Applied Mathematics and Computation 167 (2005) 153–166

www.elsevier.com/locate/amc

Spline methods for the solutionof fourth-order parabolic partial

differential equations

Tariq Aziz a, Arshad Khan b,*, Jalil Rashidinia a

a Department of Applied Mathematics, Faculty of Engineering & Technology,

A.M.U., Aligarh, U.P. 202 002, Indiab Department of Mathematics, A.M.U., Aligarh, U.P. 202 002, India

Abstract

In this paper a fourth-order non-homogeneous parabolic partial differential equa-

tion, that governs the behaviour of a vibrating beam, is solved by using a new three level

method based on parametric quintic spline in space and finite difference discretization in

time. Stability analysis of the method has been carried out. It has been shown that by

suitably choosing the parameters most of the previous known methods for homogene-

ous and non-homogeneous cases can be derived from our method. We also obtain two

new high accuracy schemes of O(k4,h6) and O(k4,h8) and two new schemes which are

analogues of Jain�s formula for the non-homogeneous case. Comparison of our results

with those of some known methods show the superiority of the present approach.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Fourth-order parabolic equation; Parametric quintic spline; Spline relations; Stability

analysis; Vibrating beam; Class of methods

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.06.095

* Corresponding author.

E-mail addresses: akhan1234in@rediffmail.com, akhan1234in@yahoo.co.in (A. Khan).

154 T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166

1. Introduction

Consider the problem of undamped transverse vibrations of a flexible

straight beam in such a way that its supports do not contribute to the strain

energy of the system and is represented by the fourth-order parabolic partial

differential equation,

o2uot2

þ o4uox4

¼ f ðx; tÞ; 0 6 x 6 1; t > 0; ð1Þ

subject to the initial conditions

uðx; 0Þ ¼ g0ðxÞ;and utðx; 0Þ ¼ g1ðxÞ for 0 6 x 6 1;

ð2Þ

and with boundary conditions at x = 0 and x = 1 of the form

uð0; tÞ ¼ f0ðtÞ; uð1; tÞ ¼ f1ðtÞ; and

uxxð0; tÞ ¼ p0ðtÞ; uxxð1; tÞ ¼ p1ðtÞ; t P 0;ð3Þ

where u is the transverse displacement of the beam, t and x are time and dis-

tance variables respectively, f(x, t) is dynamic driving force per unit mass.

Numerical solution of (1) based on finite difference and reduction of (1) intoa system of second-order equations have been successfully proposed by Collatz

[4], Crandall [7], Conte and Royster [6], Conte [5], Albrecht [3], Evans [9], Jain

et al. [14] and Richtmyer [17]. While Fairweather and Gourlay [11] derived

explicit and implicit finite difference methods based on the semi-explicit method

of Lees [15] and high accuracy method of Douglas [8] respectively. Evans and

Yousif [10] follow the Conte scheme [5] where a stable implicit finite difference

approximation is presented and this scheme was unconditionally stable, and

has local truncation error of O(h2). The approach, introduced in [10], usingthe alternating group explicit method (AGE), achieved a better accuracy level.

Wazwaz [19,20] approaches the problem by utilizing the Adomian decomposi-

tion method [1]. The solution by this method is derived in the form of a power

series but does not include numerical results. The homogeneous problem has

also been studied by Rashidinia [16].

We need to construct a direct numerical method for solution of Eq. (1).

Direct explicit and implicit difference methods have been given by Albrecht

[3], Collatz [4], Crandall [7], Jain [12], Jain et al. [14] and Todd [18]. The threelevel explicit direct method with order of accuracy O(k2 + h2) given by Collatz

[4] is stable when the mesh ratio (k/h2) 6 1/2. The three level unconditionally

stable formulas of accuracy O(k2 + h2) and Oðk2 þ h2 þ kh

� �2Þ are given by Todd

[18], Crandall [7] and Conte [5] respectively. Five level, unconditionally stable,

explicit method with truncation error of Oðk2 þ h2 þ ðkh Þ2Þ has been given by

Albrecht [3]. Direct and splitting approach finite difference methods have been

proposed by Jain et al. [14].

T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166 155

We have derived new three level methods based on parametric quintic

spline-I for the solution of fourth-order, non-homogeneous, parabolic partial

differential equation governing transverse vibrations of a flexible beam. In Sec-

tion 2, we give a brief derivation of this non-polynomial parametric spline. We

present the spline relations to be used for discretization of the given system (1).

