A deficient spline function approximation to fourth-order differential equations

A deficient spline function approximation to fourth-order differential equations M. Nabil Esmail Department

of Chemical

Engineering,

University

of Saskatchewan,

Saskatoon,

Canada

Tharwat Fawzy and Magdy Ahmed Department

of Mathematics,

Faculty

of Science,

University

of Suez Canal, Ismailia,

Egypt

Hamdi 0. Elmoselhi Department of Science & Mathematics, Canal, Suez, Egypt

Faculty

of Petroleum

and Mining,

University

of Suez

An approximate spline solution is developedfor the initial valueproblem of a fourth-order ordinary diflerential equation. The approximation is based on deJicient spline polynomials of degree m = 8 and deficiency 4. The existence and uniqueness of the solution, which satisfies a Lipschitz condition, are proved. The consistency, stability, and consequently convergence of the solution are established. Furthermore, the method is proved to be of order 9, and the errors are limited by the relation IIS”’ - y”‘(x)ll = 0(h9-‘), i = O(l)% Keywords: spline functions, numerical solutions, ordinary differential equations

Spline approximations have been used extensively in the numerical solution of problems involving differential equations. Loscalzo and Talbot’ applied a spline approximation to Cauchy’s problem of a first-order differential equation. They used splines of degrees m = 2, 3 and deficiency 1. Micula2-4 used splines of degrees m = 3, 4 and deficiency 1 to solve Cauchy’s problem of a second-order differential equation. Sallam and El-Hawary5g6 applied the spline functions’ approximation to the solution of the Cauchy problem of the first-order equation Y(l) = f(x, y). They used splines of degrees m = 4,5 and deficiency 3. They7 also approximated solutions of the Cauchy problem of the second-order differential equation y(‘) = f(x, y, y(l)), using splines of degrees m = $6 and deficiency 3. Ahmed’ applied splines of degrees m = 6,7 and deficiency 3 to the Cauchy problem of the third-order equation yc3) = f(x, y). In this work we apply the spline functions’ approximation to the solution of the Cauchy problem of the fourth-order equation Yc4’= f(x, y). We use spline Address reprint requests Engineering, University Canada. Received 1994

658

7 September

Appl.

Math.

to Prof. Esmail at the Dept. of Saskatchewan, Saskatoon 1992; revised

Modelling,

4 May

1994,

of Chemical S7N OWO,

1994; accepted

8 June

Vol. 18, December

polynomials of degree m = 8 and deficiency 4. First, we define a class of spline functions and a construction for the approximate solution. We then prove the existence and uniqueness of such a construction. This will be followed by a discussion of the consistency relations and convergence criteria for spline approximations of degree m = 8. We conclude the method by establishing the level of convergence for such solutions. A test problem is solved, and comparison is made with other methods.

The spline functions’ approximation Let (S,, C”) be a class of spline functions with the set of knots (xi). This class consists of polynomial functions of degree m, which are connected in the knots, up to derivatives of k(k < m). Consider the differential equation

respect to piecewisesmoothly the order

Y(4) = f(x, Y)

(1)

where f : [o, b] x R + R is a sufficiently subject to the initial conditions:

smooth function,

y(0) = y,, y(‘)(O) = y$,‘), y(‘)(O) = yv), y(3)(0)

Although equation,

=

and

yb3'

(2)

we will be dealing with a single fourth-order all the procedures and proofs in this work can

0 1994 Butterworth-Heinemann

Deficient

(x, Y*) E Co,

w,

SC4’((4k + q)h) = S!&)+v=

S(x) = mi4S$i (x -rYkh)’ + r=O

where q = 1(1)4. Solving C 8,k = ;

1 c 6,k = -h2

In the case of a spline polynomial equation (4) takes the form:

=

i

4k

and

5 1 - ~ sg 2

(6)

of degree

m = 8,

11 12 26 -3

C 5,k

1 =

-

$$‘+4

-

(11)

)

y

h

35 -12

1

Sk;)+ +

1 i - 4 Sk?+ 4

in terms

(7)

(x - pkh)’ r.

