The Sturm-Liouville difference problem for a fourth order equation with discontinuous coefficients

TBE STURM-LIOUVILLEDIFFERENCE PROBLEM FOR A FOURTH ORDER EQUATION WITH DISCONTINUOUSCOEFFICIENTS' KIM0 6800 (Yoscon) (Received

20 October

19621

In this paper we use the methods developed by AN. Tikhonov and A.A. Samarskii (see [ll. [21) to study the Stum-Liouville difference problem for a differential equation with discontinuous coefficients: L(Jw.q.WU =L(k.m)

LW.rMu

=

u -

Sr (2) u = 0,

gb[k(4 $$I-& [p(2) $I+ q(4 I&.

To solve this problem we use the homogeneous differeme

schemes (see

[31) Lf.mz.W

y = LFpeq)y -

l% (2) y,

We shall give the argument below for the boundary conditions u (0) = a’(0) = u (1) = u’(i) = 0. The results obtained for these bouudary .conditions csu be used without essential alteration for the class of boundary value problems with other bouudarJ conditions (see 131). We shall prove that any homogeneous difference scheme of second rauh which satisfies the necessary conditions for second order approximation has second order accuracy in the class of smooth coefflcieuts aud first order accuracy iu the class of discontinuous coefficients. The difference

’

Zh.

vych.

mot.

3,

No. 6,

1014-1031, 1363

1963.

1384

Kh tto Shou

scheme with the coefficients 1 c1=

+ \,:,_I,ds]-1, b =

[S&&s 0

0

‘8

d =

5p(x + sh)&

-1

%

I?=

g(x+Wds,

f

%

s %

r (5 + sh) ds

has second order accuracy in the class of discontinuous coefficients.

1. Statement of the Stnrm-Lionville 1. The Stura-Liaville

problem

for

a differential

We consider the homogeueous differential ~(k.v,q.ar)u

=

p.P.a)

difference

24 -

problem

equation

equation

hr(x)u

= 0,

o
with homogeneous boundary conditions u (0) = u’(0) = 2.5(1) = u’(l) = 0.

(2)

The Stuns-Liouville problem for the differential equation (1) with boundary conditions (2) consists in finding those values of the parameter h (eigenvaluesf to which there correspond non-trivial solutions (eigenfunctions) of the problem (l)a(2) and also in finding the eig~f~ctions of this problem. We shall assume that the coefficients conditions

in equation (1) SatiSfY the

where Cja j = 1, 2, 3, 4, 5, 6 are certain positive

constants.

Whenthe coefficients k(x), p(x) of equation (11 have discontinuities of the first kind at certain points of the interval 0 < x < 1, as we s8w in k3], the solutions of the Sty-Liouville problem must satisfy the coupling conditions

difference

The Sturn-Liouville

ful = 0,

lu’f = 0,

where 5 is a point of discontinuity (1). We shall call (1%.

2.

Equivalence

pu’l = 0

[(ku”)’ -

[ku”] = 0,

1385

problem

of the coefficients

the problem defined by conditions

of the &urn-Liouville

problem

for

2 =

E.

(4)

k, p of equation

(l)-(4)

the problem

and an integral

equation We consider Green’s function G(x, g) of the operator L(k*peq) for the boundary conditions (2). The definition

of Green*s, function for the case

be extended with some alteration When

k,

p,

q EQ’O)

to the case

k,

p,

k,

p,

q E

q E

C(a)

can

Qfm).

Green’s function is defined from the conditions:

1) the functions Gk 5) and GL (2, f), in the square 0
k (z) G& (5, E) are continuous

2) for 0 < x < S and 5 < x < I the function G(x, g) satisfies homogeneous equation

the

t(kz’.q) c -_ 0 and the coupling conditions; 3) for any values of E from the interval satisfies the boundary conditions G (0,E) = G; (0, E) =

G (5, El =

4) on the diagonal of the rectangle in (kG&

For

k,

-

pG;

p,

q E

[O, 11 the function G(x, 0

G’*: ($1&) = 0;

there is a sudden discontinuity

defined by the condition

Q(O)

Green’s function G(r, Q has the following

form

1386

Khao Shou

where Vi, j = 1, 2, 3, 4 are solutions #h+,,j satisfying

the coupling conditions

of the homogeneous equation = 0,

and the conditions

01(O) = 0,

0s (0) = 0,

f)a (1) = 0,

04 (1) = 0,

sr; (0) = 0,

0; (0) = 0,

0; (i) = 0,

0; (1) = 0,

(b;),

= 4,

(G),

= 0,

(G),

(b;),

(k?& = 0,

(k&

= i,

(kz& = 0,

= 1,

= 0,

(kv;); = -

1.