In Section 3, we present the formulation of our method. In Section 4, we showthat by choosing suitable values of parameters, most of the previous known

methods can be derived from our method. In Section 5, stability analysis has

been carried out. Finally in Section 6, numerical evidence is included to dem-

onstrate the practical usefulness and superiority of our schemes and confirm

their theoretical behaviour.

2. Parametric quintic spline functions

We consider a uniform mesh D with nodal points xj on [a,b] such that

D : a ¼ x0 < x1 < x2 < � � � < xN�1 < xN ¼ b;

where xj = a + jh, for j = 0(1)N. Also we denote a function value u(xj) by uj.

A function SD (x,s) of class C 4[a,b], which interpolates u(x) at the mesh

points xj, j = 1(1)N, depends on a parameter s, and reduces to ordinary quinticspline SD(x) in [a,b] as s ! 0 is termed as parametric quintic spline function.

Since the parameter s can occur in SD(x,s) in many ways such a spline is not

unique.

If SD(x,s) is a piecewise function satisfying the following differential equa-

tion in the interval [xj�1,xj],

Sð4ÞD ðx; sÞ þ s2Sð2Þ

D ðx; sÞ ¼ ðF j þ s2MjÞx� xj�1

hþ ðF j�1 þ s2Mj�1Þ

xj � xh

¼ qjzþ qj�1�z; ð4Þwhere z = (x � xj�1)/h, �z ¼ 1� z, qj = Fj + s2Mj, S

ð4ÞD ðxi; sÞ ¼ F i, i = j � 1, j and

s > 0, then it is termed �parametric quintic spline-I�, see [16].

Solving the differential equation (4) and determining the four constants of

integration from the interpolatory conditions SD(xi,s) = ui, S00Dðxi; sÞ ¼ Mi,

i = j � 1, j we obtain

SDðx; sÞ ¼ zuj þ �zuj�1 þh2

6½g0ðzÞMj þ g0ð�zÞMj�1�

þ hx

� �4 x2

6g0ðzÞ � g1ðzÞ

� �F j

þ hx

� �4 x2

6g0ð�zÞ � g1ð�zÞ

� �F j�1; ð5Þ

where g0(z) = z3 � z, g1ðzÞ ¼ z� sinxzsinx , x ¼ h

ffiffiffis

p.

156 T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166

2.1. Spline relations

Considering the spline in [xj�1,xj] and [xj,xj+1] and using the continuity of

the first and second derivatives at xj leads to the following equations.

ðiÞ Mjþ1 þ 4Mj þMj�1 ¼6

h2ðujþ1 � 2uj þ uj�1Þ

� 6h2ða1F jþ1 þ 2b1F j þ a1F j�1Þ;ðiiÞ Mjþ1 � 2Mj þMj�1 ¼h2ðaF jþ1 þ 2bF j þ aF j�1Þ;

ð6Þ

where a ¼ 1x2 ðxcosecx� 1Þ, b ¼ 1

x2 ð1� x cotxÞ, a1 ¼ 1x2

16� a

� �, b1 ¼ 1

x2

13� b

� �.

The consistency relation for (6-ii) leads to the equation x2¼ tan x

2.

From (6) we obtain

Mj ¼1

h2ðujþ1 � 2uj þ uj�1Þ

� h2 a1 þa6

� F jþ1 þ 2 b1 þ

b6

� �F j þ a1 þ

a6

� F j�1

� �: ð7Þ

Substituting for Mj+1, Mj and Mj�1 from (7) into (6) we arrive at the following

useful relation

pF jþ2 þ qF jþ1 þ sF j þ qF j�1 þ pF j�2 ¼1

h4d4uj; ð8Þ

which is alternatively written as KF j ¼ 1h4d4uj, where d is the central difference

operator and the operator K is defined by

Kwj ¼ pðwjþ2 þ wj�2Þ þ qðwjþ1 þ wj�1Þ þ swj; ð9Þwhere p ¼ a1 þ a

6, q ¼ 2 1

6ð2aþ bÞ � ða1 � b1Þ

�, s ¼ 2 1

6ðaþ 4bÞ þ ða1 � 2b1Þ

�.

When s ! 0, that is x! 0, then ða; b; a1; b1Þ ! ð16; 13; �7360

; �8360Þ and ðp; q; sÞ !