$,$+3

+;

s$,!+z

Sk”,’I

(12)

)

4 +

j

s$y+

3 -

3sg+ 2

of interval

index

To prove the existence and uniqueness of (7) let us define the operator G,: R -+ R by Cj, k = g,(Cj, k), C,, k E R, j = 5(1)8 and v = 1(1)4. Let Cj,k be CjTk E R, j = 5(1)8, and let their distances be

i = 1(1)4, ~

+

r!

(jh - 4kh) i

r=S

c,,k

r!

’

According to Lipschitz (lo), we conclude that

Sj = SCjh)

dGl(c8,k),

G,(c,*,k))

Existence and uniqueness

s

In this section we prove that there exists a unique spline polynomial S(x) approximating the solution y(x) of the problem given by equations (1) and (2) provided that the size of the subinterval h satisfies some constraints.

< 81Lh4 - 80

Theorem 1 If h = min(h,,

(10)

(13)

Cih:-4kh)

r=O

i = l(l)4

+ i c,,, r=5

and

S”’

system (9) we obtain,

{

(5)

SC4’(jh) = f(jh, SO’h)), j = 4k + i,

(7) can be written

(9)

,k 6

r=m-3

r = O(l)m - 4

S(x) = i $1 (x rfkh)’ r=O

c,

{S$4k)+4- 4~Sk4k)+~ + 6S$4k)+2

+ 9sg+

(4)

f

i

r=5

1 3 C,,, = ~ - ~ Sc4) + 7$4,)+3 - 12$4k)+z h3 i 2

and the coefficients C,,, are determined by satisfying equation (1) at the end points of the four subintervals as follows:

SO’h)

+

1 ) - 4$4k)+ + sg I

where

x = jh, j = 4k + i,

s$W

(3)

where L is a constant, and that f E C”([o, b] x R). We divide the range of solution [o, b] into intervals I, = [kH, (k + l)H], where k = O(l)N - 1, and the interval size is H = b/N. Now, let us construct a polynomial spline function of degree m(m 2 8) and continuity class Cmm4[o, b], over a single interval of the size H = 4h, where h is the size of four subintervals. The polynomial is of deficiency 4, i.e., it includes four undetermined coefficients Cr,k, r = (m - 3)(l)m. The spline function S(x), x E I, is then defined by

Equation j, where

M. N. Esmail et al.

(6)

y),

bl x R

S$i = S”‘(4kh),

approximation:

Proof First, let us determine the coefficients of the spline polynomial (7) of degree m = 8. According to equation

be carried out for a system of first-order differential equations. Equations (1) and (2) can be interpreted as the vector form of this system. We also assume that the function f satisfies a Lipschitz condition If(x, Y) - f(x, Y*)l < LIY - y*1

spline

$

dc,,,,

dc,,k,

condition

=

igl(c,,k)

c;,k)l

-

(3) and

-

equation

gltc$,k)l

4 + Q8

- 4(3)81

C8*,kk)>

if i

h,, h,, h4), where

h4L < 1

(14)

where L is the Lipschitz constant. Following steps for CT.,, C,,,, and C,,,, we find

h,
similar

and h,
defined

by equation

(7)

h,<4;

Appl.

Math.

(15)

J

Modelling,

1994,

Vol. 18, December

659

Deficient

spline approximation:

M. N. Esmail et al.

respectively. From equations (14) and (15), it follows that G,, k = l(l)4 are contraction operators. This implies the existence and uniqueness of the spline polynomial (7) under the stated conditions of theorem 1.

Consistency

It is clear that

relations and convergence

p(1) = p(‘)(l) = ~(~‘(1) = ~(~‘(1) = 0 and

It is well-known’ that a linear method will be convergent if, and only if, it is both consistent and stable. Lemma 1 The deficient spline function approximations C”-4[0, b] are consistent for m = 8.

S(x) E

Proof For x = jh = (4k + 4)h equation S

4k+4

=

Sag

~

~

(8) leads to

~’~

+

i=O

i=S

.

where sa,,=’

h3 i

= 1 h2

Hence the method is consistent for m = 8, and lemma 1 has been proved. The roots of p(i) lie in a unit circle and are simple roots. Therefore the condition of stability is satisfied. Thus the method is consistent, stable, and convergent for m = 8.