If p(x) = q(x) f 0 then all the solutions Vi, and therefore Green’ s function g(x, cl, can be calculated with the help of quadratures. Then Vj has the folloWing form: xz !! .

It 1s not difficult to see that problm homogeneous integral equation

(1% is equivalent to the

(6)

The Sturn-Liouvi

3.

equivalence of the Sturr-Liouvil variational problem

The

The Sturm-Liouville ational problem: 1) in the class of

1 le difference

functions

0,

problm

q” @‘“‘)

le problem

and a

(I&) is equivalent

of comparison

functions

satisfying

1387

problem

to the following

cp E g(*) (Q‘“’

vari-

is the class

the condition

1

bpl = s r(z)(p’(z)dz=

H

1,

cp(0) = qyo; = cp(1) = (p’(i) = 0, to find

the minimum of the functional D

bpl = Sk [cp”l’dx + j p [tp’l* dx + i qcp¶dx. 0

This

minimum defines

the first

eigenvalue:

Al = min D [VI = 2) the

other

eigenvalues

A,,

(7)

0

0

D [z+];

n > 1 are found as the minimum of

functional (‘0 in the class of comparison ing the supplementary conditions

functions

cp E Q(*)

the

satisfy-

1 H

[ql = i,

Ii lq, 4

=

cp (0) = q’(O) = &)

s

qw,,r dx = 0 for

m
= q’(1) = 0,

where uI is the m-th eigenfunction. This minimum defines

the n-th L=

4. Basic

properties

eigeavalue:

miu D Icpl = D [u,,].

of eigenfwactions

and eigenvalues

Let us give some of the basic properties values of the Sturm-Liouville problem (1% efficients

k E

Q(O), p E

Q(O), q E

Q@), r e

of eigenfunctfons and eigenfor piece-rise continuous coQ(O) (see

[41).

Khao Shou

1. There exists

a countable set of eigenvalues

to which the eigenfunctions

correspond. The eigenvalues have multiplicity not greater than two. For, as is easily seen, there exist not more than two linearly independent solutions of equation (1) for which u(0) = u’ (0) = 0. 2. All the eigenvalues are positive:

n = 1,‘2,3, . . * ,

h,>o*

3. For the eigenvalues h, we have the following 4

Q L < can’,

n=i,2,.

estimate: . . ,

where cr and ca are positive

constants which do not depend on n.

4. All the eigenfunctions with weight r(x):

un form an orthogonal system normalised

5. We have the estimates [ &lb)

b <

wz,

&wjo
for the eigenfnnctions not depend on n.

iph)~
u,, and their derivatives,

5. The Sturm-Liouvil le difference To solve the Sturm-Liouville ence scheme (see E31): (k.P. 9. W y = QPJ+/ r, where Lh(k*pDq)y =

where Ml, & and hfa do

problem

problem (IhI we use the following -

(uy;Ltp

&&

-

h
dy.

differ-

The

Sturn-Liouvi

lfe

difference

1389

problem

By analogy with [31 we shall assume everywhere that

Lj,*J’*e*&)is a

homogeneous conservative scheme of standard type. Its coefficients a(x), b(x), d(x) and e(r) are defined with the help of pattern functionals defined in the class Q(O).

by

Ah Irp (41,

--1BsQi,

Bh [cp (41,

--I<864

@

19 (41,

-+<3<+,

I?

I9 601,

-f
the formulae

ft follows

a = A’kPh’ = Ah [it (s)],

L’ (s) = k (5 + Sk),

b = B’“*h, = Bh [p (s) 1)

P (4 = p (z + sh),

a’= D’O*h’= r>h [(l ($)I,

as>

e=

T (4 = r (5 + sh).

IP)

= If [r (s)],

from the fact that k, p,

06 b6

= Q (5: -I- W,

q and r are bounded that

0 6 d6 c,,

Ctc

As the initial class of difference schemes of zero rank.