1120

ð1; 26; 66Þ, and the spline defined by (5) reduces into quintic spline and the

above spline relations reduce to corresponding ordinary quintic spline relations

[2]. In parametric quintic spline we have a greater choice and flexibility for

deriving methods suitable for various situations, which is a great advantage

of parametric quintic spline over polynomial quintic spline. For ðp; q; sÞ !136� ð1; 56; 246Þ parametric quintic spline relation (8) reduces to the correspond-ing sextic spline relation.

3. The method

Let the region R = [0,1] · [0,1) be discretized by a set of points Rh,k which

are the vertices of a grid of points (xj, tm), where xj = jh, j = 0(1)N, Nh = 1 and

T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166 157

tm = mk, m = 0,1,2,3. The quantities h and k are mesh sizes in the space and

time directions respectively [13].

We next develop an approximation for (1) in which the time derivative is

replaced by a finite difference approximation and the space derivative by the

parametric quintic spline function approximation. Eq. (1) is then discretized

and written in the form

k�2ð1þ rd2t Þ�1d2t u

mj þ F m

j ¼ f mj ; ð10Þ

where F mj ¼ Sð4Þ

D ðxj; tmÞ is the fourth spline derivative at (xj, tm) with respect to

the space variable, f mj ¼ f ðxj; tmÞ, umj is the approximate solution of (1) at

(xj, tm), dt is the central difference operator with respect to t so that d2t umj ¼

umþ1j � 2umj þ um�1

j and r is a parameter such that the finite difference approx-

imation to the time derivative is O(k2) for arbitrary r and of O(k4) for

r = 1/12. Operating both sides of (10) by Kx and using (8) we obtain

d2t fpðumjþ2 þ umj�2Þ þ qðumjþ1 þ umj�1Þ þ sumj g þ r2ð1þ rd2t Þd4xu

mj

¼ k2ð1þ rd2t Þfpðf mjþ2 þ f m

j�2Þ þ qðf mjþ1 þ f m

j�1Þ þ sf mj g; ð11Þ

where r = k/h2 is the mesh ratio and p,q, s are parameters.

After simplifying the above equation we obtain

fð2p þ 2qþ sÞ þ ð4p þ qÞd2x þ ðp þ r2rÞd4xgd2t u

mj þ r2d4xu

mj

¼ k2ð1þ rd2t Þfpðf mjþ2 þ f m

j�2Þ þ qðf mjþ1 þ f m

j�1Þ þ sf mj g;

j ¼ 2; 3; . . . ;N � 2: ð12Þ

The final scheme (12) is finite difference in time and spline scheme in x variable

which may be written in schematic form as

P 2 Q2 S2 Q2 P 2

�2P 2 þ r2 �2Q2 � 4r2 �2S2 þ 6r2 �2Q2 � 4r2 �2P 2 þ r2

P 2 Q2 S2 Q2 P 2

9>=>;umj

¼K1P K1Q K1S K1Q K1P

K2P K2Q K2S K2Q K2P

K1P K1Q K1S K1Q K1P

9>=>;f m

j ;

where P2 = p + rr2, Q2 = q � 4rr2, S2 = s + 6rr2, K1 = rk2, K2 = (1 � 2r)k2.For a fixed time level tm the solution at the grid points (xj, tm) are obtained

by solving a system of linear equations and then the solution at any intermedi-

ate point may be calculated by using the spline (5). Of course, for a suitably

chosen parameter, Fj�s are given by (8) and then Mj�s by (6).

158 T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166

4. Class of methods

By choosing suitable values of parameters p, q, s and r we obtain various

methods for homogeneous case i.e. f(x, t) = 0 and non-homogeneous case, i.e.

f(x, t)5 0. The truncation error and stability analysis of these methods are gi-

ven in Section 5.Homogeneous case: i.e. f(x, t) = 0.

(1) If we choose p = q = 0 and s = 1 and put r = 1/4,1/2 in (12) we obtain the

unconditionally stable formulas of Todd and Crandall respectively.

(2) For p = 0, q = �r2, s = 1 + 2r2, r = 0 in (12), we get the unconditionally

stable Conte formula

ð1� r2d2xÞd2t u

mj þ r2d4xu

mj ¼ 0 ð13Þ

with truncation error Oðk2 þ h2 þ ðkh Þ2Þ.