The convergence of the method has been proved. However, it is also essential to determine the order of convergence. This will be accomplished in the following set of lemmas and theorems. Lemma

- 3M,

-5s,,+,

+ 3M, - M,

,

(17)

+ 4s,,+z

2

If ISfjh) - yfjh)IkhP, and Sc4’(jh) = fkh, Stjh)] for j = 4k + E, E = l(l)4 and k = O(l)N - 1, then there exists a constant k* such that IS( - y(jh)( s k*hP and 1St4’Cjh) - yc4)(jh)I I k*hP Proof

- Sdk+3 + 2S4k + g h4Sk4k)

The Lipschitz +5M,-4M,+M,

3S4k+l -- 3 h

S4k+2

1

+

-

2

S4k+3

-

11

-

3

condition

llS’4’0’h) - y’4’Cjh)/I =

(18)

(3) can be written in the form,

II f Cjh, Wh)l - f IA yO’h)lII

I LIISO’h) - yCjh)/I I LkhP

I

sa,,=’

~(~‘(1) = 4!a(l)

3S4k+l-3S4k+2+S‘tk+3-S4k

- 3 h4S$

sg

or

The level of convergence

(16)

Ci,kr

o(l) = 1,

p”( 1) = 4!,

If we let k* = max[k, Lk] then,

6

ISc4’0’h) - yc4’0’h)I s k*hP

1 x S,, - - h4S$$’ 4

Lemma (19)

3

Let y E C”‘+i[O, b] and let S(x) be the vector valued spline function defined in (4). Furthermore, let the following conditions be satisfied:

and

IIS”‘(x,,) - y(‘)(x,,)ll = O(hP’), Y = O(l)m - 4, a (rlh) M, = 1 yl Cr,k, r=s .

(20)

‘1 = l(l)4

k = O(l)N - 1, IICm_i,k - y’“-“(x,,)ll

Substituting equations (10)--(13) into equation (16), and using equation (6) the system of equations (16)-(20) yields S 4ic+4

-

4S4,c+3 + 6S4k+1-

4S4,c+1 + S,,

(22)

IIS(m-iI(x)_ y(m-i)(x)/I = O(h’+‘), i = 0(1)3, x4k < x < x4k+4, k = O(l)N - 1

(23)

then, IIS

+

(21) = 0(hP+8), i= 1, 2, 3

- y(x)ll = W’),

i7r,j,,+2

x E Co,bl,

(24)

where, p = Following Henrici’ p(i) and a(5) are

the

characteristic

polynomials

Appl.

Math.

Modelling,

1994,

Vol. 18, December

(25)

and, IIS@‘-‘)

p(c) = l4 - 4c3 + 6c2 - 41 + 1 = ([ - 1)4,

660

min (Y + p,), pm_1 = i+ 1, i = 0(1)3, r=O(l)m

- y(“-‘)(x)11 = O(h’+‘), x E [o, b], i = O(l)3 (26)

Deficient Proof

spline

approximation:

M. N. Esmail et al.

and therefore,

The proof is by induction. Let xbk < x < x4k+4 and 0 = x - x,,; then Taylor’s expansion leads to m-l

’

,zoF y”‘(x,,)

y(x) =

S’“-“(x4,) Then,

$ y’“‘(h X4k < i < x,

+

lIS’“-i’(x) - y’m-i)(X)I~ = C&h’+‘), i = O(l)3

XE [o, b-J, (27) m-4 s(x)

=

r

1 r=o

+

” r.

s”‘(x4k)

+

ril

&

Cm-r,k

w” P)(r) m!

(28)

-

y(x)

m-4

m’

1

yl

// s

11s’r’(x4k)

+,$, fi

-

+

4

-

Y”‘(x4k)

iIS

iI&r,k

-

- y(jh)I/ < Kh9,

m = 8, and

y(m-r)(X4k)11

S(x)=

w” IIS@Yi) - Y'"'KI/ m! dx)II

s

m-4

m’

rzo

2

,il&

(21H23).