0 < ci 6 e < c8.

schemes L~k~p~q~Xy) we shall take

Now let us go on to formulate the Sturm-Liouville

difference

problem:

to find those values of the parameter Ah (difference eigenvalues) for which there exist non-trivial solutions (difference eigenfunctlons) of the holaogeneous difference equation h
with homogeneous difference

boundary conditions

0,

yl=

Yx.0

h -2

Yxx.0

(8)

r=

0,

and also to find the difference We shall call the difference Liouville problem (11 $.

Y&ml=

0,

Y;,N+~Y~,N=OR

(9) (W

eigenf~~tio~s,

problem defined by (8)-(10)

Here we use (9) and (101 Instead of the natural difference

the Sturmanalogues

1390

Khao Shou

of the boundary operators (see E31)

6. Equivalence on intcgrat

of the &urn-Liouvilk equation

difference

problem

and

Using Green’s difference formula (see [3’I) we can easily see that the Stun-Liouville difference problem (II?9 is epuivalent to the solution of the difference snalogue of the integral equation Y = I h CC@? ey)),

which, with the help of the substitution cp (4 = m;Y

(r)*

reduces to the difference

p

integral

(29 E) = Ve (4 e (E) @ (r, E) equation

9 = khCC with symmetric “kernel”

Kh(z, E), where c” (5, g)

ence function of the operator

7.

Equivalence

variational

911

ak* ” *’

of the Sturdiouville problem

is the Green differ-

with conditions

problem

(9) snd (IO).

und a difference

Using Green’8 difference formula by analogy with Section 2 it is not difficult to see that the Sturm-Liouville dffference problem (IfA) is equivalent to the following difference variational problem: 1) in the class of net comnarison functions q satisfying tions HN

iv]

‘PI = Q,*.a-2

h

=

((ee, q”))

0,

9 ax.0 = 0,

=

1, 0,

(P&I=

e)jiJv+

-+(pTd;,N = ‘*

to find the minimumof the fuuctional

this minims defines the first

difference

eigenvalue

the condi-

difference

The Stun-Liouvillc

1391

probler

2) the other difference eigenvalues h:~ n > 1 are found as the minimumof the functional 11) in the class of net functions q~satisfying the supplementary conditions

HN bpl = 1,

BN $=

P r,o-2 where

ya

is

the

n-th difference

h

= 0,

cpa.0

m-th difference eigenvalue c=

8. Properties

[qb y,,,] = ((e%y,,# = 0, TN-1 = 0,

eigenfunction;

eigenvalues

and difference

n=1,2

,...,

3. All the difference eigenfunctlons normalised with weight e(x) 1 HNIy,.Y~l={O

Iar,Oo
eigcnfunctions

eigenvalues with corre-

eigenvalues are positive:

C>o,

4. If k, p, q, r E

this minimumdefines the

min &I [VI = &V [y,,].

of difference

All the difference

%

~,N+‘+~&,N=O’

1. For each N there exist N - 4 difference sponding eigenfunctlons

2.

o for m <

yn

N-4.

form an orthogonal system

for m-n, form#n.

Q(O) then we have the estimates II b&II

o<

M* OF*

II b/n)&

< M,

(&“a,

W

for the difference eigenfunctions yn and their difference derivatives, where Ml, MS, and Ma are certain positive constants which do not depend on the step h or the number n. For let x and x’ be any two points of the net Oh. We have

1392

Khao

Y2(4 - Yaw

[a(4 y;;x(412 -

= ad+h i

b(~‘)YT&wla =

Shou

h [Y (4 + i

Y;

(4,

h [~(S)y~~(S)+~(s-h)y~(S-h)lX

[a(4 y&)1;.

From the normalisation condition fore

(s - hII

d+h

x there exists

Y

HN [y, yl = 1 it follows that

at least one point x’ at which e (s’) y2 (cc’) <

1

and there-

yB (5’) < l/c,.

Further, it follows from the boundary conditions

Y&J =

YX,6 +

that

fihhY&(S) =-&(h) + 5hhy&(s).

s=h

r=h

Then using the Cauchy-Bunyakovskii inequality

we obtain

and therefore

IIY,

no < M, (W’*,

0(YJ;IIo < M, (W.