(3) For p = 0, q = 1/6 and s = 2/3 and r = 1/4 we obtain unconditionally stable

formula of Jain et al. [14] and r = 1/12 gives the conditionally stable for-

mula of accuracy O(k4 + h4) with r2 ¼ 18.

(4) For the choice p ¼ 512r2, q ¼ 1

6� 8

3r2 and s ¼ 2

3þ 9

2r2, we get the uncondi-

tionally stable method with accuracy Oðk2 þ h2 þ ðkh Þ2Þ.

1þ 1

6� r2

� �d2x þ rþ 5

12

� �r2d4x

� �d2t u

mj þ r2d4xu

mj ¼ 0 ð14Þ

for r = 1/12 we obtain Jain�s formula [14].

Non-homogeneous case: i.e. f(x, t) 5 0.

(1) If we choose p = q = 0 and s = 1 in (12) we get the scheme (15) with trun-

cation error O(k2 + h2) which is unconditionally stable when rP1/4.

ð1þ rr2d4xÞd2t u

mj þ r2d4xu

mj ¼ k2ð1þ rd2t Þf m

j : ð15Þ

(2) For the choice p ¼ 512r2, q ¼ 1

6� 8

3r2 and s ¼ 2

3þ 9

2r2, we arrive at the uncon-

ditionally stable method with accuracy Oðk2 þ h2 þ ðkh Þ2Þ.

1þ 1

6� r2

� �d2x þ rþ 5

12

� �r2d4x

� �d2t u

mj þ r2d4xu

mj

¼ k2ð1þ rd2t Þr2

12f5ðf m

jþ2 þ f mj�2Þ � 32ðf m

jþ1 þ f mj�1Þ þ 54f m

j g�

þ 1

6ðf m

jþ1 þ f mj�1 þ 4f m

j Þ�: ð16Þ

T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166 159

(3) For p = 0, q = 1/6 and s = 2/3 we get the formula with minimum truncation

error among the class of formulas with truncation error O(k2 + h4).

1þ 1

6d2x þ rr2d4x

� �d2t u

mj þ r2d4xu

mj ¼ 1

6k2ð1þ rd2t Þðf m

jþ1 þ f mj�1 þ 4f m

j Þ:

ð17Þ(4) If we choose (p,q, s) = 1/144 (1,20,102), we get

1þ 1

6d2x þ

1

144þ rr2

� �d4x

� �d2t u

mj þ r2d4xu

mj

¼ 1

144k2ð1þ rd2t Þ½ðf m

jþ2 þ f mj�2Þ þ 20ðf m

jþ1 þ f mj�1Þ þ 102f m

j �: ð18Þ

For r = 1/4,1/12 in (17), we get the unconditionally stable formula which is

the analogue of Jain�s formula [14]. The formula has minimum truncation

error among the class of formulas with order of accuracy O(k2 + h4) andO(k4 + h4) with r2 = 1/6.

(5) If we choose (p,q, s) = 1/720 (�1,124,474), r = 1/12 we obtain a new high

accuracy method with truncation error O(k4 + h6).

1þ 1

6d2x þ

1

12r2 � 1

60

� �d4x

� �d2t u

mj þ r2d4xu

mj

¼ 1

720k2 1þ 1

12d2t

� �½�ðf m

jþ2 þ f mj�2Þ þ 124ðf m

jþ1 þ f mj�1Þ þ 474f m

j �; ð19Þ

which is conditionally stable for r2 6 760and for the particular case r2 ¼ 1

84,

we obtain the scheme of O(k4 + h8) which has highest accuracy among all

the known schemes.

5. Truncation error and stability analysis

Expanding (12) in Taylor series in terms of u(xj, tm) and its derivatives, we

obtain the following relations

d4xuðxj; tmÞ ¼ h4D4x þ

h6

6D6

x þh8

80D8

x þ17h10

3024D10

x þ 62h12

10!D12

x þ � � �� �

uðxj; tmÞ;

d2t uðxj; tmÞ ¼ �r2h4D4x þ

1

12r4h8D8

x �1

360r6h12D12

x þ 1

20160r8h16D16

x � � � �� �

� uðxj; tmÞ;ð20Þ

where ðD2t þ D4

xÞuðxj; tmÞ ¼ f ðxj; tmÞ. Using (12) and (20) we obtain the trunca-

tion error

160 T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166

Tmj ¼ fð2p þ 2qþ sÞ þ ð4p þ qÞd2x þ ðp þ r2rÞd4xgd

2t u

mj þ r2d4xu

mj

� k2ð1þ rd2t Þfpðf mjþ2 þ f m

j�2Þ þ qðf mjþ1 þ f m

j�1Þ þ sf mj g

¼ p1k2D2

t þ p1k4D4

t

12þ p1

2k6D6t

6!þ p1

2k8D8t

8!

