some Global Steepest Descent Results for Nonlinear Systems." Trends in Theory and Pract ice of Nonlinear Pi fferent ial Equat ions. edited by V. Lakshmikantham. New York, Marcel Dekker, 1984, pp. 413-418.

14

15

10. ; . "Steepest Descent and Time-Stepping for the Numerical Solution of Conservation Equations." Proceedi nqs of the Conference on Physical Math and Nonli near Different ial Equat ions. Marcel Dekker. ( To apptear ).

11. , "Steepest Descent for General Systems of Linear Differential Equations in Hilbert Space," Proceedings of a Symposium held at Dundee. Scot land March-Julu 1982. Lecture Notes in Mathemat ics. Vol. 1032. New York, Springer-Verlag. 1983, 390-406.

12 . , "Steepest Descent for Systems of Nonlinear Partial Differential Equations." Oak Ridqe Nat ional Laboratory CSD/TM-1 & 1, (1981).

13. Rosen. G.. "Minimum Value for c in the Sobolev Inequality 11*11 :£ c-ll7#il ." SI AM Journal on Appl led

6 Z Mathemat ics. 21 (July, 1971 ), 30-32.

14. Rosen, J. B.. "The Gradient Projection Method for Nonlinear Programming. Part I. Linear Constraints," Journa1 of the Society for Indust rial and Applied Mathemat ics. VI I I (March, i960). 181-217.

15. Rosen, J. B., "The Gradient Projection Method for Nonlinear Programming. Part II. Nonlinear Constraints." Journa1 of t he Soci et y for Indust rial and Appl ied Mat hemat i cs IX (December, 194.1), 514-532.

16. Rosenbloom, P. C., "The Method of Steepest Descent." Sympos ia i n Appli ed Mat hemat i cs. Providence. Rhode Island, American Mathematical Society, 1956, 127-176.

17. Sobolev. S. L.. "On a Theorem of Functional Analysis," Matematlceski i Sbornlk. 4 ( 1938). 471-479. American Mathematical Society Translation Series 2, 34 (1963), 39-68.

18. Talenti, Giorgio. "Best Constant in Sobolev Inequality," Anna 1i di Mat hemat i ca Pura ed Appli cata. Series 4. Vol. 1 10. ( 1976). 353-372.

19. Wismer, David, and Chattergy. R., Int roduct ion t o Nonlinear Opt imi zation. New York. North-Holland, 1 978 .

CHAPTER 1 I

DUAL STEEPEST DESCENT

From a theoretical standpoint the most difficult aspect

of steepest descent is finding conditions for convergence.

In an excellent survey article, P. C. Rosenbloom (6, p. 128)

establishes the following fact. If X is a Hilbert space,

$ :X —* IR is C* 2*. z(0) = x Q and z'(t) = - v*(z<t>) for t 2

0. and there is an A > 0 such that for all x.y in X.

2 <H(x)y,y> > A• IIyII where H denotes the Hessian of • , then

lim z(t) s u exits and is the unique minimum of *. If * t-+oo

2

arises from linear least squares, i.e.. *(x) = 1/2 IIF (x > II

for some positive self-adjoint F. then <t> will satisfy

Rosenbloom's condition. Neuberger (2. 3. 4) gives another

sufficient condition for convergence of a variational If

ft is an open subset of X, z(0> = x Q and z(t) = - r*(z(t)),

Range(z) c a, and there exists a C > 0 such that for all x

in X. llv*(x)il £ C*llF(x)ll, then the lim z(t) s u exists and t-*oo

F(u) = 0. The following theorem connects the two

conditions.

(2) Theorem I If *:X — • IR is C . a is a positive number so

2 that for all y in H, <H#(x)y.y> > a IIyII and for x in X.

2 f (x} = 1/2 II v*ll . then i satisfies Neuberger's gradient

1 6

17

inequality, i.e., there is a positive C so that lly?(x)ll £

C*llv + (x)ll for all x in X.

Proof: We use primes to denote Frechet derivatives and

H #(x) to denote the Hessian of • at x. Let each of x.h. and

| <(v*)(x+h),k> - <(v*)(x).k> |

k be in X.

<(?•)•(x)h.k> « | <(v*)(x+h),k> - <(v*>(

= * • (x + h)k - • •< x)k

« * • ' ( x) (k. h)

= <H.(x)k.h> . P

So. Tf'(x)h a <(v*)•(x)h,v«(x>> = <H (x)(v*(x)).h> and v-*(x) P

= ( x ) ( v4> ( x ) > . Now <H^(x)y,y> a*llyll2 just says H^(x) Js.

a-I which implies H^(x)2 > a2•I . P

IIVfl|2 = <H(X) (V*(X) ) ,H fX)<7#(X> )> v v

2. a 2 • II v# ( x ) II 2

o 2 , X

= 2 • a • 7 ( x ) .

Taking square roots we have the desired inequality.

Both Rosenbloom's and Neuberger's conditions for

convergence of the steepest descent process are far from

necessary. Work remains to foe done to find large classes of

problems for which they would give existence results not

already obtained by other methods. Also, their connection

with the well known Palais-Smale (S. 1. p. 353) property

should be investigated.

In the dual steepest descent process, it is natural to

ask if the gradient inequality applies to both of the

18

functions * and $, will it apply to $ ? Not without some

additional condition, for example, that the angle between

the vectors v*(x) and v$(x) be bounded away from 7r .

Theorem I I If fi is an open subset of X, x e a, and

(1) Range(z) c a where z Q = x, z'(t)= -v£(z(t>)

(2) there is a c > 0 and a d > 0 such that for all y in it.

Ilv#(y)ll > c • IIF ( y) II and II ( y) II > d • IIB (y) II

(3) there is an a in {-1.1] such that for all y in a

< v # ( y > . ( y ) > > a

II v*(. y) II • l|v$ ( y) II

Then 1im z(t) s u exists and F(u) = 0 = B(u).

t -*<x>

Proof: Clearly, we may assume a € {-l.o) .

II v C ( y) II2 = llv*(y) + v ( y) II2

= II v* ( y) II 2 + 2 • < y ) . v$ (y) > + II v^ <y) II2

2 2 S II v* (y) tl + 2-a-l!v*(y) II • llv$ (y) II + llv£(y)ll

= -a | II v* ( y) II2 - 2 • llv*(y) II • II v-5 ( y) II + llv${y)ll2 j

( 1+a) • | II v# ( y) II 2 + llv$(y)ll2 j.

£ {l+oO»| llv*{y)ll2 + llv£(y)ll2 j

| IIF ( y) II2 + IIB { y) II 2 | > (l+a)* min<c 2.d 2>

(1+a) min<c2.d2> IIG (y) II2

Letting k = J l+a min<c,d), we have a constant K such that

for all y in a, II v C C y) II > K-IIG(y)ll and Neuberger's theorem

assures that lim z(t) s u exists and satisfies G(u) = 0. t->«o

For first order problems nonlinear boundary conditions

which involve only the values of the unknown function at the

1?

endpoirits of the interval can be put in the following

2 ( 2)

general framework. Given f: IR — • R . c ; define a Type I

boundary condition on H n[0,l] to foe a functional B(y) =

f(y(0),y(l)). f is said to generate B. To investigate

convergence of the boundary part of the steepest descent

process we need to compute v$(x) = B«(x)*B(x). A simple

argument shows

B'(x)y = f 1(x(0),x(l))-y(0) + f(x(0),x(l))«y(l) .

and to compute the adjoint, we seek <g,h> in IR x H 1 such

that for all y in H 1,

<y.h >Hl = <B'(x)y,g> . i.e., so that

/J y - h + y ' - h ' = <f 1 (P x)•y<o) + f 2 < P > • y < l) > • g

where P x = <x(0),x(l)>. Supposing for a moment that h has a

second derivative we can use integration by parts to write

the above as 0 = Jn yth-h*') + y(l>-< h'(l) - f (P )•g >

' ° 2 X

y(o)•< h 1(0) + f,(p )-g > 1 X ^

Which leads to the boundary value problem: find h in

H ^ O . l ] so that

h' • = h

h'(l) = f <P ).g and h'<0) = - f <P )-q 2 x 1 x ^

The solut ion is

h ( t ) = s i nh ( 1) { V P x > - c ° s h d - t > + f 2(Px>•cosh(t> } .

and the following theorem can now be proved.

20

Theorem 11 I Let B be a type I boundary condition on H i[0,l]

2 ( 2 ) with generating function f:!R —1> IR. f C and not

identically 0. The Following statements are equivalent:

2 (1) There is a d > 0 such that for (a.b) in R .

Ilvf (a.b)ll i d .

(2) There is a c > 0 such that for all x in H 1[0,1J,

it v <; ( x) II 2: c • IIB ( x) ti.

Proof: iiv?(x)n2 = iib • ( x) *B < x) ill2

H H

- I B'»>| 2-{VV slnh(l) {vPx'COSh<1> + f 2 C P x ) } «•

W { W + W C O S H ( N } }

l B ( X ) I < A Vf(P ),Tf(P )> x X 2

where A =

sinh( 1) IR

cosh(1) l is a positive definite

1 cosh(1)

matrix with lower bound m = cosh{l) - 1 and upper bound M =

cosh(1) + 1.

(l) —• (2) Let x e H 1.

1 2 IIB '.( X) B ( X > tl t = s l n h ( 1 ) <A7f(P x).rf<P x)> |B(X)|

H IR

m 2 2 II v f ( P ) II . I B ( X ) I

x 2 ' I s i nh(1) x R 2

m 2 2 d IB(x)I

sinh(1)

and there exists a global c for B.

(2) -• (1) Let G = <(a.b) e IR2: f(a,b) * 0 > and (a.b) e G.

For t in [0.1], define x(t) = (l-t)a + tb. Now x € H*[0,1 J

M and I B ( x) I 2 . . llvf(P ) it 2 £ llv$(x)ll2 > C 2-|B(x)| . 1 1 sinh(l) x 1 1

21

C 2 sinh(l) Since B (x) * 0. llvf <P ) II > and letting d be

X M

the square root of the right hand quantity, we have that for

all (a,b) in G. Ilvf(a,b)ll £ d. To show that the inequality

2 must hold on all of R . let S be the interior of the

2

complement of G and suppose that S is nonempty. Since R is

connected, there is a limit point p of S in the complement

of S. Let <qR> be a sequence of points of S and <rR> be a

sequence of points of G. both of which converge to p. Since

q belongs to the open set S on which f is zero, vf(q.) = 0, K K

and since the gradient is continuous, vf(p) = 0. However,

r in G implies llvf(r ) II > d and therefore llvf(p)ll 2: d > 0. K *\

2 The contradiction shows that G is dense in R .

CHAPTER BIBLIOGRAPHY

Berger .Melvyn, Nonl ineari ty and Funct ional Analysis . New York, Academic Press, 1977.