We can put the solution of the problem (IIA) in the form Y (4

=

y (3h) +

y,- (3h) +

+dlxgh

i

_

h

m=ah

m-h

=(m+h)

+d2=sh

i h _,, _+,

1

=(m+Wr

where dl and d2 are determined from the boundary conditions yN_1 = 0, It

is

not difficult

y;, N + +y;;.N

to see that d/ = 0 (Ah),

It follows

=o.

i = 1.2.

from equation (8) that

[a (4 YG WI,-

= b (4 Y,-(4 + i

h I-

d (4 y (4 + hhe (4 y (4

1 + d,

The Sture-Liouville

difference

problcr

1393

and therefore

Since

we obtain

and therefore

2. Convergence of the nolution of the Sturu-Liourillc difference problem In the

work 111 it

tion

that

(II)

converge’to

Liouville

for

(1)

theorem using

epuaproblem

of the initial

of discontinuous

convergence

differential

of the difference

and eigenf~ctions

in the class

8n analogous

a second order

8nd eigenf~ctions

the eigenvalues

problem

we prove

~8s proved

the eigenvalues

coefficients.

the method given

StuwHere in

El].

We omit the proof. We first

consider

Let y = ytx, fWiCtiOn81

DN

the case of the first

hf be the net function

with

the nowslisstion functions

values

which realises

(n = 1). the minimum of

the

[q] x” =

net

eigenvalue

{y(z,

condition

h))

DN

[j/l

HJV [y] =

i.

Consider

and the corresponding

the sequence of

sequence of

first

{Ah? on some sequence of nets.

Lemma 2, For each

positive

E there

??<&$-a where h is the first

eigenvalue

exists

an ho such that

for h
(1%.

eigen-

1394

Proof.

Khoo Shou

Let u’(z)

be some function of G(2) for which J#== i*
and let

From Leppraa 1 we have

for A
a* (A) < a + s Therefore,

from the principle

of the minimumof a functional

kh
h
+ s,

Lemma3. The sequences of functions equicontinuous and bounded.

we have

{y (z, A)} and (y; (2, A))

are

For if x4 and x **are any two points of the net Oh then 9

Using the estimates (12) we find from Lemma2 that {y(x, continuous: I y W,

where M is some positive

h) -

y (s’, W I < M 1 2” -

A)) is equi-

5’ 1,

constant which does not depend on h.

From the normalisation condition at one point x = x’ the inequality

HN [yl = 1 it follows that at least

y (z’, h) G---L v C6 holds. It follows bounded:

from (13) that the sequence {y(x,

1 Y (G h) I <

Similarly

I Y (~3 h) -

Y (2’9 h) I +

/I)) is uniformly

I y W, h) I < M.

we can Prove that {y,-(s, A)} is equicontinuous and bounded.

It follows from the Arzela theorem that there exists some sub-sequence (Y (5, hk)) which converges uniformly to some function G(r) continuous on the segment [O, 11:

0Yt5, hk)

-

G (2) no =

P @kh

where p(hk) +O We shall

as hk-

assume that

{hhk = h (hk)}

difference

Sturn-Liouvillc

The

0.

the

converges

1395

problem

sequence {hk? is such that the sequence

to sooie limit

[ll)

“h (see

lim 1 (hk) = x, hk+Q

the sequence

and

converges uniformly to some function w(x).

(y,-(z,hk)}

Lemna 4. If for some sequence {hk)

1Y,-(2, k) -

‘;; (5) tb = P (hk),

1 Y (G hd -

W (z) g= P (hk),

then ;fi (z) = tD(z). We omit the proof. Lemma 5. If for some sequence {hk)

lim 1 (hk) = x, hk-“’

then

We omit the proof (see [ll). Lemma6. Let gh (z, E) be the Green difference ator

ak’ y = (a&);,

for the boundary conditions

gtx,
function of the oper-

(2).

If

kE

(9) and (lo), operator

Q(O) and the difference

= (ku’)

scheme

has zero rank ilghtt, E) -

g (z, E)ib = P (h),

@(z,

E) -+g

(2, E)C = P (h).

To prove this it is sufficient to note that the functions and v(j)(x) in the expressions of the Green functions g(x, cl and

Proof.

vi(x)

and let

L”‘u

gh(x,
form:

cz

Vl(4

=

k+ ss

ddz,

11

cl

(4 =

zz va (;st-

Adz, o 1so k (0

&jdtdz, ss

XL

00

11

v4 (4 =

ss xz

$$

dtdz;

1396

Khao Shou

We nor consider the equation Y =

W’, , - dy +

Jh)).