�þ � � �

þ q1h2k2D2

xD2t þ q1h

2D2x

k4D4t

12þ q1h

2D2x

2k6D6t

6!

þ q1h2D2

x

2k8D8t

8!þ � � � þ q1

12þ s1

� h4

� D4x k2D2

t þk4D4

t

12þ 2k6D6

t

6!þ 2k8D8

t

8!þ � � �

� �

þ 2q16!

þ s16

� �h6D6

x k2D2t þ

k4D4t

12þ 2k6D6

t

6!þ � � �

� �

þ 2q18!

þ s180

� �h8D8

x k2D2t þ

k4D4t

12þ � � �

� ��umj

þ r2 h4D4x þ

h6D6x

6þ h8D8

x

80þ � � �

� �umj

� p1 k2 þ rk4D2t þ r

k6D4t

12þ 2r

k8D6t

6!þ � � �

� ��

þ 2q1h2D2

x

2!k2 þ rk4D2

t þ rk6D4

t

12þ 2r

k8D6t

6!þ � � �

� �

þ 2ð16p þ qÞ h4D4

x

4!k2 þ rk4D2

t þ rk6D4

t

12þ 2r

k8D6t

6!þ � � �

� �

þ 2ð64p þ qÞ h6D6

x

6!k2 þ rk4D2

t þ rk6D4

t

12þ 2r

k8D6t

6!þ � � �

� �

þ2ð256p þ qÞ h8D8

x

8!k2 þ rk4D2

t þ rk6D4

t

12þ � � �

� ��

� ðD2t u

mj þ D4

xumj Þ;

T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166 161

which may be written as

Tmj ¼ ð1� p1Þr2h4D4

x þr2h4

6� q1k

2

� �h2D6

x

�

þ r2h4

20� ð16p þ qÞk2

3

� �h4

4D8

x þ17r2h4

2!� 2ð64p þ qÞk2

5

� �

� h6

144D10

x þ � � � þ 1

12� r

� �p1k

4D4t þ

1

12� r

� �q1h

2k4D2xD

4t

þ q112

þ s1�

h4 � p1rk2 � ð16p þ qÞh4

12

� �k2D2

t D4x

þ 2q16!

þ s16

� �h4 � q1rk

2 � 2ð64p þ qÞh4

6!

� �h2k2D6

xD2t

þ 2

6!� r12

� �p1k

6D6t þ

q112

þ s1�

h4 � r12

p1k2

�

� rð16p þ qÞh4

12

�k4D4

xD4t þ

1

360� r

2

� �q1h

2D2xk

6D6t

þ 1

8!� r6!

� �2p1k

8D8t þ

2q8!

þ s180

� �h4 � rð16p þ qÞk2

12

�

� 2rð256p þ qÞh4

8!

�k2h4D8

xD2t þ

2q8!

� 2q1r6!

� �D2

xD8t k

8h2

þ q112

þ s1� 2

6!h4 � 2p1r

6!k2 � rð16p þ qÞh4

144

� �k6D4

xD6t

�umj þ � � � ;

ð21Þwhere p1 = (2p + 2q + s), q1 = 4p + q and s1 = p + rr2.

For various values of parameter p, q, s and r, the truncation errors men-

tioned in Section 4 may now be obtained.

Using Von Neumann�s method [12] the characteristic equation of the

scheme (12) is obtained as:

n2 þ 2cnþ 1 ¼ 0; ð22Þwhere c ¼ 8r2sin4/

16ðpþrr2Þsin4/�4ð4pþqÞsin2/þð2pþ2qþsÞ � 1, / ¼ 12hh, where h is the variable in

the Fourier expansion.