Neuberger, J. W. , "Gradient Inequalities and Steepest Descent for Systems of Partial Differential Equat ions."Proceedings of the Conference on Numerical Analysis. Texas Tech. University. 1983.

"Steepest Descent for General Systems of Linear Differential Equations in Hilbert Space."Proceedings of a Sumposlum held at Dundee. Scot land March-Julu 1982.. Lect ure Notes in Mathemat ics. Vol. 1032. New York, Springer-Verlag. 1983, 390-406.

4. , "Steepest D«iscent for Systems of Nonlinear Partial Di f f erssnt ial Equations," Oak Ridge Nat ional Laboratory CSD/TM-161. (1981).

5. Palais, R., and Smale, S., "A Generalized Morse Theory." Bui let in of the American Mathemat ical Society. 70 (1964), 16S-171.

6. Rosenbloom, P. C.. "The Method of Steepest Descent," Sumposia in Applied Mathematics. Providence. Rhode Island, American Mathematical Society. 19S6, 127-176.

22

CHAPTER III

TYPE II NONLINEAR BOUNDARY CONDITIONS

An important class of boundary-value problems is the

one in which the differential equation is ordinary of order

n and the boundary conditions involve the unknown function

but none of its derivatives. Even this class is so large

that we restrict ourselves to the following subset, where

without loss of generality all work is done on the interval

[0,1]. Let n be a positive integer, K-. [0,l]xlR — • IR be

continuous and such that is continuous, and g:[0,ll — • IR

be a function of bounded variation. A Type II boundary

condition with generating kernel K is B(u) = 0 where

B : H n [ 0 . 1 ] —» IR is given by

(*) B(u) = K(x,u(x)) dg(x )

If K is defined by K(a.b) = (l-a)b - a b 2 , and g by

, o x = o g ( x) = < l 0 < x < l

. I 2 x = 1

2

then the boundary condition y(0) = y(l) of our sample

problem can be put into this framework. Once again, to

investigate steepest descent for the functional $(u> = 2

1/2 IIB (u) II , one seeks an explicit form for v£(u) =

B'(u)*B(u). Theorem IV If y 6 H n [ 0 . 1 ] and 8: H n[0.13 —» IR is defined by

23

24

(*). then

(a) B • (y > f • / J K (t.y(t))•f(t> dg(t)

^ 1 sin(kfl)

(b) ( B 1 { u) l)(t) > — — X, n, — — : — r r S T T K,(X.y(x)>

* n+1 k = l sinh(sin(kfl)) > 0 2 3

| cosh((l-x-t)sin(ke))* cos((t-x)cos(kS))

cos(2k9)-cosh(<l+t-x)sin(k8))*cos((t-x)cos(kfl))

sin(2k0)•sinh((1+t-x)sin(kS))•sin((t-x>cos(ke)) j>dg(x)

2 n t + T m 2 k = i s i n ( k e > / 0

K2<*,y<x>>

| cos(2k0)'sinh((t-x)sin(ke))'cos((t-x)cos(kfl))

+ sin(2k0)•cosh((t-x)sin(k0))•sin((t-x)cos(kfl)) j dg(x)

Proof: (a) This follows directly from the definition of

Frechet derivative.

(b) We first prove the theorem in the special case that

both K and g are c * 2 n ) and that for k=l 2n, both g*fc*(0)

(k >

and g (1) are 0. Using the definition of adjoint

(4, p. 30?). we seek a member h of H n [ 0 . l ] such that for

every f in H n [ 0 , l ] , <B'(y)f.l> = <f,h> n .i.e., so that iR H

/» f d 9 - ik 2 o ilf

(k) _(K) h

Supposing that h has 2n derivatives (Later this will be

verified), integration by parts can be used to obtain

lo t K 2 [ I . y ] 9 ' - / o f-0 <" * /J f-ft < 2 R ) «I -

I*"' ( - D ' - ' - J f < J » . h <2 R - » - J > ,» I

j=0 lo /

" lo '•{ *k = 0 n } d I • i..". f ( J ) h'2*-'-.!' |» . n _k- i •*.1 *j=0

25

Using Fubini's Theorem for finite sums, the double sum can

n-1 „ n . . ik-l-j ^(j) ( 2k— 1 — j) ,1 be rewritten as £ j _ 0

ZK-j+l ( - 1 ) f h '0

_ _. ,, _n-1 f < j) _n-i , . x m - j . (2m-j + l) .1 ) and finally as. 2 j _ 0 { m-j * 'o j '

By a standard argument, for the above equation to hold for

every f in H n[0.l], h must satisfy the boundary-value

problem :

. k .(2k) „ , r ( 1 ) z

k _ 0 ( _ 1 ) h = K 2CI.yJ g

m= j

< 2) j=0.....n-l

I n - ' + = 0 m.j

For convenience, let v denote K [ I . y 3 • g ' . The m

n k 2k characteristic equation. Z (-1) w = 0 . can be

n » v

rewritten as

, 2. n+ i - . 1 - (- w )

n . 2. k 0 = r, (- w ) = .

k = ° 2. 1 - (- w )

Let S be the set of solutions to (- w 2 ) n + 1 = 1 . excluding

* 0 i + i. Let 9 st — a n d a = e . An enumeration of the — n+l

members of S is given by > where IV n> •" i

Ak -i a K for 1 £ k £ n

-(k-n) „ . ^ ^ ^ i a for n+l 5 k 2S 2n

2n K

z(t) ss 2:^^ cfc e is the general solution of the

homogeneous problem. Variation of parameters yields a

2b

A, t 2n k

particular solution Zp<*> = U R l t > e T h e

differential equation in standard form is

2 n ( - l ) n - k h ' 2 1 " - ( - ! > " V

k=o

and so denoting the mth order Vandermonde determinant by

m Vand</Jj >

/ , v n / , x 2n+k r j*k Akj t Vand< A . > . (-1) v(t) (-1) e L J 1 J

V ( t > -£ A Xj=l Aj

vand< A j >

-A, t . . . k-1 / ^ • ,N 2"+k K (-1)

= (-1) v(t) (-1) e .and n (A -A, )

m k m*k

A, t . A, (t-x) _ , > «2n k „2n - /1 k , . . h < " • IK=l Ck 6 + ZK = 1 -P* )o 0 V < X > d X

( - 1 ) "

where /3 = ' . Now. using the fact that K and g n (A -A, )

m*k / a * \

are C . we have that for 1 < m £ 2n, ,.<«&>/*.> ~ 2 n , m h = ZR=1 C» \ e + 1 4>'n ./

k = 1 \

,t m k . . . _m j - l B-i .. . f A, e v x dx + Z . , A,J v J (t) > o k j = l k

and h has the 2n derivatives supposed earlier.

The boundary conditions specify that for q = 0 n-1,

r (t) I r n " ' (-l)P-'» h ( 2 P " c ' + 1 >( t )

q p=q

„n- 1 . ,^p-qf „ 2n „ 2p-q+l Ak* „2n

" p = q ( " I > P q { r K . l s *

Z k . l ' V

{/o A2?"''*1 / " " " " ' . [ . I d . • Z*P-01 AJ V ^ P - ^ ' c t ) U

should be zero when evaluated at 0 and 1. Using the fact

27

(k) (k) that v (0) = o = v <1) for k = 0 2n-l. the above

simplifies to

t , * » *.2n k

r (t) = Zt 4 e q k = 1 • 1 ) P - q a ^ + 1

<=k + ( z n _ 1 (-: \ p=q

* k V > ° k ^ fo e ^ ^ v c x j d x . / x " - 1 aJP-1 + 1\

k= 1 k ' 0 1 p=q k f and the boundary conditions can be written as A*c = b where

A is the 2n x 2n matrix defined by

.n

A JK

Z" . (-l)P-J AfP-J p=J It

j-n.k

and 3 is the 2n vector given by

0

(B) , = J _ -A. x

J*

1 £ j £ n

n+l £ j £ 2n

1 S k S 2n

1 < j S n

Z k = l ^k Aik /J e R v < x ) d x n + 1 * j * 2n

It is shown in Lemma I that A is non-singular, so

h (t ) = X 2 n ( A~ 1 • 3 ) . e j + X 2 n

J = 1 J J=1 A j t ~ n « r' ^ < t _ X ) , ^

=1 ~ Pk >0 ® v(x)dx k > 0

Further investigation of the matrix A and the sequence

2n will give a more explicit form for h.

n a rl

=l^rn Jo -1 T+ 2n

A • b = x m

~ A x l e m v(x)dx A ( 0 , . . , 0 . A A_ )

n+1 , m 2n ,m

Define M to be the 2n x 2n matrix in which the nonzero

entries are

28

M J *

M J *

A A k k + n

e - e

A 4 2 k 1 + A, e k + n

A A 2 k k + n 1 +

e - © k

A, % 2 k 1 + A e k-n

A A, 2 k k-n 1 + A

e - e k

A k

e

J«k

1 S k i n

j=k+n

j = k-n

n+1 < k £ 2n

A A, k k-n

e - e

j=k

Examining the product A • M, one finds that

0 0

0 0

A • M = n+ 1 . 1

2n. 1 -1

n+1,2n

2n, 2n

and then applying A to both sides.

-1 T A CO 0, A A_ ) = mth column of M.

n+1.m 2 n, m

a •* 2n , 4 -1 -•. j

Z . (A b) . • e J = J =1 J

= Z 2 n e A j t lz2n p -(l e""kAv(x)dx-M 1 j=l \ k=l pk Jo VVJWUX ™jk /

_ / ~ A x _ A . t _2n J . ,1 k , , . _2n j „ j

- « s P. • J ® vCx)dx •Z. , e J M., > k = 1 | *k •> 0 j = 1 jk J

29

" Z k = l " k f l ' * v l x ) d x {® " N k k * * "** V k . k } *

zk"» ^n+k f l * "** v ( x > d x {" " V n * k + a """ M n * k . n . k }

From Lemma II, we have that for 1 £ k £ n.

sin(k9) sin(k9> 2

^k n+l Ak a n d ^n + k ~ n+l An+k

Also the expressions for the nonzero entries of the matrix

can be simplified, for example, if 1 £ k £ n.

A, , 2 k l + A

e n+k

M n+k.k » A 2

e - e k

1 1 + ( i a " k ) 2

A - A, . , . k, 2 n+k k l + (la )

1 - e

" 2 k

1 1 - a

2sin(kd) . 2k l - e l - a

-sin(kfl) -k k -k -s a (a - a )

2 sinh(sin(k®)) k, -k k. a (a - a ;

-sin(kfl )

" * 2sinh(sin(ke)) k

After such simplification we have

1 s i n ( k 9 > T ^ — — — — — — ( V ( X ) k= 1 sinh(s in(ka)) > 0 n+l

f , " V - < 2 V < 1/2 e V k 8 + 0 I

, A,t , A , t v

n " } }

+

- A . x + s i n ( k a ) - . A . t A ,s n+k I k ,2 n+k . .