Given

that

$jh)

Taylor’s

=

i

y

I/s’r’(x4k)

-

y”‘(x4k)11

Y(m-r)(X4k)ll

yb”

expansion +

!T!$

yields #9’({),

(30)

h)],

k=l(l)N-1,

(C,,, - yy’)

+ q +

Vhi (G3.0

- y$?) - o’h)9 ybg’(Q (31) 9!

gr

0<[<4h To complete the proof of lemma 4, we have to show that Ci,o, i = 5(1)8 are uniformly bounded as h + 0. The function gl(C,,,) is a contraction if h < qm. In particular, for h < lm we have - sl(G,o)l

< 0

However, y(m-i)(x4k)

x = jh, j = l(l)4

cr.02

S(jh) - yO’h) = !Y@ (C 5.0 - y2’) + !J!!lT (C,,, - Jg’) 5! 6!

lgi(Cs.0)

=

_

f

hp+

1 -‘)({I),

h

<

0 f

81 so Lh4/G,o,

GZo)

K3,o- G,ol

If we set C$,, = 0, we obtain 191(G3,o)I

-

ISl(O)ls lSI(G.0) - Sl(O)l

+ ; hy’“+ 1-i)(cz),

(32)

h X4k

<

12

the

Therefore,

i = O(l)3

y(m- 0

(4) takes

j = 1(1)4, 0 < [ < 4h

(x4,+;,>

+S+)(X~~-;

equation

.

r=O

IICm-r,k-

1 2[SW-9

x E [0,4h],

(29)

Equation (24) has been proved. In view of equation (24), to prove equation (26), it is sufficient to consider the nodal points x4k, k = l(l)N - 1. We define the functions S(m-i)(~), which are piecewise continuous on [0, b], using equation (22), and the arithmetic mean, =

j = l(l)4

‘to5 Yb" +,t55

The truncated

+ 0” llS(m)(5) - Y’“‘(i)II m!

s(m-i)(x4k)

K such that

Proof

I/

For form:

if we make use of conditions w = x - x4k I 4h, then II s(x)

Lemma

S(“)(x) is

.

r=O

+

This completes the proof of lemma 3. Similar lemmas were proved under different conditions by Loscalzo,’ Micula,4 and Sallam and El-Hawary.’

Let m = 8; then there exists a constant

Subtracting (27) and (28), and noting that constant over x4k < x < x4k+4, we obtain /I s(x)

+ O(h’+ ‘)

= y’“-“(x4,)

<

X4k

+

g IG,o)I 5 Islm

i

Appl.

Math.

Modelling,

*

IC,,o)I

1994,

I ;

ISI(O)l

Vol. 18, December

661

Deficient

spline

approximation:

However,

gl(Cs,J

= C&;

M. N. Esmail et al.

therefore,

and equation

&IC,,o)I -2

IC8,O)I - ISl(O)l5

(37) becomes,

IISW - YO’W= we

ICs,o)l5 Sl(O)

Similarly,

applying

lemma

c.5 ,,- yf’ = O(P), and the convergence Using equations (10) and (31), and applying Taylor’s expansion, we get 1 gl(0) = ; { -4y’4’(h) + 6yc4’(2h) - 4yc4’(3h) + yc4’(4h) + yg’ + O(h’)} I M

(33)

for some constant M. Because a uniform spacing is required over the interval [0, b], there is only a finite of possible values of h between lm Therefore C,,, is uniformly bounded

number dv=. h < gm

as h + 0. Similarly,

and for all

and

C,, 0 - Yf) = O(h)

level improves

to

llw4 - YO’h)II= 0(h7) Lemma C

2 with u = 4 leads to I

5,0

-

C,,,-

yb5’ = 0(h3),

yb6’ = O(h’),

and C.7 0 - yb7’ = O(h) and the level of convergence,

IIWd - yW)II = 0th’) Finally,

we can prove that

for p = 5, we find that,

C.5 0 - ybs’ = O(h4), C,,,

9yc4’(h) - 12yc4’(2h) + 7yc4’(3h)

gz(O)= $

2 to p = 3, we find that

- yr’ = O(h3),

C 7.0 - yb7’ = O(h’)

and

C8

o -

yf)