Applying the Routh–Hurwitz criterion to (22) we get the necessary and suf-

ficient conditions for (12) to be stable as:

�1 6 1� 8r2sin4/

16ðp þ rr2Þsin4/� 4ð4p þ qÞsin2/þ ð2p þ 2qþ sÞ6 1:

Simplifying and putting 2p + 2q + s = 1 we obtain from the left inequality

4½4p þ ð4r� 1Þr2�sin4/� 4ð4p þ qÞsin2/þ 1 P 0: ð23Þ

162 T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166

We deduce that the scheme (12) is unconditionally stable if

r P 1=4; q < 1=4 and p ¼ 0 or p P ð4p þ qÞ2=4;and conditionally stable if

(i) r < 1/4, q < 1/4, p = 0, 0 < r2 6 1�4q4ð1�4rÞ or

(ii) r < 1/4, (4p + q) < 1/4, p ¼ 14ð4p þ qÞ2, 0 < r2 6 ½1�2ð4pþqÞ�2

4ð1�4rÞ .

6. Numerical illustrations and discussions

In this section we consider the numerical results obtained by the schemes

discussed above by applying them to the following fourth-order initial bound-

ary value problems.

Example 6.1. We consider a fourth-order non-homogeneous parabolic partial

differential equation introduced by [10].

o2uot2

þ o4uox4

¼ ðp4 � 1Þ sin px cos t; 0 6 x 6 1; t > 0; ð24Þ

with the initial conditions

uðx; 0Þ ¼ sin px; utðx; 0Þ ¼ 0; 0 6 x 6 1; ð25Þ

and the boundary conditions

uð0; tÞ ¼ uð1; tÞ ¼ o2uox2

ð0; tÞ ¼ o2uox2

ð1; tÞ ¼ 0; t P 0: ð26Þ

The exact solution of the above problem is

uðx; tÞ ¼ sin px cos t; ð27Þ

For solving (24) we use scheme (12). The first two boundary conditions in (26)

are replaced by

um0 ¼ umN ¼ 0; t P 0: ð28ÞWe discretize the last two boundary conditions in (26) by the following

equations

ðiÞ 2um0 � 5um1 þ 4um2 � um3 ¼ h2ðum0 Þ00;

ðiiÞ � umN�3 þ 4umN�2 � 5umN�1 þ 2umN ¼ h2ðumN Þ00:

ð29Þ

For high accuracy formulas of O(k4 + h6) and O(k4 + h8) we use the following

equations for approximating the boundary conditions:

T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166 163

ðiÞ 45um0 � 154um1 þ 214um2 � 156um3 þ 61um4 � 10um5 ¼ 12h2ðum0 Þ00;

ðiiÞ � 10umN�5 þ 61umN�4 � 156umN�3 þ 214umN�2 � 154umN�1 þ 45umN ¼ 12h2ðumN Þ00;

ð30Þwhere ðum0 Þ

00 ¼ o2uð0;tmÞox2 , ðumN Þ

00 ¼ o2uð1;tmÞox2 .

We solved example 6.1 with h = 0.05 and k = 0.005 giving r = 2, and by

choosing r = 1/4 and various values of parameters p,q, s presented in

Table 1. The errors in the solutions computed by our method (12) and the

AGE method [10] have been presented in Table 1 for 10 time steps andx = 0.1(0.1)0.5 and in Table 2 for x = 0.5 and larger time steps. In a second ser-

ies of experiments, calculations are carried out for h = 0.05 and k = 0.00125

giving r = 0.5, and by choosing suitable values of parameters p, q, s and results

presented in Table 1. The absolute errors in the solution are shown in Table 1

for 16 time steps and x = 0.1(0.1) 0.5 and in Table 2 for x = 0.5 and larger time

steps. From Tables 1 and 2, it is evident that our method is superior. Moreover,

we solved the same problem with various values of parameters carrying out the

computations for different time steps. In Tables 3 and 4 we have tabulated theabsolute errors at x = 0.5 for different combination of parameters p, q, s, r and

Table 2

Absolute errors at mid points, x = 0.5, Example 6.1, h = 0.05

Our method p, q, s, r r = 2 r = 0.5

No. of time steps No. of time steps

25 75 100 32 48 64

(0, 0, 1, 1/4) 2.7 · 10�3 7.8 · 10�3 3.0 · 10�3 3.0 · 10�4 7.0 · 10�4 1.2 · 10�3

(0, 1/6, 2/3, 1/4) 1.0 · 10�4 3.4 · 10�4 1.3 · 10�4 1.8 · 10�6 2.7 · 10�5 5.2 · 10�5

(1/144, 5/36, 17/24, 1/4) 1.0 · 10�4 3.5 · 10�4 1.3 · 10�4 2.1 · 10�6 2.7 · 10�5 5.2 · 10�5