1/2 e < e + A e >> dx

The remaining calculations are routine, for example.

-A, x + A , t - sin(kfl) -A x + A t + sin(kfl) , • K n + K . , • n + k k

l/2>e + 1/2-e

30

, x<-icos(k0)+sin(kfl>> + tCicos(k0)+sin(k0>> - sin(kfl) = l / 2 ' < e {'

x<-icos(k0)-sin(k0)> + t(icos(k0)-sin(k0)> + sin(k0)\

) , -(l-x-t)sin(k0> < l-x-t >sin( k0 ) \ i<t-x)cos(ke)

= 1 / 2 • i e + e } - e {' f = cosh((l-x-t)sin(kfl))|cos(C t - x ) c o s ( k 9 ) ) + i s i n ( ( t - x ) c o s ( k ® ) ) |

-A x - s i n ( k e ) 2 R e i Afct

After simplifying the terms 1/2© ( - e )e and

-A x+sin(k0) . . A .t l/2e n + ( - • " > • and w o r k i n g in similar

, t A (t-X)

fashion w i t h ~^K'/O e v(x)dx. o n e has the

following expression for h<t):

l s i m < k 0 ) h ( t ) = r— Z n :——;—:—., _ . . f . V ( X )

n+1 k = l s l n h ( s i n ( k 0 ) ) ' 0

cosh((l-x-t)sin(kfl))* cos((t-x)cos(kfl)) {{

"{

- cos(2k9)•cosh((l+t-x)sin(kfl)>-cos((t-x)cos(kfl))

- sin(2k0)•s1nh((1+t-x)sin(kfi))•sin<(t-x)cos(k0))|

+ i*^ c o s h ( ( 1 - x - t ) s i n ( k 0 ) ) • s i n ( < t - x > c o s ( k 0 ) )

- cos(2k9)•cosh((l+t-x)sin(k0))'sin(<t-x)cos(kfl))

+ s i n ( 2 k 8 ) • s i n h ( ( 1 + t - x ) s i n ( k 0 ) ) • c o s ( ( t - x ) c o s ( k 0 ) ) | | dx

1 + 2 — 2r.n, sin(kfl) ( n v(x)

n+l k=1 ; 0

c o s ( 2 k 0 ) • s i nh((t-x)sin(k0))•cos(<t-x)cos(kfl))

+ s i n ( 2 k 0 > • c o s h ( ( t - x ) s i n ( k 0 ) ) * s i n ( ( t - x ) c o s ( k 0 ) ) |

+ i•|-s i n ( 2 k 0 ) • c o s h ( < t - x ) s i n ( k 0 > ) • c o s ( ( t - x ) c o s ( k 0 ) )

+ cos ( 2k0 ) • s i nh( < t-x > s in< k0 ) ) • s in( (t-x > cos (Jt0 ) ) | | dx

Since h(t) is real, the m u l t i p l e s of i must sum to zero and

w e a r e left w i t h the desired e x p r e s s i o n (recall that v(x) =

K ( x , y ( x ) ) • g 1 ( x ) ) . The Theorem is proved in the case that

{{

31

K and q are and g^^(O) = 0 = 9 (1). K = 1 2n. 2

Surprisingly, the general case follows from the special

one. Let K:C0.13xlR —» IR be continuous and such that K 2 is *

continuous. g:[0.U —• R be increasing, and h = B'(y) 1.

There exists a sequence of functions such that

is C ( 2 n ). K —• K and <Kj>2 —• * 2 uniformly on £ 0. U x C-i.t ]

where y = sup<|y(t)J:t e [0,1]>. Let < g j > j = i a sequence

(2n) (I j is C . g^

(2n) (k) . (k) . b_. yn and such that g is C . g, = 0 = g^ for k-l.....2n, and

g i g pointwise. With each pair <Kj,gj> there is an

associated h^ = B^'(y)*l which has been shown to have the

desired form. Now. <K.)_ —» K uniformly, implies for fixed J £

w and r.

f1 (K .)_{x,y(x))*w(x)dr(x> —» f* K (x.y(x))•w(x)dr(x) ) o j 2 ' u z

Also, since each gj is nondecreasing and g^ —• g pointwise,

(4• p. 121), for fixed w,

J* w(x)•g^'(x) dx = J* w(x> dg^(x) —» /J w(x) dg(x).

This shows that <h^.>^1 must converge pointwise to a

function of the desired form. Now. <kj>2 —• K

2 uniformly implies B.'(y) —• B*(y) in norm, and using the continuity of

* * .

the adjoint operation, h = B'(y) 1 = 1im B ' ( y) 1 » lira h . n-#«> n->°°

The case where g is of bounded variation follows immediately

from the case in which g is increasing*

Lemma I The matrix A defined in Theorem IV is nonsingular

and if Vand<d >" denotes the nth order Vandermonde K K ~ i

determinant, then

32

det A = i n < n - 1 ) |vand< 2cos(k®)> n 2 n

(e ia

- e ia

-k ) .

Proof: Let w be a complex number such that (- w 2 ) n + 1 = 1.

w * + i - -- , . *®-J 2m-J j _ n . , xm-j 2(m-J) Z . ( - 1 ) J w J = w J £ _ , ( - ! ) J w J =

n m = j m=j

2.n-j + 1 j . 2. - ( - w ) J w J - {- w )

2. n+1 (-1) Jw w J- ( - l ) J w - j

W-1 + w

2n so, det A » n

R = 1

1 + w

det

1 + A,

1 + w

k=l 2n

A, e k v.

where v is the n x 1 vector whose jth component. <v )., is

Aj| - ( - i ) J A ~ J

k k For 1 £ m.j <n.

(v ). = (ia"™)-' - (-1) J ( ia~ m)~j n+m j

= iJa- mJ - (-l)Ji-Ja mJ

* J

= ( - l ) J ( i a m ) - j . . m . j (ia )J

= -(v ). . m j

so adding column m to column n+m. det A is

„2n ,4 ,2. 27, , (1 + A ) k = 1 k

det A k=1...n K

e vk

- A . - 1

k=l (e

A.

- e ) • v.

-1

- 2 1 1 Jlt 4 C 1 + A ) k = 1 k

_ n . k

n

where A = dettv k = l n] . For 1 <, k < n, (v ). = J

<ia k)j - (-1 )-•( iak)~.-' = i-Ua kj - a~k-'> = i < 2 i • s i n ( k j 9 ) > .

J > A = i (2i) det M .

where M. = sin(kj0). According to Muir (3. p. 187). J • k

determinants of this type were first investigated by Prouhet

33

in 18S7. He used the trig identity

2 k _ 1 cos*'1* sin* = ZplJ ( k p S sin((k-2p)*)

and elementary properties of determinants to show det M =

det M where M. « 2 k _ 1 cos k - 1(je> sin(je) and then factored J * K

sin(je> from the jth row to arrive at the nth order

Vandermonde determinant Vand<2cos(j0)>j_^

If l £ k £ n. then there is an m, 1 £ m £ n. such that

'2 n 2 n 2 IK 2 A = - A . Using this fact, IT' (l + A ) = FF . ( L - A ) = n+k m K = I K K - I

n n (-a k (a k-a~ k)) 2 » J7 " <-ak 2i sin(k0))2 . which is R=1 k=l

{2 (2i) 2 n a 2 • < ffR21

sin(k0) > 2-

Noting that an ( n + 1 > - i 2 n

a n d assembling the pieces,

1 k -k „ n , ia la

det A — — — x • " e }

(2 i > i < J7." sin(kfi) > K — 1

r ) 2

,n(n+l) { 2 i )2 n

{ ff^^sintkfl) >2 ivand< 2cos( je ) >j = 1 |

and the proof is complete.

Lemma I I If 1 £ k 5 n . then

(-l) n sin(kfl) A2

5 " ( V \ > = " n t l * m*k

(-l) n sin(k9> a2

( b > ^n + k = It (A -A . . " n+l n + k ib n-i- K /

m*n + k

,. m . k„ 2 ,. -m . k. Proof: (a) n (A - A ) = n (i« -i<* > ® < I A _ L C T >

m*k m 1<m<n m=l m*k

n (- 1aR)<l-a m- R) ff (-ia k)(l-a~ m - k)

1<m<n m=1 m*k

34

. . k . 2 n - l l t ~ 1 , , - ( k - m ) . " . . m-k. " - ( m + k ) . = ( - i a ) 27 ( l - a ) 27 ( l - a ) 27 ( l - a > m=1 m=k+1 m=1

k - l n -k n+k * ( - i a ) 27 ( l - a ~ J > 27 ( i - a J ) 27 ( l - a ~ J )

j = l j . i j = k + l

. -» . t - 1 , n+k . n -k _ . . , . k . 2 n - l „ , „ - 1 . _ , , - j , „ , 4 - ( 2 n + 2 - j ) . = ( - 1 0 ) 27 ( l - a J ) 27 ( l - a J ) 27 ( l - a J ) j = 1 j =k+1 j = 1

, _ . 2n+l , . k . 2 n - l „ . . - j , „ , 4 - j , = ( - i a ) 2 7 ( 1 - a J ) 27 ( l - a J )

l £ j £ n + k j = n+k+2 j*k

. k . 2 n - l - j . = ( - i a ) 27 ( l - a J ) lJSjS 2n+l j*K.n+k+l

. _ , 1 2n+l . ( - i a ) , „ „ 27 ( 1 - a - J )

( l - a ) ( i - a ~ ) j = l

k 2 n l 1 n i 2 n + 1 i = ( - i a ) - 27 ( l - a " J > -2 - 27 ( l - a " J >

( l - a ) ( l + a > j = l j =n + 2

, . k , 2 n - i 1 „ 2 - h 2 - ( 2 n + 2 - i K = ( - i a ) -2* 27 ( l - a J ) 27 ( l - a J )

*" fa l\ , . l - a j = 1 j = 1

= ( - i a k > 2 n _ 1 — — — - — 2 7 ( l - a - - * ) ( l - a - * ) a ( a - a ) j = l

k 2 n - 1 2 n

= ( - i a ) — — 27 (2 — 2 c o s ( j0 ) ) a 2 i s i n (k 0 ) j = l

i. 2 ^ ^ ^

( - i a ) 2 27 ( l - c o s ( j e ) ) . s i n ( k 0 ) j = i

2 1 n = A —• 2 27 ( 1 - C O S ( j 8 ) )

s i n ( k 0 ) j = 1

In h i s book P l a n e T r i q o n o m e t r y . Hobson ( 1 ) d e r i v e s t h e

_ _ _ _ _ ( n / 2 ) n 2 7C e q u a t i o n J n + l = 2 s i n — s i n :— * • • where

n+1 n+1 n - l k

t h e l a s t f a c t o r i s s i n — - — -— i f n i s odd and 2 n+l

35

n ik

s i n i f n i S even. Using this fact, the last factor

2 n+l

in the last equation can be simplified.

n (i) n odd 2 n n (l-cos(je)) =

J " * n-1

2 n+l n 2 n it <i-cos(j«>) • ( i - c o s { — e ) ) • n (i-cos(je>)

j=l J = n + 3

2 n-1

n \ 2, . „. = 2 IT sxn (j 9 )

j = i

= n+l

(ii) n even - similar to (i).