=

O(h)

and - : yc4’(4h) - ; y&+’+ 0(h5)

g3(0) = $

i

yc4’(h) + ;

- p

+ ;

yc4’(4h) + ::

1

5 M,

yc4’(2h) - ;

IISW4- H.34 = 0(h9)

(34)

This completes the proof of lemma 4. We have shown that the starting point S(4h) has an error of the order 0(h9). Indeed the following relations are supported by lemma 2 for p = 9:

yc4’(3h)

yg’ + O(h5)

S(jh) = y(jh) + O(h9), and

(35)

Sc4’(jh) = yc4)(jh) + O(h9) (38)

g4(0) = ; -;

yb4’ + 0(h5) I M

(36)

and consequently Ci,?, i = 5(1)8 are uniformly as h + 0. From equation (31) we obtain, SCjh) _ y(jh)

=

Cjh)’

i

(c.

i!

i=5

W9

- 91 Because Ci, o, h -+ 0, then,

i = 5(1)8

bounded

LO

y(i))

(37)

uniformly

bounded

as

Proof Consider

=

y,

q = l(l)4

Then the constants as follows:

Ci,k, i = 5(1)8 can

be determined

(29) and (30) we obtain yf')

+

i

i=2

y

.

Ci,o = 0(h2)

8,k=

+8

i

- 672o(S,, + 1 - Al) + 630(S4k+2 - A2)

-~(S,,+,--A3)+~(S4k+4-A4),

Then,

1 (39)

C,5 o = yv’ = O(h)

662

S,,,,

r=s

c -

the equation

S((4k + rl)h) =

Sc4’(jh) - yc4’(jh) = O(h’)

jh(C5.0

i = 5(1)8

(lS”‘(x) - Y”‘(x)II = O(h9-‘),

0

In further development toward the proof of lemma 4, we will determine that the level of convergence is of the order O(h’). If we set p = 2, lemma 2 yields

equations

2

Let f~ C8([0, 61 x R); then, _

~‘~‘(0, j = l(l)4

are

E = l(l)4

The purpose of our discussion is to prove that S(x) and its derivatives converge to y(x) and its corresponding derivatives. Theorem

SO’h) - y(jh) = O(h’)

From

j = 4k + E, and

4yc4’(h) - 3yc4’(2h) + ‘: yc4’(3h) - $ yc4’(4h)

Appl. Math.

Modelling,

1994,

Vol. 18, December

Deficient

G,, =

$

75WS4,+1 - Al) - 63O(S,,+,

1960 + 27 (KM+ 3 - A3) - g

Let then,

C 5,k

=

~~

h5 i

A3) + g

@4k+4

A4) ,

-

-

for i = l(l)4

further

that

- A3) - :2’x

4k+3

(24),

we consider

the

conditions

are

i = 1, 2, 3,

(48)

the following

r = O(l)m - 4, k = O(l)N - 1, ((Cm-i,k - y’m-i’(X4k)(/ = O(h’+‘),

81 (s

equation

= O(hm-‘+ ‘),

IlS”‘(x,,) - y”‘(x,,)ll

A21

@,k

f4

-

(47)

I~S’“-i’(x) - y’“-“(x)11 = O(h’+‘),

A4) >

i = 0(1)4, x4k < x < x4k+4,

k = O(1)N - 1,

(42)

where,

(49)

then,

A, = i y $1, r=O . Using constants

r/ = l(l)4

I(S”‘(x) - y”‘(x)ll = O(hm)

Taylor’s expansion and can be expressed as

c,,,

= yi”k’+ O(h), c,,,

c,,,

= y$j + 0(h3),

k

(38)

these

Proof Let m = 8 in equations 7 eIr y(x) = r;. -;

C,,, = y$“k’+ O(h4)

and

(6) and (29) into equations expansion, we obtain,

,g5(5:-;;- j

=

equation

= yi’k’ + 0(h2),

If we substitute equations (lOH13) applying Taylor’s

cj

in

5

Assuming satisfied:

160 +

m = 8

and

(&k+z - A2)

- Al) - “2’ @,k+,

480(S4k+l

p = 9,

To continue the proof following lemma.