Evans [10] 3.3 · 10�3 4.1 · 10�3 3.9 · 10�3 3.1 · 10�4 6.9 · 10�4 1.2 · 10�3

Table 1

Absolute errors, Example 6.1, h = 0.05

Our method p, q, s, r r Time

steps

x = 0.10 0.20 0.30 0.40 0.50

(0, 0, 1, 1/4) 2.0 10 1.5 · 10�4 2.8 · 10�4 3.7 · 10�4 4.2 · 10�4 4.4 · 10�4

(0, 1/6, 2/3, 1/4) 1.8 · 10�5 2.0 · 10�5 1.4 · 10�5 8.3 · 10�6 5.7 · 10�6

(1/144, 5/36, 17/24, 1/4) 1.8 · 10�5 2.1 · 10�5 1.5 · 10�5 8.8 · 10�6 6.2 · 10�6

(0, 0, 1, 1/4) 0.5 16 3.2 · 10�5 5.1 · 10�5 6.2 · 10�5 6.9 · 10�5 7.2 · 10�5

(0, 1/6, 2/3, 1/4) 9.3 · 10�6 8.0 · 10�6 2.8 · 10�6 1.0 · 10�6 2.7 · 10�6

(1/144, 5/36, 17/24, 1/4) 9.2 · 10�6 7.9 · 10�6 2.8 · 10�6 9.8 · 10�7 2.5 · 10�6

Evans [10] 2.0 10 2.2 · 10�4 4.1 · 10�4 5.4 · 10�4 6.2 · 10�4 6.5 · 10�4

0.5 16 2.5 · 10�5 4.7 · 10�5 6.6 · 10�5 7.8 · 10�5 8.2 · 10�5

Table 4

Absolute errors at mid points, x = 0.5, Example 6.1, h = 0.02

Our method p, q, s, r r No. of time steps

10 20 30

(0, 1/6, 2/3, 1/4)p1/6 2.53 · 10�12 7.60 · 10�12 2.60 · 10�10

(1/144, 5/36, 17/24, 1/4)p1/6 1.51 · 10�11 9.40 · 10�11 1.30 · 10�9

(1/144, 5/36, 17/24, 1/12)p1/6 1.51 · 10�11 1.52 · 10�10 2.69 · 10�11

�1720

; 124720

; 474720

; 112

� � p7/60 5.66 · 10�15 1.45 · 10�10 3.37 · 10�10

�1720

; 124720

; 474720

; 112

� � p1/84 8.88 · 10�15 4.14 · 10�14 7.17 · 10�14

Table 3

Absolute errors at mid points, x = 0.5, Example 6.1, h = 0.1

Our method p, q, s, r r No. of time steps

10 20 30

(0, 1/6, 2/3, 1/4)p1/6 4.63 · 10�5 1.50 · 10�3 2.92 · 10�3

(1/144, 5/36, 17/24, 1/4)p1/6 4.13 · 10�5 1.52 · 10�3 2.95 · 10�3

(1/144, 5/36, 17/24, 1/12)p1/6 2.99 · 10�5 1.50 · 10�3 2.98 · 10�3

�1720

; 124720

; 474720

; 112

� � p7/60 1.12 · 10�4 9.78 · 10�4 2.17 · 10�3

�1720 ;

124720 ;

474720 ;

112

� � p1/84 1.14 · 10�5 9.40 · 10�5 8.92 · 10�5

164 T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166

varying values of the mesh ratio r for h = 0.1 and h = 0.02 respectively. The

errors in displacement function u(x, t) at midpoint of the interval [0, 1] are givenin Tables 2–4.

Example 6.2. We consider a homogeneous fourth-order parabolic equation

o2uot2

þ o4uox4

¼ 0; ð31Þ

with the initial conditions

uðx; 0Þ ¼ x12

ð2x2 � x3 � 1Þ; utðx; 0Þ ¼ 0; 0 6 x 6 1; ð32Þ

and the boundary conditions

uð0; tÞ ¼ uð1; tÞ ¼ o2uox2

ð0; tÞ ¼ o2uox2

ð1; tÞ ¼ 0; t P 0: ð33Þ

The exact solution is given as

uðx; tÞ ¼X1s¼0

ds sinð2sþ 1Þpx cosð2sþ 1Þ2p2t; ð34Þ

where ds = �8/[(2s + 1)5p5].