(14)

Using the Green formula we cau reduce equation (14) to the equation Y = -

((b,$?I<,+ w9

ve- 4 Yh

where gh(x, Q is the Green difference function of the operator LAk) with conditions (9) and (101. Passing to the limit we obtain 1

L(2) = -

1p (8 g; (2. 8 2 (6)dE+ 5g(5, El iiir(E)0

where g(x,

q (EJI ii tE) d&v (15)

0

Q is the Green function of the operator

(ku”)”

tion (2). It follows that the solution ii(x) of the integral (15) satisfies the differential equation L”.

P. u) ;

-Xr

for condiequation

(z)iY=O

and the boundary conditions E (0) = s (0) = L(1) = 2 (1) = 0. Since A = hl is the least eigenvalue of the problem (IA) and “h < A we have i = A. Because of the multiplicity following lenma.

of difference

eigenvalues we have the

The Sturn-Liouvillt

difference

probltr

139’7

Lemma 6. The difference eigenvalue Ah has multiplicity than the multiplicity of the corresponding eigenvalue A.

for

not greater

Proof. We shall prove the lemma by assuming the contrary. Suppose, example, that h is a simple eigenvalue and that hh is a double eigen-

value of the “kernel” G”(5, t). Let i (5, h) and y (2, h) be orthogonal and normalised eigenfunctions belonging to the eigenvalue Ah. As we have shown there

exists

{hk)

a sequence

sponding sequence {g (x, hk)} converges ul(x) belonging to the eigenvalue A: II i (xv hd -

uniformly

such that

the corre-

to the elgenfunction

~1 (4 jio = P (hk)

and

lim X (hk)

=

x

=

&.

hk*

We shall

assume that

the sequence 19 (G hk) -

{hh)

is

such

that

the equation

% (2) gb = P (hk)

holds.

tion

If not we could pick cut a sub-sequence and restrict our considerato only those suffixes k which corresponded to this sequence. We will

then have

and therefore

11 l-3h@ (X) p

(2, hk);

-*. but this

contradicts

(X&k)-‘2 hfl (‘) “f b) II,=

P thkh

-k

the limit

equation

In exactly the s&a8 nay It can be proved that when h is a double eigenvalue the difference eigenvalue hh has multiplicity not greater two.

than

1398

Khao Shou

Lemma7. The difference eigenvalue Ah has multiplicity the multiplicity of the corresponding eigenvalue A. To prove this it is sufficient form an orthogonal system

to note-that

H [unt 4

The following Theorem

of

u(x)

eigenvalue Al.

to the other eigenvalues A,h, n > 1.

theorem is a consequence of the above argument. If the difference

Convergence.

rank, the solution of nets as h -

the eigenfunctions

n+m.

= 0,

The reasoning above applied to the first It is also applicable

not less than

scileme

L~kvP~q*Ar) has zero

(c, y,,j of problem (IIA) converges on auy sequence

0 to the corresponding solution 11I:--&,=p(h),

for any coefficients

u, IJo= P (4

IIl/n -

k, p, q, r E

(A,,, u,,) of problem (1%:

Q(O).

3. On the accuracy of the solntions of the Sturr4.ionville difference problem I.

The equation

for

the error

Let (Ah, y) be a solution (A, IL) be the corresponding problem (I’). The error z = w.r. t. the solution u(x) of

of an eigenfunction

of the difference problem (11% and let solution of the initial Sturm-Liouville y - u of the solution y(x) of problem (IIh) problem (1% satisfies the conditions W.P>~.W~ = y

Lh q

=

f

VI,

VT&-l =

(W

(47)

p1

(W where Y = $ + (A” I# ---

h) eu,

[(Id)” - (au;;), 1 - [(pu’)’ - (~~;),I + (q - 4 u + h (e - 4 u9 (1% p1 = - u (1 - h), VI = - u (h), h 3

=

-

ux.0 -

2

uu,o

,

IL2=-

UG,N

+$uz,,}.

The Stura-Liouvi

We call

Zle

di,fference

the problem defined by (W-(18)

2. Formula for

the error

1399

problem

problem (111%.