2n 2 For further simplification, note that Afc *Afc =

k.2n,. k.2 ,2(n+l). 2(n+l),k , ..n+l 2n ( i a ) d a ) = i (a ) = ( - 1 ) . s o A R =

. „.n+l -2 ( - 1 ) •A . Finally.

IV

( - 1 5 n ( - 1 ) " sin(k0) 2

^k = 27 (A -A, ) ~ ~~ 4 ,n+l ~-2 n+l ~ ~ n+l Ak mat[f m k ( - 1 ) A nirK K , / , v

sin<k9)

(b) The techniques used to prove this are essentially the

same as those used to prove (a).

Corollary If for each positive integer n. c^is the least

number k such that for all f in H n[0,l], Hfll ^ /c * II f II , <x> T\ , 2

t hen

1 2 s i n 3 ( k 0 ) v l / 2 k

I where Q — ————« n+l k=l tanh(sin(k0>) | n+l

Moreover. li« e - {2-/J t a n h ( 5 t n ( K « > n-*<» v '

s i n 3 ( * x) v1/2

36

Proof: Let n be a positive integer. The least k such that

for all f in H n[0.1], llfli < «• II fll _. is just the norm of <*> n, 2 J

the imbedding operator J:H n[0.l] — • CEO.l]. For a in [0.13.

define B : H n [0.1] — • IR by B (f) = f(a). Now. |B I £ IJ | for a a 1 a • ' '

all a. Let fi = sup<|B |:' a € CO. :L 1 > and suppose p < |J|-

Then there is an f in H n[0.l] such that llfil > a'llfll <*> ^ n. 2

Let a € [0,1] such that |f(a)| = HfH • Then

Hfll = != I B C f > I S I B I • II f II S /3 • II f II , < llfll 00 1 1 1 a i ' a ' n.2 ^ n, 2 oo

and we have a contradiction, so p = |j|. Now. finding |Ba|

directly from the definition is as difficult as finding |j|

directly. Instead, we use the fact that JB I = IB * I and 31 ' <31 '

Theorem IV. Define •tCO.l] — • IR by *(a) = | B^ * | 2 - From the

definitin of adjoint, for all f in H n[0,l], <f,B * 1> n = 3 H

<B f , 1 > = f (a) . a IR

* ( a ) = l B**| 2 == • B *l> u n = { B * 1) (a) . 1 a 1 a n,z a a H a

Applying the theorem with K defined by K(a,b> = b and g the

step function with unit jump at a , we obtain

1 s i n ( k 9 ) • (a) = — x. n

n+l k=l sirih(sin(kfi ) )

cosh ((l-2a)sin(ke>) - cos ( 2k6 ) cosh ( s i n ( ke )) j> .

* is symmetric and since

1 4 sin 3(k8)

* " ( a ) = ~n+T~ Zk=l sinh(sin( k©T7 (1 - 2a ) s i n ( kfl ) )

is greater than 0. 4> is convex. Therefore it must assume

its maximum at 0 and 1. Evaluating at 0 and doing some

algebra, the first part of the theorem is proved. The

second part follows from the first and the definition of

37

Riemann integral.

As mentioned in Chapter I, the determination of the

best c for the case n = 1 was made by Marti (2) using

methods from the calculus of variations. The Corollary

extends significantly Marti's result by giving explicitly

the optimum c for all n. Table I lists the values of c for n

1 S n £ 10.

CHAPTER BIBLIOGRAPHY

1. Hobson. E., A Treatlse on Plane Trigonometry. Cambridge, Cambridge University Press, 1928.

2. Marti. J. T.,"Evaiuation of the Least Constant in

Sobolev's Inequality for H 1(0.s)," SIAM Journal on Numerical Analysis. 20 (December. 1983), 1239-1242.

3. Muir. Thomas, The Theory of Determinants. Vol.11. The Period 1841 to i860. London, Macmi11an and Co.. 1911.

4. Riesz. Frigyes, and Sz-Nagy„ B., Funct ional Analysis. New York, Frederick Ungar Publishing Co.. 1955.

38

CHAPTER IV

NUMERICAL SOLUTIONS OF SECOND ORDER PROBLEMS

WITH NONLINEAR BOUNDARY CONDITIONS

As mentioned in the introduction, one of the most

attractive features of steepest descent is that, in addition

to giving a valuable theoretical framework, the method is

amenable to modern numerical methods. This chapter

discusses the practical aspects of implementing dual

steepest descent to solve second order problems and explains

the FORTRAN program FH2.CGR. A listing of the code is given

in the Appendix.

4 4 ( 2) Suppose F: IR — • IR and B: IR —» IR, both C , are

2

given. Find a member u of H Ca.b] such that for x in Ca.b],

F(x.u(x),u'(x),u' ' <x)) = 0

B ( u(a) . u( b) . u ' (a) . u 1 ( b) ) = 0

We approximate the general problem using finite differences

(finite elements could also be used, giving more accuracy at

the expense of a more complex code). Let n be a positive

integer: find which satisfies within a preassigned

tolerance the. nonlinear difference equations: u - u u - 2u + u i + l i-l i + l i i-l

(1) F(6-( i-l) .u. . — . - ) = 0 i 2-6 s 2

3?

40

U. - U. U - U' 3 I n n-2

(2) B (u . u . — , r—r ) = 0

1 n 2 • o 2 • o

where 2 £ i £ n-1 and 5 = l/(n-:L). Note that CI) is the

only natural difference scheme to use if the equation to be

solved is truly second order, i.e., is not identically

zero, and the standard finite-difference approximation to

u''(x) is used. However, for the boundary condition portion

of the problem there are several other reasonable choices.

for example.

u, - u, u - u B ( u . u , . n n ) = 0 . or

1 n 5 U- - u4 u - u 3 I n n-2

B ( u > . u . . . ) = 0 . 2 n _ 1 j 7 i — —

Computations should show if one is superior.

To soive (*) by steepest descent, first define

T:R n (lR4)n~2 by (Tu) i = (6-( i-1) ,u. . ( Dlu) . . ( D2u) t ) .

2 :S i £ n-l. where Dl and D2 are the standard difference

operators defined above. Define H: lRn — • lRn 2 by H(u> i =

F ( (Tu) . ) and # : R n — • R by *<u) = 1/2 HHull2 „ : * measures i 2

IR

how close a vector u comes to satisfying (1). Similarly,

define $:IRn — • R by

u - u u - u . , , _ io/ 3 * n n-2. ,2

< (u) = 1/2 B (u, . u , . )| I i j-^ 11 • 1

2-6 2-5

Finally, define f:IRn —» (R by $(u) = s *(u) + s $(u), where

Sj and s 2 are positive scale factors or weights which allow

the boundary conditions to be emphasized over the

4 1

d i f f e r e n t i a l e q u a t i o n o r v i c e v e r s a . I f u € R n s u c h t h a t

£ ( u ) = 0 . t h e n u i s a s o l u t i o n t o ( 1 ) a n d ( 2 ) . a n d e v e r y

s o l u t i o n i s a z e r o o f £ .

F o l l o w i n g N e u b e r g e r i n ( 7 ) . f o r 2 £ i S n - 2 .

( H ( V ) - H ( U ) ) » F ( ( T V ) ) - F U T u ) ^

= F ( ( T u ) . ) ( V . — U . ) + F , ( ( T U ) . ) ( D i ( v - u ) . ) + 2 1 1 1 3 1 i

F ( ( T u ) . ) ( D 2 ( V - U ) . ) + <b( ( T V ) . . ( T U ) . ) 4 1 1 T i 1

w h e r e t h e l a s t t e r m g o e s t o z e r o a s ( T v K —• ( T u ) . .

S O . H ' ( u ) ( w ) = F ( < T u ) ) - W + F . t ( ( T U ) ) • ( D1W) . +

F ( ( T u K ) * ( D2w) . U s i n g t h e c h a i n r u l e ,

* ' ( U ) W » <HU . H ' ( U ) W> _ a Z 0 * ( H U ) . ( H ' ( U ) W ) . „ n - 2 1=2 l l IR

= Z n ~ \ F ( T u . ) * F _ ( T u . ) • w , + F ( T u . ) • F _ ( T u . ) • D l w . + 1 = 2 1 2 1 1 1 3 1 1

F ( T u . ) • F ^ ( T U . ) • 0 2W. 1 4 1 1

= <H_U Hu . Jw> _ + <H . U H U , D 1V/ > _ + < H U H u . D2w> 2 „ n - 2 3 „ n - 2 4 n - 2

IR IR R

w h e r e ( J r ) . s < 0 ? P ^ n . • T h e l a s t s u m c a n b e l 1 r . 2 S i £ n - l

l

w r i t t e n a s < J H _ u Hu + Dl H_ Hu + D2 H u Hu , w> . s o 2 3 4 n

IR

( v * ) ( u ) - J * H „ u Hu + Dl H u Hu + D 2 H u H u . N o t e t h a t t h e 2 3 4

a d j o i n t s a b o v e r e f e r t o IRn a n d IRn 2 e q u i p p e d w i t h t h e

E u c l i d e a n i n n e r p r o d u c t .

T o c o m p u t e v $ , d e f i n e S : fRn —• IR4 b y

u „ - u u - u 3 i n n - 2

S u = ( u . u 1 ' n ' 2 - 6 ' 2 - 6

a n d Gu = B ( s u ) . T h e n

42

Gv-Gu * B.(Su)(v - u 4 ) + B^(Su){v -u > + i 1 1 2 n n

( v — u ) - C v - u > 3 3 1 I7

B3<SU)-

So

(v -u ) - (v _ — u ) n n n-2 n-2

2 * 5 + B (Su )-

4 2 * 6

V G ( u ) . = I

B (SU) - — — B (Su) i = 1 1 2 • o 3

B ( S u) 2*5 3

0

B (Su> + — — B (S U) 2 2 * o 4

2 7 6 V S u )

i = 3

l =n

otherwise

i =n-2

Using the chain rule. v$(u) = B(Su)•vG(u).

At this point we depart from the standard steepest

descent method by employing the Sobolev inner product.

Widely used in theoretical work, the Sobolev inner product

has been used by Neuberger (3, 4, s. 6. 7) and by Glowinski

(8) to give markedly superior results in the numerical case.