T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166 165

We solved example 6.2 by using scheme (12) together with Eqs. (i) (28),(29)

and (ii) (28),(30). By choosing suitable values of parameters p, q, s presented in

Table 5 with r = 1/4, we carried out the computations over 50 time steps with

h = 0.1 and k = .02 giving r = 2. We repeat the computations for 100 time steps

with r2 = 1/6. We also include results given by conditionally stable method

Table 5

Absolute error in displacement function u(x, t), h = 0.1, Example 6.2

Parameters

p, q, s, rr2 Time

steps

0.1 0.2 0.3 0.4 0.5

0, 0, 1, 1/4 4 50 3.21 · 10�4 5.77 · 10�4 7.24 · 10�4 7.89 · 10�4 8.10 · 10�4

Todd 4 50 3.19 · 10�4 6.19 · 10�4 8.81 · 10�4 1.07 · 10�3 1.15 · 10�3

1/6 100 3.81 · 10�4 3.33 · 10�4 7.74 · 10�4 7.81 · 10�4 7.66 · 10�4

Todd 1/6 100 2.61 · 10�4 4.43 · 10�4 5.47 · 10�4 6.08 · 10�4 6.33 · 10�4

0, 0, 1, 1/2 4 50 1.00 · 10�5 5.00 · 10�5 1.73 · 10�4 3.33 · 10�4 4.10 · 10�4

Crandall 4 50 4.32 · 10�4 8.34 · 10�4 1.18 · 10�3 1.42 · 10�3 1.52 · 10�3

1/6 100 3.52 · 10�4 6.30 · 10�4 7.77 · 10�4 7.72 · 10�4 7.38 · 10�4

Crandall 1/6 100 2.30 · 10�4 4.08 · 10�4 5.40 · 10�4 6.56 · 10�4 7.02 · 10�4

�1720

; 124720

; 474720

; 112

� �7/60 100 1.38 · 10�4 1.74 · 10�4 9.05 · 10�5 3.4 · 10�5 9.6 · 10�5

1/84 100 3.53 · 10�5 6.22 · 10�5 7.11 · 10�5 6.11 · 10�5 5.53 · 10�5

Table 6

Absolute error in displacement function u(x, t) at mid points of interval

r2 r h Time steps

10 20 30 40 50

For parameters ðp; q; sÞ ¼ �1720

; 124720

; 474720

� �1/84 1/12 0.02 1.2 · 10�12 3.5 · 10�11 7.7 · 10�11 2.2 · 10�10 1.2 · 10�9

1/32 9.0 · 10�11 2.4 · 10�11 4.0 · 10�9 3.1 · 10�8 6.4 · 10�8

1/16 2.8 · 10�8 6.7 · 10�8 7.0 · 10�7 3.2 · 10�6 4.1 · 10�6

7/60 1/12 0.02 2.1 · 10�10 2.4 · 10�11 1.6 · 10�8 1.2 · 10�8 3.0 · 10�8

1/32 1.3 · 10�8 1.1 · 10�7 2.3 · 10�7 2.2 · 10�7 2.7 · 10�8

1/16 2.7 · 10�7 6.3 · 10�7 5.0 · 10�6 1.2 · 10�5 6.6 · 10�6

For parameters ðp; q; sÞ ¼ 1144

; 536; 1724

� �1/6 1/12 0.02 2.5 · 10�10 6.2 · 10�9 2.4 · 10�8 5.6 · 10�8 2.5 · 10�9

1/32 2.6 · 10�8 9.3 · 10�8 3.8 · 10�7 7.4 · 10�7 1.3 · 10�6

1/16 2.3 · 10�6 4.2 · 10�6 1.7 · 10�5 4.8 · 10�5 4.7 · 10�5

4 1/4 0.02 4.0 · 10�8 4.6 · 10�7 4.9 · 10�7 3.3 · 10�7 5.4 · 10�9

1/32 2.1 · 10�6 2.7 · 10�6 8.9 · 10�6 7.2 · 10�6 2.3 · 10�6

1/16 3.3 · 10�5 1.1 · 10�4 8.1 · 10�5 1.1 · 10�4 3.9 · 10�4

166 T. Aziz et al. / Appl. Math. Comput. 167 (2005) 153–166

obtained by p, q, s = 1/144 (1,20,102) and high accuracy schemes obtained by

p, q, s = 1/720 (�1,124,474), r = 1/12 with r2 = 7/60 and r2 = 1/84. The results

are shown in Table 6.

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