AA = hh - A.

since the parameter hh is an eigenvalue of problem (II’) the difference problem (IIIh) is soluble only when the following equation holds:

From (19) and (20) we obtain a formula for the error Ah = hh - A: W, $4) -t V -f+{-+

i

-

-$-

A) ((eu+ YH + -$ h ~~~~~~+UN-~~N-~~~ alp,

i

&%-&N-2

(Za, -b

+

2al

@aN-%

+

i-

PbJ

2aN+

+

Y% -

function

jj = Cy

w&B

hbN-2)

As in the work iJ.1 we choose a multiplier

-I-

)

%

f

YN-% -

pl

=

0.

C in such a nay that the

is orthogonal to the difference

2 = y--

u:

((6, +,, = 0. We then obtain a formula for estimating the error Ah = hh - h:

aud a formula for estimating c2 -

f C- -

1 = (@b z’>>+

1 1: (HN iuf

-

H !ul),

where cz41

as h-0.

Introducing the function q(xt with the help of the conditions tt;,=g**

ql=rit.l=o

and then using Green’ s formula we find that for k, p, q, r E c’( 2 * I)

(211

1400

Khoo

Shou

where

I#@)= ku” q,’ = (q -

IIS’ II*=

q(b) = bu, -

ataG,

4 u + A (e - r) u + q (2) =

II71b + h I rl(1 - 4

$,

0 (W,

5 h i hqqz”), i=lh z&ah

pu’ ,= (pu’) l p;, This gives

the following

lemas

Lemma8. If y(x) has the form

then

IAll=

iA”- 1 I < M (9 Illlva) ll, + lwb) III + IIlb*lls+ I Vl +

I3

I+

IIL1

I+

I + I PaII,

and hma

9. If y(x) has the form

at all points &+s, then

of the net q, Wart from the points

AX 1 < M @I #i’,‘“’ II; f

+h*

n+S 7A Iel+q

8-n-l

IWb)II;+ US’a;+ l#$_, I + I$$,+,l + z

w-3--s)*,I+hl

l-n-1

grp./+ r-n-1

+

where

x = xn+, G, til,

I Yl

I + I VI I +

IIL1 I

+ I CL%’ IL

The

3.

A

priori

Sturw-Liouville

difference

1401

problcn

estimates

By analogy with [I] we estimate the error z = y - u. We first the problem (IIIA) to the analogue of the integral equation z = h” ((Ch, ei)) +

reduce

((Gh, !I!)) + [p,(t~(-l)G-,-)~a(-lfG~p,l~ -

(22) - [vl(d+')Gk)r - a(+'~Gh,v,l,-, - bi+IpIG;,N_I + b,~,Gh*,~,

where G%x, @ is the difference function of the operator the boundary conditions (9) and (10). The eigenfunction tion

T of problem (IIA) satisfies

Lik*p* 9) for

the “integral’

epua-

i = Ah((Gh, &)), which, with the help of the substitution Kh (5, E) = I/e (4 e (F;) G” (JG EL reduces to the difference

“integral”

with symmetric “kernel” Khfx, @;

elation

and with the help of the substitution

Kh(5, E) = v’e(4 e (%)Gh(2, %I,

--

v (4 = Ve h-42 (4

equation fZZt reduces to the equation E,= hh ((P,

v)) + f,

(231

where

Equation (23) is soluble and so we shall have au o~hogonality tion for the solution rg(x) and the right-hand side f: W

condi-

f)) = 0.

Now let us go on to construct the resolvant

R of the “kernel” Kh(x,Q

ilhao Shou

through

which the solution WV(X)of equation (23) can be expressed

by

formula Q= f + W

01.

It follows from the conditions that

t(G* 3, = 0,

ml m = 0

The resolvant R is found from the equation

and can be written in the

form (see El])

R = K, + R,, where

and the sm is taken over all qpkwith the exception of those to which the eigenvalue %,hcorresponds. Be have the following

estimate (see

[II ):

From this we obtain

Themera 1. Let yn(r) be the n-th normalised eigenfunction of the difference problem (IIA) and let u,(z) be the n-th normalised eigenfunction of the problem CIA. If y(x) has the form

the

The

Sturm-Liouuillc

difference

problem

i4nn

then

where M(A) is some positive Proof.

constant which does not depend on

h.