Consider lRn equipped with the inner product

<U,V> = z" 1 u V + (Dlu) (DlV). + ( D 2 u ).(D2v). ^ 1-iS 1 1 1 1 1 X

Recalling that both gradients and adjoints depend on the

inner product, we propose to move in the direction -v f(u) s

to seek a decrease in £. The given gradients are converted

to Sobolev ones by noting that <u,v> $ = <u.Av> where A = J*J

* *

+ 01 01 + D2 02 and using the standard result; If <X,(*#«)>

is a Hilbert space, A € L(X,X) is positive and self-adjoint,

((x,y)) » (x,Ay), and f:X — • ir has gradient at x, then the

43

gradient of f in the space <X.((•,•)>>. {v f)(x), is just

A - 1vf(x). As noted earlier, it is useful to compute the

cosine of the angle between the vectors v #(u) and v $(u) in

the Sobolev space. When the angle is small, the process

moves in a direction that better satisfies both the

differential equation and the boundary condition, whereas if

the angle is large, the two "pull in opposing directions."

In any descent algorithm, one moves from the present

position u in a prescribed direction z (usually the negative

gradient) to u + a-z; there are many methods of determining

how far one should move, i.e.. of minimizing the function

h:CO,<») —• IR given by h(a) = $(u + a-z). Following Hestenes

(1. p. 59) we use the secant approximation to Newton's

method: to find a t such that h'(i) = o. try

h' (0) h'(0)

i = h' 1(0) h'(e) - h»(0)

- 3 -4

where e is a small positive number, say 10 or 10

Defining L: IR —• X by L(a) = u + a-z. h = ?CL] and h'(a)/3 =

C • (L(a) )L 1 (a)/3 = < ? s $ ( L (a) ) . L ' ( a ) fS> s = <?se<u + a- z) .flz>s. so

e u).z>s

i = <vse(u+6Z> - vsC<u). z>s

Many numerical studies indicate that a nonlinear

version of Hestenes' original conjugate-gradient algorithm

gives superior results to the basic steepest descent

process. Following Glowinski. Keller, and Reinhart (8)

44

we shall use the Polak-Ribiere (2) variant of the conjugate-

gradient algorithm:

(1) Pick u° s g° = v s$(u°) : z° = g°

(2) For n greater than 0, p minimizes h(a) = $(un-azn), n

n+1 n n n+1 „, n+1. u = u - p nz g = v s *

< u )

, n+1 n n+l <g - g -g > s

, . — Z n + 1 = g n + 1 + , Z n

n "gn|,s n

one of the most attractive features of a conjugate-gradient

algorithm is that in the special case where $ is quadratic,

i.e., the underlying equations are linear, it is well known

that convergence theoretically must occur in fewer that n+1

steps, where X = lRn. In the nonlinear case, as a solution

is approached, the ? function will increasingly approximate

a quadratic, giving good convergence. In numerical

applications it is difficult to maintain the

conjugate-orthogonality, on which the success of the method

depends, over a large number of iterations. For this

reason, one should restart the algorithm every ten or

fifteen iterations.

The program FH2.CGR is a FORTRAN implementation of the

above ideas; the code is briefly reviewed. Section 1

dimensions the needed arrays - 2200 8-byte words are

required - and defines the functions F and B, together with

their partials. Note that the construction allows for very

general nonlinear differential equations and boundary

45

conditions and that the storage requirements are quite

moderate. The various parameters used by the program are

set up in Section 2 - number of iterations, scale factors,

interval of solution, etc. In Section 3, the main and 2

super diagonals of the pentadiagonal, positive definite,

A * *

symmetric matrix ABD = J J + Dl 01 + D2 D2 are computed and

stored in a format to be input to the LINPACK subroutine

OPBFA which factors ABD so that later the linear system

(ABD)w = ?£<x) can be solved.

The remaining code is within a DO loop which controls

the steepest descent iteration. Sections 4 and 5 compute

the Euclidean gradients v*(u) and v$(u). Mention should be

made of the method in which adjoints are computed: in the

ODE case it is not critical, but careful coding now will pay

off in the PDE case. We take the operator D2 as an example:

in FORTRAN it is implemented by

DIMENSION U(N). D2U(N) DO 100 1=2.(N-l)

100 D2U(I) = (U(I+1) - 2*U(I) + U(1-1))/(DX**2)

Where the n-dimensiona1 array D2U is treated as a

(n-2)-vector by considering that only the entries 2 through

• (n-l) contain data. To compute D2 , one could find the

matrix representation of D2t take its transpose, and

*

determine that D2 is implemented by

46

DIMENSION V(N).D2A(N> D2A<1) = V(2)/DX**2 D2A(2) a <-2*V(2) + V(3))/DX**2 DO 100 1=3.(N-2)

100 D2A(I) = CV(I+1) + 2*V(I) + V(1-1))/DX**2 D2A(N-1) = (V(N-2) - 2*V(N—1)) / DX* * 2 D2A(M) = V(N—1)/DX* * 2

*

A much easier way of computing D2 is obtained by

considering how it acts on a unit vector e . 2 < j £ n-1 in

the space lRn 2 . For all u in lRn we want <u,D2*e.> = J S?n u

J + 1 - " j * uj-1 <D2u.e.> = r , so we must have

J fR11"2 6 2

(D2*e . )i J

6 2

6 2

i = j — 1 or j + l

i = j

0 otherwise

Using this and the linearity of D2*. the following

•k

particularly simple code computes D2 :

DIMENSION V(N), D2A(N) DO 100 1=2.(N-1) D2AC1-1) = V(I)/DX* * 2 D2A(I> = -2*V(I)/DX* * 2

100 D2A{I + 1) = V( I )/DX* * 2

This difference in perspective - taking V(I) and seeing

which elements of D2A it affects rather than taking D2A(I)

and using elements of 'V to compute its value, results in

substantial simplification in the corresponding PDE code

where five adjoints are computed. Moreover, rather than

computing the adjoints separately the work can all be done

47

in one DO loop.

Section b computes the gradient v£(u) and the value

f(u). If ?(u) is less than the pireassigned tolerance, a

solution has been found to the desired accuracy and the

program exits. The LINPACK subroutine DPBSL is called twice

in Section 7 to compute v c*(u) and v_$(u), and then the

cosine of the angle between them is found. Section 8

implements the conjugate-gradient algorithm, determining the

direction z in which to move. Note the variable IREST which

controls how often the CG algorithm is restarted. The

secant formula is used in Section 9 to determine the optimum

distance to move.

Finally, in Section 10, we move to the trial vector w =

n n * C w > u - R -z and compute f(w). The ratio RR = should

^ * • n x £ (u )

be < 1 . but because of the nonlinearities we may have moved

too far. If RR > l, we go back and move just half as far in

the direction z. The process continues up to ten times

until £(w) < £(un), whereupon u n + L <— w, and the ratios n+l-> n+1, n+1,

t(u ) M u ) £ (u ) , and are printed out

r\ *

t(u n) *(u n) ?(u n)

Table II lists the output of several runs of FH2.CGR

for the test problem y' = y, y<0) = y(l) 2. with an initial

guess of the constant 1 function, the program quickly

x — 2 converges to the solution e . Starting at -1 or 0.1, the

48

other solution, 0, is reached. In runs #4 and #5> different

weights are attached to the differential equation and

n+1. «(u )

boundary conditions. Note that the ratios and . . n. *< u )

^. n+1» . . n+1. $ (u ) $ (u )

can vary widely, but stays below 1, ^ • ft x * • n. $ < u ) $ C u )

indicating continued descent. Finally in run #6. the

conjugate gradient method is restarted at every step, so

regular steepest descent is done. With IREST = 1.

convergence is slower, but much less erratic.

CHAPTER BIBLIOGRAPHY

1. Hestenes, Magnus. Con jugate Direct ion Methods in Opt imizat ion. New York. Springer-Verlag. 1980.

2. Polak, E., Computat ional Met hods i n Opt imi zat ion. New York, Academmic Press. 1971.

3. Neuberger. J. W.. "Gradient Inequalities and Steepest Descent for Systems of Partial Differential Equations," Proceeding of the Conference on Numerical Analysis. Texas Tech. University. 1983.

"Some Global Steepest Descent Results for Nonlinear Systems." Trends in Theory and Pract ice of nonlinear Different ial Eauat ions. edited by V. Lakshmikantham, New York. Marcel Dekker. 1984. pp. 413-418.

'Steepest Descent and Time-Stepping for the Numerical Solution of Conservation Equations." Proceedings of the Conference on Phusleal Mat h and Nonli near Di f ferent ial Equat ions. Marcel Dekker,, C To appear ).

. "Steepest Descent for General Systems of Linear Differential Equations in Hilbert Space." Proceedings of. a Sumposium held at Dundee. Scot land March-July 198 2. Lecture Notes in Mathemat ics. Vol. 1032. New York. Springer-Verlag, 1983, 390-406.

. "Steepest Descent for Systems of Nonlinear Partial Differential Equations," oak Ridge Nat ional Laboratory CSD/TM-161. (1981).

Glowinski. Roland. Keller, H.. and Reinhart, L., Cont inuat ion-Con jugate Gradient Methods for the Least Square Solut ion of Nonlinear Boundary Va1ue Problems. Rapports de Recherche. Institue National de Recherche en Informatique et en Automatique, 1982.

49

s o

TABLE I

Least C For Sobolev*s Imbedding inequality

F SI < C * II f II <x> n„ 2

1 2 3 4 5 6 7 8 9 10

14587751766902701 11280304092518190 11229455523920118 11228314131571282 11228287031259754 11228286380032815 11228286364338547 11228286363960049 11228286363950901 11228286363950679

51

TABLE I I

Program : FH2.CCR

Equat ion u u ; u (0) a u (1)

On the interval C0.0, 1.01

Run # 1

N s 15 ITER = 2 0 TOLERANCE a 0.0000001 EPSILON a 0,001

SCALE 1 = 1. SCALE2 a 1. RESTART INT. a S

Initial guess : u(x) a i.