(41) 1

i = 0,

IS(X) - Y(X)// = W9).

-Al)+?

-

M. N. Esmail et al.

- A4) )

(S,,,,

Lemma 1680 -__ 4k+3 81 (s

approximation:

Proof

- A2)

(40) -3120(S4k+l

spline

(27) and (28), then 0P

Y’r’(x4k)

+

$ + ‘x”: S@‘(1),

‘(‘) = i

r=O

y"'(X)

Y’9’K), r

2j

(43)

(6), assuming

that

X E tx4k,

X4k

+ 41,

S(‘)(x4k)

IlS”‘(4 - Y”‘(X)ll s ’

+

(43) into

(s

c,

-‘,

k

<

i

(

x

(51)

11 s”‘(x4k)

S”‘(x) - y”‘(x) = 0(h2),

-

,tl&

Y”‘(x4k)

11

‘il

(7-F

ilC8-‘,k

-

Y’8-r)(X4k)lI

the corresponding

Sc5’(x) - y”‘(x) = 0(h4), S’@(x) - y’@(x) = O(h3)

+ ;;

(45)

llS@‘K) - YW)I/

where o = x - x4k I 4h. From equations (47H49)

and

S@‘(x) - y@‘(x) = O(h)

IIS”’ (46)

that

((S”‘(x) - y”‘(x)I/ = 0(h9 -‘), i = 5( 1)8

we have

- y”‘(x)ll = 0(h8)

This completes the proof of lemma 5. Now, if we subtract the second to fourth derivatives of equation (50) from the corresponding derivatives of equation (51), we obtain, IlS2’(x) - ~‘~‘(x)Ij = O(h’),

3

If f E C”([O, b] x R) then there exists a constant such that IIS”’

‘il

o7-’ +

Substituting equations equations (44), we obtain,

Theorem

x,

Subtract the first derivatives of equation (50) from the first derivatives of equation (51); we have

(44)

Then we conclude

<

.

where i E (x4,, X), IX - X,,14h. equation

i

x)9-i

(9 -j)!

From we get

<

($+r

X4k

+

X4k

.f%)?

(50)

Y

(x L!!!

-z

- y”‘(X)/( < Ki(h9-‘).

Ki

IlSC3’(x)- yC3’(x)II= 0(h6) lISC4’(x)- ~‘~‘(x)lI = O(h’)’ Theorem

3 has been proved.

APPl. Math. Modelling,

1994,

Vol. 18, December

663

Deficient spline approximation:

Absolute maximum

Table 1. h I.5 x I.4 x I.2 x 1.1 x 8.0 x 7.0 x 6.0 x 5.0 x 4.0 x

M. N. Esmail et al.

IIS-

IO-' IO-' IO-' IO-' IO-2 Io-2 1O-2 Io-2 IO-2

1.75 5.22 9.32 3.09 3.66 2.62 I.68 4.93 3.24

Table

2.

x x x x x x x x x

YII,

Il(S -

Y(‘)ll,

IIts-

IO-8 IO-' IO-'0 IO-'0 IO-" IO-" IO-" IO-'2 IO-l2

2.04 5.19 9.15 2.85 5.43 4.27 3.04 I.26 9.16

x IO-' x10-s x 10-s x IO-' x IO-'0 x IO-'O x IO-'0 x IO-lo x IO-"

2.36 5.41 9.61 2.76 5.35 5.16 4.11 2.38 I.92

Absolute maximum

IO-' IO-' IO-' 10-l IO-* IO-2 IO-* IO-2 IO-2

2.08 5.10 I.36 5.30 I.76 1.30 4.80 3.90 2.73

x x x x x x x x x

IO-3 IO-“ 1O-4 Io-4 IO-5 10-s IOm6 Io-7 IO-'

wx x x x x x x x x x

IO-6 IO-' 1O-s IO-8 10-s 10mg 10-s IOmg 10-s

Il(S - Y@II

IIts-

2.57 x IO-5 5.64 x IO-' 1.07 x10-s 2.99 x IO-' 3.76 x 1O-s I.73 x 10-s 1.69 x 1O-8 I.56 x IO-* 1.04 x IO-8