We consider

f = ((fYh, Fj) -!- (A” -

A)) ((Kh, fi

+ J& { (pL1(a(-l)G&

-

u)) +

a(-l)G~q,I~ -

Iv1 (a(+l)G$), -

a(+W&v210} -

- 1/- eh-rCrlG$,,,

-

b~G~,dy

where I=

--L-Y. I/c

Introducing the function q(x) with the help of the conditions q& = \p’*

rll =

rlr.1 = 0

we have

II4 G df (V W”’ IL + IIWb)III + IIS’ Ila+ I Vl I + I v2

I+

IPl I +

I P2 I>*

Using (21) and !Z=

y---u=- ;

++z,

we obtain (24). Theorem 2. Let y,(x) be the n-th normalised eigenfunction of the difference problem (11% and let u,(x) be the n-th normalised eigenfunction of the problem (IA). If Y(X) has the form ‘Ip = 9::

-f- Sib’ -!- 9’

at all points of the net

1404

Khao Shou

of

4. The order

accuracy

of

in the class

smooth coefficients

from Theorems 1 and 2 that the order of accuracy ot the solution of the Sturm-Liouville difference problem WA) depends on the approximation error of the difference scheme y and also on the approximation error of the normalising functional HN, i.e. on the quantity It

is clear

X = HN [ul -

H [u].

In order to study the question of the accuracy of tile solution problem (IIh)

it is sufficient

to estimate

We shall prove the following

of

/I$@)l/r, llp(b)j/l, 11 $*]I, and x.

theorem.

Theorem 3. If the difference scheme ~~~‘p~q,~r’ of second rank satisfies the necessary conditions for second order approximation, then the solution r E

(G, &,)

C( 2 * ‘)

of the difference

problem (IIA) in the class k, p, q,

has second order accuracy,

I 1Lt:-

5

I<

M &J

h2,

where M&J is some positive

i. e.

II yn (4 -

un (4 lb <

&if (W h2,

constant which does not depend on h.

Proof. Carrying out the seme reasoning as in the proof of Theorem 3 in 131 we obtain

gcar = ku” -

au- = xx

0

(h2),

+a) = bn, -

AS we have shown the difference order approximation:

v1 = 0 (h%), It is not difficult k, P, Q,

F E

c’2n”

y1 = 0 (P),

operators

ptr’ = 0 (h2),

$,’ = 0 (h”).

(9) and (10) have second

l.~ = 0 (ha),

pS = 0 (h2).

to see that the normalising function N,,, for h&M

S econd

order approximation w.r. t. h (see

~11~:

7he

.5. The order

Starm-liouo~lle

of accuracy

in the class

difference

problem

of discontinuous

14(25

coefficients

In this section we discuss tne auestion of the accuracy of the solution of the Sturm-Liouvifle difference prObleM UIh) in the class of disconti~uo~~scoefficients Qini). To avoid complicating the explanation we assume that the coefficients of equation (1) have the same point of discontinuity: E = xn -!-oh, 0 < 8 < 1. We note that for a second rank scheme satisfying the necessary conditions for second order approximation in the class k, p, q, r E fy2,” n-a

h2 2

n+2 (n -i-3 -

sf $1 = 0 (h),

ain-1

k

2

** = 0 (k);

m-1

for the scheme Lik*p’Q*“” with coefficients

(25)

cm (271

w3)

in the class k, p, q, r E

n+z

Q(‘*‘)

n+e

n+a

This gives the following theorem. Theorem 4.

Any difference scheme

L~kip*q*x” of second rank satisfying

1408

Khao

Shou

the necessary conditions for second order approximation has in the class first order accuracy: k, p, (I, r E P*l)

and the difference scheme with coefficients accuracy in the class k, p, q, r E Q’ ’ * I’:

where U(h,) is some positive

(25)-(28)

gives second order

constant which does not depend on h.

In conclusion, I express my sincere gratitude to A.A. Ssmarskii for his valuable advice and help in the completion of this work, and also to A.N. Tikhonov for his interest in it.

REF'ERENCES 1.

TI~ONOV, A.N. and SAMARSKII, 805, 1981,

2.

SAMARSKII, A.A.. 2%. wych. Rat., 1. No. 6. 972-1000, 1961.

3.

KHAO SHOU, Zh.

4.

COURANT. R. and HILEERT, D., Metody

vych.

mat.,

of Mathematical Physics), dat. 1951.

A.A.

t

Zh.

uycft.

aat.,

I.

No,

5. 784-

3, No. 5, 841-860, 1963.

Vol.

aatcnatichaskoi

fiziki

(Methods

1. Moscow and Leningrad, Gostekhiz-