£ new * new new Distance

Moved Cos ine

* o l d * o l d ^ O l d

1 0 .022237 0 .016 0 .0 0 .507 0 0 2 0 .055957 0 .061 0 .042 5 .250 0 . 436 3 0 .944718 0 .969 0 . 848 0 .733 0 .648 4 0 .337422 0 .311 0 .458 5 .403 -0 , .119 5 0 .514833 0 . 488 0 .598 0 .514 0 . 850 6 0 .456134 0 .514 0 .306 9 . 146 -0 268 7 0 . 175810 0 .213 0 .015 50 .312 -0 .589 8 0 .921007 0 .913 1 .456 0 .571 -0. .815 9 0 .675432 0 .686 0 .241 24 .612 0 292

10 0 .959044 0 .954 1 .566 0 .924 -0 .572

0 . 134455 0 .206175 0.316819

APPROXIMATE SOLUTION AFTER 10 ITERATIONS 0.144840 0.221248 0.340533

0.155511 0.237454 0.364837

0 . 166852 0.254949

0.179032 0.273920

0 . 192125 0 . 294529

11 0. , 773107 0 .784 0. . 042 6 .200 0 .695 12 0. 545754 0 .537 11 . .492 58 711 -0 .260 13 0. .964337 0 .970 0 . 653 8 .117 -0 . 253 14 0. ,491388 0 .464 2 . . 928 7 . .769 -0 8 88 15 0 , .810646 0 862 0 .082 52 .412 -0 638 16 0. 930735 0 .929 1 . 117 4 . . 835 -0 .581 17 0 . .772209 0 . 777 0 . 124 2 . . 197 0 . 759 18 0 . 901011 0 , 902 0 .259 44 488 -0 . 004 19 0 , .870454 0 . 864 17 . 129 1 .110 -0 . 893 20 0. .935846 0 .936 0 .973 7 . 972 -0 .220

APPROXIMATE SOLUTION AFTER 20 ITERATIONS 0.135973 0.146267 0.156990 0.168578 0.131033 0.194378 0.208699 0.224108 0.240700 0.258540 0.277667 0.298164 0.320253 0.344197 0.369135

5 2

Run # 2

N a 15 ITER * 20 TOLERANCE = 0.0000001 EPS1LON a 0.001

SCALE1 = 1. SCALE 2 = 1. RESTART INT. = 5

Initial guess : u(x) = - 1 .

new

'old

4> new

* old

'new Distance ; Moved 'old

Cosine

1 0 .024627 0 .011 0 .069 0 . 419 2 0 .270031 0 .353 0 .227 4 . 101 3 0 .384403 0 .267 0 .478 0 .498 4 0 .192316 0 .066 0 .249 6 . 337 5 0 .943803 1 .937 0 . 826 1. . 179 6 0 .147016 0 , . 195 0 . 134 10 .560 7 0. 897035 1 .236 0 759 1 .424 8 0 . 715835 1 . 125 0 . 445 2 .440 9 0 . ,424352 0. . 191 0 . 816 0 . . 740

10 0. 762809 1. . 101 0 .630 5 .825

597 603 810

-0.625 -0. 854

879 606

-0.921 -0 .923 -0 .714

a APPROXIMATE SOLUTION AFTER 10 ITERATIONS -0.010818 -0.012083 -0.013298 -0.014534 -0 015814 •0.018495 -0.019897 -0.021364 -0.022940 -0 024683 -0.028880 -0.031336 -0.033854

-0,017136 -0.026652

11 12 13 14 15 16 17 18 19

0.753074 0.164670 0 .697761 0.475691 0.934011

345026 561565 930327 106747

TOLERANCE MET 20 0.106747

0 .497 0 .923 0 .921 0 .501 0 .042 54 .514 0 .600 1 . 130 0 .611 0 .535 0 .336 7 .775 0 .977 0 .775 0 .545 0 ,371 0 , .222 42 .955 0. ,525 0 , 845 12 , .380 0 . 886 1, . 148 6 . .740 0 . 120 0 056 6 . . 748

0. 120 0 . 056 6 . 748

-0 .922 -0.902 -0.943 -0.170 0.833

-0 .784 -0.538 -0.819 0,850

SOLUTION TO TOLERANCE AFTER 20 ITERATIONS -0.000140 -0.000202 -0.000203 -0.000212 -0 000225 -0.000248 -0.000266 -0.000295 -0.000329 -0 000357 -0.000389 -0.000432 -0.000464

-0 .000236 -0 .000373

53

Run # 3

N = 15 ITER « 20 TOLERANCE = 0.0000001 EPSILON = 0.001

SCALE 1 a 1. SCALE2 a 1. RESTART INT. a 5

Initial guess : u(x) = 0.1

e new

old

new

old

new

'old

Distance Moved

Cosine

1 0 .018135 0 .016 0 .045 0 .491 0 .987 2 0 .106167 0 .017 0 .628 8 .948 -0 .309 3 0 .988131 0 .689 1 .035 1 .050 -0 .963 4 0 .317371 2 .782 0 .057 61 .373 -0 .947 5 0 .951598 0 .905 1 . 190 1 .730 -0 .785 6 0 .531684 0 .530 0 .540 6 .751 0 .547 7 0 .273018 0 .098 0 .939 19 .800 -0 .642 8 0 .972673 1 .206 0 . 880 2 .979 -0 .934 9 0 .025587 0 .071 0 .001 21 .369 -0 .949

10 0 .848208 0 .855 0 .555 4 .434 -0 . 101

-0.000060 -0.000091 -0.000183

11 0,808667 TOLERANCE MET

12 0.806667

APPROXIMATE SOLUTION AFTER 10 ITERATIONS -0.000045 -0.000049 -0.000101 -0.000103 -0.000196 -0.000188

0.775 2.880

0.775 2.880

-0.000052 -0.000105

1.083

1.083

-0.000060 -0.000118

-0.937

-0.000075 -0.000148

SOLUTION TO TOLERANCE AFTER 12 ITERATIONS -0.000103 -0.000093 -0.000102 -0.000110 -0.000122 -0.000139 -0.000157 -0.000169 -0.000172 -0.000175 -0.000187 -0.000215 -0.000249 -0.000260 -0.000250

54

Run # 4

N a IS ITER » 20 TOLERANCE = 0.0000001 EPSILON « 0.001

SCALE 1 a .8 SCALE2 a .2 RESTART INT. a S

Initial guess : u(x) a i.

new

C old

new

old

new

'old

D istance Moved

Cosine

1 0 . 0 1 7 8 4 0 0 . 0 1 6 0 . 0 0 . 6 3 5 2 0 . 0 1 3 6 6 8 0 . 0 1 0 0 . 0 5 9 10 . 0 2 2 3 0 . 9 7 9 7 4 3 0 . 9 9 2 0 . 9 5 7 1 . 0 7 8 4 0 . 7 4 8 6 1 7 0 . 7 4 3 0 . 7 6 0 10 . 4 2 5 5 0 . 8 7 0 3 4 1 0 . 8 4 9 0 . 9 1 0 0 . 6 5 8 6 0 . 2 4 0 6 6 0 0 . 2 3 9 0 , . 2 4 4 8 1 . . 2 9 5 7 0 . 7 2 2 3 4 6 0 . 6 4 7 0 . 8 4 8 2 7 , . 0 1 5 8 0 . 9 5 9 8 6 1 0 . 9 2 7 1 , . 0 0 1 1 . 4 9 9 9 0 . 3 1 1 4 1 3 0 . 3 5 2 0 . 2 6 3 18 . 9 5 8

10 0 . 7 8 4 8 0 9 0 . 8 4 0 0 . 6 9 7 6 4 . 0 0 4

0 4 6 9 622

- 0 . 3 2 4 0 . 7 2 2

- 0 . 4 8 4 - 0 . 5 5 8

5 8 2 7 0 7

- 0 . 6 9 5

0 . 1 4 4 9 2 0 0 . 2 2 2 6 4 9 0 . 3 4 2 0 7 6

APPROXIMATE SOLUTION AFTER 10 I T E R A T I O N S 0 . 1 5 5 8 5 7 0 . 2 3 9 1 9 8 0 . 3 6 7 8 2 5

0 . 1 6 7 2 0 1 0 . 2 5 6 8 8 6 0 . 3 9 4 4 3 3

0 . 1 7 9 4 6 8 0 . 2 7 5 8 1 7

0 . 1 9 2 7 9 2 0 . 2 9 6 1 6 6

0 . 2 0 7 1 8 6 0 . 3 1 8 1 7 4

1 1 0 . 8 6 7 9 0 1 0 . 9 4 2 0 . 7 2 6 0 . 9 9 0 12 0 . 6 5 8 7 1 2 0 . 6 6 6 0 . 6 4 0 2 0 . 7 5 0 1 3 0 . 9 3 4 1 5 3 0 9 8 2 0 . 8 1 1 10 . 7 6 0 1 4 0 . 3 7 9 0 3 2 0 . 4 4 8 0 . 1 6 5 6 . 5 1 4 15 0 . 8 8 1 1 8 5 0 . 9 0 3 0 . . 6 9 9 1 7 . 5 7 0 16 0 . 9 8 8 4 0 2 0 . 9 8 5 1 . 0 2 6 1 . 8 1 5 17 0 6 0 8 5 0 0 0 . 6 4 1 0 , 2 7 1 3 3 . 4 9 6 1 8 0 . 8 2 9 8 8 1 0 . 8 5 4 0 . 2 4 3 1 2 5 . , 3 7 5 19 0 , 9 6 3 7 8 8 0 , . 9 6 4 0 . 9 3 3 0 . 7 6 4 2 0 0 . 9 6 6 0 4 0 0 . , 9 6 9 0 . 7 2 9 1 8 . 9 2 8

0 . 7 3 3 - 0 . 6 6 9 - 0 . 3 5 8 - 0 . 8 4 6 - 0 . 1 0 1 - 0 . 7 4 8

0 . 4 6 6 - 0 . 4 3 0 - 0 . 8 7 9 - 0 . 0 9 6

0 . 1 3 6 2 0 4 0 . 2 0 9 0 3 6 0 . 3 2 0 6 4 3

APPROXIMATE SOLUTION AFTER 2 0 I T E R A T I O N S 0 . 1 4 6 4 1 3 0 . 1 5 7 1 0 0 0 . 1 6 8 7 1 5 0 . 1 8 1 2 2 3 0 . 2 2 4 4 9 9 0 . 2 4 1 1 1 7 0 . 2 5 8 9 6 3 0 . 2 7 8 0 9 2 0 . 3 4 4 6 3 6 0 . 3 6 9 7 0 4

0 . 1 9 4 6 3 9 0 . 2 9 8 5 7 9

55

Run # 5

N = 15 ITER a 20 TOLERANCE a 0.0000001 EPSILON a 0.001

SCALE 1 a .2 SCALE2 * .8 RESTART INT. a 5

Initial guess : u(x) a i.

new

'old

new

old

new

'old

Distance Moved

Cos ine

1 0 . 0 3 9 4 1 0 0 0 1 7 0 . 0 2 . . 5 1 1 0 . 0 2 0 . 2 2 4 5 9 8 0 . 3 6 7 0 . 1 2 2 8 . . 0 6 9 0 . . 2 9 2 3 0 . 8 5 5 7 7 3 0 . . 9 7 8 0 . 5 9 0 3 , . 6 2 2 0 . 6 3 0 4 0 . 2 2 0 8 5 0 0 . 2 3 4 0 . 1 7 5 1 6 . . 2 1 0 - 0 . 3 3 2 5 0 . 5 6 6 5 4 4 0 . 6 3 8 0 , . 2 2 5 2 . . 5 5 8 0 . . 8 9 4 6 0 . 0 8 6 4 0 3 0 . . 0 8 5 0 . 1 0 6 3 1 , . 7 1 5 - 0 , . 2 8 2 7 0 . 9 5 7 2 4 2 0 . 9 1 7 1 . 3 9 9 1 3 . 6 8 8 - 0 . 8 3 0 8 0 . 5 7 1 3 1 5 0 . 6 2 4 0 . 1 9 3 2 6 . 9 3 8 0 . 3 4 4 9 0 . 6 9 3 0 7 5 0 . . 6 5 1 1 . 6 6 8 4 . . 7 8 0 - 0 . . 8 9 5