2.47 5.46 1.16 3.35 5.03 1.05 1.02 1.01 1.00

yt4)IIco x x x x x x x x x

IO-“ IO-5 IO-5 Io-6 IO-' IO-7 IO-' IO-' IO-'

errorsof the numerical solutionand itsderivatives

II@ - y(5)llm

h I.5 x 1.4 x I.2 x 1.1 x 8.0 x 7.0 x 6.0 x 5.0 x 4.0 x

errorsof the numerical solutionand itsderivatives

tits1.57 5.13 2.05 I.23 5.30 4.62 2.57 8.17 2.01

y(6)Ilm x x x x x x x x x

Theorem 4

IO-2 10-s IOm3 IO-3 IO-4 IO-4 IO-4 IOm5 10-s

IIts7.05 x 5.05 x 2.93 x 2.24 x I.24 x 1.12x 8.20 x 8.38 x 4.21 x

Y(‘)II

ll(S - Y@lI a

IO-' IO-2 IO-2 IO-' IO-2 10-2 IOm3 IO-4 IO-4

6.82 5.01 4.16 3.77 2.96 2.61 2.27 I.78 I.42

x x x x x x x x x

IO-' IO-' IO-' IO-' IO-' IO-' IO-' IO-' IO-'

Concluding remarks

Let m = 8, then /IS”‘(x) = Y(~)(X)II = O(hg - ‘), i = 0( 1)8.

(52)

Proof It follows clearly from theorems

2 and 3.

We proved the existence and uniqueness of the spline polynomials’ approximation (4H6), for polynomials of degree m = 8 and deficiency 4 as a solution for the Cauchy problem (1) and (2) of a fourth-order differential equation. We proved that the method is consistent, stable, and consequently convergent for m = 8. We also established a level of convergence for this solution, which is expressed by equation (52).

A test problem The method can be illustrated in the solution of the following fourth-order linear differential equation. Consider the initial value problem, YC4)= I: with the initial

- 1I x I 1 conditions,

References Loscalzo, F. R. and Talbot, T. D. Spline function approximation for solutions of ordinary differential equations. SIAM J. Numer. Anal. 1961, 4,433445 Micula, G. Approximate solution of the differential equation @;_;A(x, y) with spline functions. Math. Comput. 1973, 27, Y

Y(0) = Y”‘(0) = Y@)(O)= Y@‘(O)= 1 The exact solution

of this problem

is

Y(x) = eX The application of an approximate spline solution of degree m = 8 and deficiency 4 yields the numerical results summarized in Tables I and 2. Sallam and El-Hawary reported a deficient spline function approximation of degrees m = 4,5, and deficiency 3. In their numerical solutions, they showed that the method yielded errors of the order of magnitude of O(h’) and 0(h6) respectively. Tables I and 2 illustrate that the error generated in numerical solutions based on our method is of the order of magnitude of O(h’) as stated in lemma 4. The numerical solution (9) satisfies the consistency and stability conditions as stated in lemma 1.

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Appl. Math. Modelling,

1994, Vol. 18, December

Micula, G. Deficient spline approximate solutions to linear differential eauations of the second order. Muthematica 1974. 16(39), 1, 65-j2 Micula, G. The numerical solution of nonlinear differential eauations bv suline functions. Z.A.M.M. 1975,55,254255 Sallam, S. and El-Hawary, H. M. A deficient spline function approximation to systems of first-order differential equations, 1. Appl. Math. Modelling 1983, 7(5), 380-382 Sallam, S. and El-Hawary, H. M. A deficient spline function approximation to systems of first-order differential equations, 2. Appl. Math. Mode&g 1984, S(2), 128-132 Sallam, S. and El-Hawary, H. M. A deficient spline function approximation of second-order differential equations. Appl. Math. Modelling 1984, 8(6), 408-412 Ahmed, M. Approximation solutions of the initial value problem of y’3’ = f(x, y) using spline functions. Ph.D. Thesis, Suez Canal University, Ismailia, Egypt, 1989 Henrici, P. Discrete Variable Methods in Ordinary Differential Equations. John Wiley and Sons, New York, 1962