1 0 0 . 7 5 6 4 4 5 0 . 8 3 4 0 . 0 5 0 3 2 . 1 0 5 - 0 . 3 8 8

0 . 1 3 2 6 3 6 0 . 2 0 4 5 6 3 0 . 3 1 6 2 0 6

APPROXIMATE SOLUTION AFTER 1 0 I T E R A T I O N S 0 . 1 4 3 2 2 3 0 . 2 1 9 7 3 6 0 . 3 3 9 5 0 2

0 . 1 5 4 0 3 1 0 . 2 3 6 1 6 2 0 . 3 6 3 2 2 2

0 . 1 6 5 3 8 9 0 . 2 5 3 9 6 0

0 . 1 7 7 5 0 5 0 . 2 7 3 2 3 1

0 . 1 9 0 5 2 4 0 . 2 9 4 0 1 0

1 1 0 . 8 7 8 3 0 4 0 . 8 7 5 1 . 3 2 1 3 . 2 9 9 - 0 . 8 8 7 1 2 0 . 3 5 4 1 0 5 0 . 3 1 1 4 . 7 4 6 4 2 0 . 3 7 4 - 0 . 4 9 6 1 3 0 . 8 6 9 7 0 5 0 . 9 2 9 0 . 4 7 7 2 . 8 1 8 0 . 8 9 5 1 4 0 . 6 4 2 2 8 3 0 . 6 9 2 0 . 0 0 6 4 4 . 7 4 7 - 0 . 5 7 5 1 5 0 . . 9 8 4 1 8 8 0 . 9 8 0 7 5 6 5 17 . 1 1 1 - 0 . 5 3 3 1 6 0 . 8 6 8 6 7 9 0 . 8 6 0 2 . 5 8 1 2 4 . 2 8 1 0 . 7 5 2 17 0 . 7 4 7 4 4 9 0 . 7 5 6 0 . 1 8 6 1 1 6 . 9 0 8 - 0 . 3 4 7 1 8 0 , . 5 8 0 8 7 7 0 . 5 8 3 0 . 0 3 1 4 6 8 . 0 5 6 - 0 . 3 2 1 1 9 0 . 6 4 3 9 0 2 0 . 5 6 7 3 8 9 . 2 4 3 3 . 4 7 7 - 0 . 9 0 1 2 0 0 . 8 6 9 4 4 3 0 . 9 8 6 0 . 0 1 3 1 3 4 . 3 2 1 - 0 . 9 1 1

APPROXIMATE SOLUTION AFTER 2 0 I T E R A T I O N S Q . 1 3 5 0 7 0 0 . 1 4 5 3 2 7 0 . 1 5 6 0 0 5 0 . 1 6 7 5 5 3 0 . 1 7 9 9 7 7 0 . 1 9 3 2 9 0 0 . 2 0 7 5 8 2 0 . 2 2 2 9 7 2 0 . 2 3 9 5 4 5 0 . 2 5 7 3 3 2 0 . 2 7 6 3 5 0 0 . 2 9 6 7 0 6 0 . 3 1 8 6 8 3 0 . 3 4 2 5 5 2 0 . 3 6 7 4 2 3

5 6

Run # 6

N - 15 ITER = 20 TOLERANCE == 0.0000001 EPSILON » 0.001

SCALE 1 « 1. SCALE2 = 1. RESTART INT. = 1

Initial guess : u(x) = 1.

new

'old

<#> new

4> old

new

'old

Distance Moved

Cos ine

1 0 . 0 2 2 2 3 7 0 . 0 1 6 0 . 0 0 . 5 0 7 2 0 . 2 2 6 4 5 2 0 . 2 9 2 0 . 0 4 5 4 . 4 0 8 3 0 . 4 7 7 4 6 0 0 . 4 0 5 1 . 7 9 1 0 . 5 3 3 4 0 . 5 0 1 0 3 2 0 . 5 7 2 0 . 2 0 9 3 . 8 7 1 5 0 . 5 4 3 3 7 2 0 . 4 4 7 1 . 6 2 8 0 , . 5 3 4 6 0 , . 5 8 6 1 0 6 0 . 6 6 7 0 , . 3 3 7 3 . . 8 7 5 7 0 . 6 5 0 5 0 1 0 . 5 2 5 1 . 4 1 6 0 . . 5 3 6 8 0 , 7 1 5 7 0 9 0 8 2 4 0 . 4 7 0 3 , . 9 4 2 9 0 . 7 8 7 8 7 4 0 6 6 8 1 . . 2 6 5 0 . 5 3 9

10 0 . 8 3 7 4 9 1 0 . . 9 6 2 0 . 5 7 7 4 . 1 3 5

0.0 0 . 4 3 6

- 0 . 5 6 3 0 . 4 0 1

- 0 . 6 5 4 0 . 3 1 6

- 0 . 7 0 5 0 . 1 7 1

- 0 . 7 4 2 - 0 . 0 7 1

0 . 1 5 6 9 9 9 0 . 2 3 8 3 2 1 0 . 3 6 6 7 5 9

APPROXIMATE SOLUTION AFTER 10 I T E R A T I O N S 0 . 1 6 8 9 2 8 0 . 2 5 5 8 7 2 0 . 3 9 2 6 6 1

0 . 1 8 1 0 7 8 0 . 2 7 4 9 8 1 0 . 4 1 8 8 8 5

0 . 1 9 3 8 2 4 0 . 2 9 5 6 9 1

0 . 2 0 7 4 6 2 0 . 3 1 7 9 8 0

0 . 2 2 2 2 3 2 0 . 3 4 1 7 4 2

1 1 0 . 8 8 0 8 3 7 0 . 7 9 3 1 . 1 8 8 0 . 5 4 4 12 0 . 9 0 0 3 0 4 1 . 0 1 1 0 . 6 4 2 4 . 4 3 8 1 3 0 . 9 1 7 0 7 4 0 . 8 5 0 1 . 1 6 3 0 . 5 4 8 1 4 0 . 9 2 2 7 2 0 1 . 0 1 3 0 . 6 8 1 4 , . 6 3 8 1 5 0 . 9 2 7 9 7 7 0 . 8 6 9 1 . 1 6 1 0 . 5 5 0 1 6 0 . 9 3 0 0 8 0 1 . 0 0 6 0 . . 7 0 3 4 , . 7 0 0 17 0 . 9 3 2 1 7 5 0 . 8 7 8 1 . . 1 6 4 0 5 5 1 1 8 0 . . 9 3 3 5 2 0 1 . 0 0 1 0 . 7 1 5 4 , . 7 1 4 1 9 0 . 9 3 4 9 0 7 0 . 8 8 4 1 . 1 6 6 0 . 5 5 1 2 0 0 . 9 3 6 0 7 2 0 . 9 9 9 0 . 7 2 1 4 . 7 1 7

- 0 . 7 7 5 - 0 . 3 2 3 - 0 . 8 0 1 - 0 . 4 4 6 - 0 . 8 1 7 - 0 . 4 8 2 - 0 . 8 2 5 - 0 . 4 9 2 - 0 . 8 3 1 - 0 . 4 9 4

* APPROXIMATE SOLUTION AFTER 2 0 I T E R A T I O N S 0 . 1 4 8 4 3 7 0 . 1 5 9 7 3 4 0 . 1 7 1 2 7 2 0 . 1 8 3 4 2 6 0 . 1 9 6 4 5 3 0 . 2 2 5 8 4 3 0 . 2 4 2 4 7 7 0 . 2 6 0 5 5 3 0 . 2 8 0 1 5 8 0 3 0 1 3 4 2 0 . 3 4 8 2 4 0 0 . 3 7 3 4 6 6 0 . 3 9 9 1 0 2

0 . 2 1 0 5 4 3 0 . 3 2 4 0 8 3

APPENDIX

C PROGRAM NAME: FH2.CGR STEEPEST DESCENT, SOBOLEV NORM C SOLVES F(X,U,U',U*')=0, B(U(0),U(1),U'(0),U'(1)) C BEGIN DATE: 22 FEB 1984 C REVISED: 24 OCT 1984 ..

SECTION 1

IMPLICIT REAL * 8 (A-H,0-Z) . REAL * 8 U(200),GR(200), BGR(200),XIGR(200),SXIGR(200) REAL*8 WR1(200),Z(200),SGOLD(200) REAL * 8 ABD(3,200) REAL*8 F,F2,F3,F4,B,B1,B2,B3,B4

C

C###

*

F(X,U,U1,U2)= U1 - U F2(X,U,Ul,U2)= -1.D0 F3{X,U,U1,U2)= 1.D0 F4(X,U,U1,U2) = 0 . DO B(A,B,C,D)= A - B**2 B1(A, B, C , D ) = 1 .DO B2 (A,B,C,D)= -2.DO *B B3(A,B,C,D)= 0 • DO B4(A,B,C,D)= 0 .DO WRITE(6,1) FORMAT(IX,'PGM=FH2.CGR '/,

IX, 'EQ: IT ' = U ' {' IX,'BC: U(0) = U(1)* *2 ' )

SECTION

N=15 IF(N.GT.200) CALL EXIT NMl=N-l NM2=N-2 NM3=N-3 NM1SQ=NM1* * 2 PI = 4.DO *DATAN(1.DO) ELFT= 0.DO ERHT= 1.DO DX=(ERHT-ELFT)/(N-l) ITER =20 ITMOD=l MODPNT=10 IREST=1 . RIN =1.0 TOLER=l.0D-7 EPSILN= 1.0D-03 SCALE1=1.0 SCALE2=1.0

57

58

WRITE(6,10) ELFT,ERHT,N,ITER,TOLER,EPSILN, * SCALE1,SCALE2,IREST

10 FORMAT(IX,'ON INTERVAL ,,F5.2,' TO *,F5.2 /, * IX,'N= *,15,' ITER= ',110, * ' TOLERANCE= ',E15. 7, ' EPSILON=',F10.7/, * IX,' SCALEl= ,F5.3,' SCALE2= ',F5.3, * 1 RESTART INT.=',14/, * 20X,' INITIAL GUESS: ') DO 20 1=1,N

20 U(I)=RIN WRITE(6,990) (U(I),I=1,N)

c SECTION 3

ALP= 1./(4•*DX**2) BET= 1./(DX**4) ABD(1,1)=0. ABD(1,2)=0. DO 35 J=3,N

35 ABD(1,J)= -ALP + BET ABD(2,1)= 0. ABD(2,2)= -2.*BET DO 40 J=3,NM1

40 ABD( 2 , J) = -4.*BET ABD( 2 , N) = -2.*BET ABD(3,1)= ALP + BET ABD(3,2)= 1. + ALP + 5.*BET DO 45 J=3,NM2

45 ABD(3,J)= 1. + 2.*ALP + 6.*BET ABD(3,N-l)= 1. + ALP + 5.*BET ABD(3,N)= ALP + BET CALL DPBFA(ABD,3,N,2,INFO) IF(INFO.NE.0) CALL EXIT

c BEGIN S.D. ITERATION DO 1000 KK=1,ITER

c SECTION 4

PHIOLD=0.DO DO 80 1=1,N

80 GR(I)=0. DO 100 1=2,NM1 CX=DX*(I-1.)

100

CU=U(I) CU1= (U(I+1)-U(I-1))/(2.*DX) CU2= (U(1+1)-2.*U(I)+U(1-1))/(DX* *2) CF=F(CX,CU,CU1,CU2) CF2=F2(CX,CU,CU1,CU2) CF3=F3(CX,CU,CU1,CU2) CF4=F4(CX,CU,CU1,CU2) PHIOLD=PHIOLD + CF*CF GR (1-1) ==GR( 1-1) - ( CF 3 * CF) / ( 2 . * DX) + ( CF4 * CF ) / (DX 2 ) GR(I)=GR(I) + CF2*CF - (2.*CF4*CF)/(DX**2) GR(1+1)=GR(1+1) + (CF3*CF)/(2.*DX) + (CF4*CF)/(DX**2) PHIOLD=.5*PHIOLD

59

SECTION 5 DO 380 1=1, N

380 BGR(I) = 0 Bli-K { 1 ) = U . Cl= (U(3)-U(1))/(2.*DX) C2 = (U(N)-U(N-2))/(2.*DX) CB = B{U(1),U(N),CI,C2) CB1= B1(U(l),U(N),CI,C2) CB2= B2(U(1),U(N),C1,C2) CB3= B3(U(1),U(N),CI,C2) /-.-O A _ nil/Tr/1 \ TT / XT V n 1 \

BGR(N-2)= - ( C B 4 X C B ) / ( 2 . * D X ) BGR(N) = CB2*CB + (CB4*CB)/(2. BCOLD = .5*CB* *2

SECTION 6 DO 400 1=1,N

400 XIGR(I)=SCALE1*GR(I)+SCALE2*BGR(I) XIOLD = SCALE1*PHI0LD + SCALE2*BCOLD IF(XIOLD.GT.TOLER) GO TO 550 WRITE(6,992)

992 FORMAT(IX," TOLERANCE MET") GO TO 1100

550 CONTINUE c SECTION 7 C SOBOLEV GRADIENT

CALL DPBSL(ABD,3,N,2/GR) CALL DPBSL(ABD,3,N,2,BGR)

C** COMPUTE COSINE IN H2 SUM1=0.DO SUM2=0.DO DOT=0.DO DO 625 1=2,NM1 CGI = (GR(1+1)-GR(1-1))/(2.*DX) CG2 = (GR(1+1)-2.*GR(I)+GR(1-1))/(DX* *2) CB1 = (BGR(1+1)-BGR(1-1))/(2.*DX) CB2 = (BGR(1+1)-2.*BGR(I)+BGR(1-1))/(DX* *2) SUM1=SUM1+ GR(I)* * 2 + CG1**2 + CG2**2 SUM2=SUM,2 + BGR( I) * *2 + CB1**2 + CB2**2

625 DOT=DOT + GR(I)*BGR(I) + CG1*CB1 + CG2*CB2 DENOM=DSQRT(SUM1* SUM2) COSINE=0.DO IF(DENOM.GT.0.) COSINE= DOT/DENOM DO 635 1=1,N

635 SGOLD(I)=SXIGR(I) DO 640 1=1,N

640 SXIGR(I)= SCALE1*GR(I) + SCALE2*BGR(I) c SECTION 8

IF(MOD(KK,IREST).NE.0) GO TO 644 DO 642 1=1,N

642 Z(I)=SXIGR(I)

60

646

GO TO 650 644 CONTINUE

DO 645 1=1,N 645 WR1(I)=SXIGR(I)-SGOLD(I)

DOT=O.DO DO 646 1=2,NM1 CW1= (WR1(1+1)-WR1(1-1))/(2.*DX) CW2= (WR1(I+1)-2.*WR1(I)+WR1(I-1))/(DX**2) CS1= (SXIGR( 1+1) -SXIGR( 1-1) ) / (2 . *DX) % w / . _ . CS2= (SXIGR(1+1)-2.*SXIGR(I)+SXIGR(1-1))/(DX 2) DOT = DOT + WR1(I)*SXIGR(I) + CW1*CS1 + CW2 CS2 SUM1=0.DO DO 647 1=2,NM1

647 SUM1=SUM1+ SGOLD(I)**2 + ((SGOLD(1+1)-SGOLD(1-1))/(2.*DX)) *2 +

* ((SGOLD(I+l)-2.*SGOLD(I)+SGOLD(I-l))/(DX**2))**2 GAMMA=0.DO IF(SUM1.GT.0.DO) GAMMA= DOT/SUM1 DO 648 1=1,N

648 Z(I)=SXIGR(I) + GAMMA*Z(I)

SECTION 9 COMPUTE OPTIMUM DISTANCE - SECANT FORMULA

C C 800 Rl = 0.DO

805 1=2,NM1

DX * * 2) /v 2.*DX)

. . +SXIGR( 1-1) ) / (DX* *2 ) 805 Rl"= Rl" + Z(I)*SXIGR(I) + CZ1*CS1 + CZ2*CS2

DO 810 1=1,N 810 GR(I)=0.

DO 850 1=2,NM1 CX=DX*(1-1) CUMl=U(1-1) + EPSILN* Z(1-1) CU= U(I) + EPSILN* Z(I) CUP1=U(I+1) + EPSILN*Z(I+1) CU1= (CUP1-CUM1)/(2.*DX) CU2= (CUP1-2.*CU+CUM1)/(DX**2) CF=F(CX,CU,CU1,CU2) CF2=F2(CX,CU,CU1,CU2) CF3=F3(CX,CU,CUl,CU2) CF4=F4(CX,CU,CU1,CU2) v , GR(I-1)=GR(I-1) - (CF3*CF)/(2.*DX) + (CF4*CF)/(DX* 2) GR(I)=GR(I) + CF2*CF - (2.*CF4 *CF)/(DX* *2)

850 GR(1+1)=GR(1+1) + (CF 3 * CF)/(2.* DX) + (CF4*CF)/(DX 2) DO 890 1=1,N

890 BGR(I)=0. CWl=U(1)+EPSILN* Z(1)

\ « - r r i t * rr / 1 \ CW3=U(3)+EPSILN*Z(3) CWNM2=U(N-2)+EPSILN*Z(N-2)

61

CWN=U(N)+EPSILN* Z(N) Cl= (CW3-CW1)/(2.*DX) C2= (CWN-CWNM2) / (2 .*DX) CB = B(CW1,CWN,CI,C2) CB1= B1(CW1,CWN,CI,C2) CB2= B2(CW1,CWN,CI,C2) CB3= B3(CW1,CWN,CI,C2) CB4= B4(CW1,CWN,CI,C2) BGR( 1 )=CB1*CB - (CB3*CB) / ( 2 . *DX) BGR ( 3 ) == (CB3*CB)/(2.*DX) BGR(N-2)= - (CB4*CB)/(2.*DX) BGR(N) =CB2*CB + (CB4*CB)/(2.*DX) DO 900 1=1,N

900 WR1(I)=SCALE1* GR(I) + SCALE2*BGR(I) CALL DPBSL(ABD,3,N,2,WR1) R2= 0.DO DO 920 1=2,NM1 CW1= (WR1(1+1)-WR1(1-1))/(2.*DX) CW2= (WR1(1+1)-2.*WR1(I)+WR1(1-1))/(DX**2) CZ1= (Z(I+1)-Z(I-1))/(2.*DX) CZ2= (Z(I+1)-2.*Z(I)+Z(I-1))/(DX**2)

920 R2 = R2 + WR1(I)* Z(I) + CW1*CZ1 + CW2*CZ2 R3 = (EPSILN*R1)/(R2-R1)

c SECTION 10 ICOUNT=0

925 DO 930 1=1,N 930 WR1(I) = U(I) - R3*Z(l)

PHINEW=0.DO DO 935 1=2,NM1 CX=DX*(I-1) CU=WR1(I) CU1= (WR1(1+1)-WR1(1-1))/(2.*DX) CU2= (WR1(1+1)-2.*WR1(I)+WR1(1-1))/(DX* *2)

935 PHINEW==PHINEW + F (CX, CU, CU1, CU2 ) * *2 PHINEW=.5*PHINEW Cl= (WR1 ( 3 ) -WR1 (1) ) / (2 . *DX) C2=(WR1(N)-WR1(N-2))/(DX**2) BCNEW= .5*B(WR1(1),WR1(N),CI,C2)* *2 XINEW=SCALE1*PHINEW + SCALE2*BCNEW RR= 0.DO RPHI= 0.DO RBC= 0.DO IF(XIOLD.NE.0.0) RR=XINEW/XIOLD IF(PHIOLD.NE.0.0) RPHI=PHINEW/PHIOLD IF(BCOLD.NE.0.0) RBC=BCNEW/BCOLD IF(RR.LT.0.999999999) GO TO 950 WRITE(6,993) KK,RR

993 FORMAT(IX,110,' RR = ',F12.6,', .GT. 0.999999999') R3=.5*R3 ICOUNT=ICOUNT+l IF(ICOUNT.LT.5) GO TO 925 WRITE(6,996)

62

996 FORMAT(IX,' STALLED ') 950 ICOUNT=0

DO 955 1=1,N 955 U(I)=WR1(I) C * *

IF(MOD(KK,ITMOD).EQ.0) * WRITE(6,994) KK,RR,RPHI,RBC,R3,COSINE

994 FORMAT(IX,17,F10.6,3F10.3,5X,F10.3) IF(MOD(KK,MODPNT).EQ.0) WRITE(6,990) (U(I),I=1,N)

990 FORMAT(IX,6F12.6) SECTION 10-

1000 CONTINUE C** 1100 CONTINUE

WRITE(6,994) KK,RR,RPHI,RBC,R3 WRITE(6,990) (U(I),I=1,N) CON=0.62737639353D0 CALL EXIT END

/INC LINP.DBL

BIBLIOGRAPHY

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inequality for H ^ O . s ) , " SI AM Journal on Numerical Analusis. 20 (December. 1983). 1239-1242.

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6S

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Reports

Glowinski, Roland, Keller. H., and Reinhart. L.. Cont inuat ion-Con jugate Gradi ent Met hods for t he Least Square Solut ion of Nonlinear Boundaru Va1ue Prob1 ems. Rapports de Recherche, Institue National de Recherche en Informatique et en Automatique, 1982.