A survey on parametric spline function approximation

Applied Mathematics and Computation 171 (2005) 983–1003

www.elsevier.com/locate/amc

A survey on parametric splinefunction approximation

Arshad Khan a, Islam Khan b,*, Tariq Aziz b

a Department of Mathematics, A.M.U., Aligarh 202 002, Indiab Department of Applied Mathematics, Faculty of Engineering and Technology, A.M.U.,

Aligarh 202 002, India

Abstract

This survey paper contains a large amount of material and indeed can serve as an

introduction to some of the ideas and methods for the solution of ordinary and partial

differential equations starting from Schoenberg�s work [Quart. Appl. Math. 4 (1946)

345–369]. The parametric spline function which depends on a parameter x > 0, is

reduces to the ordinary cubic or quintic spline for x = 0. A note on parametric spline

function approximation, which is special case of this work has been published in [Comp.

Math. Applics. 29 (1995) 67–73]. This article deals with the odd-order parametric spline

relations.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Cubic spline; Parametric cubic spline; Quintic spline; Parametric quintic spline;

Numerov�s method; Spline relations; Diagonally dominant

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.01.112

* Corresponding author.

E-mail addresses: [email protected] (A. Khan), [email protected] (I. Khan).

mailto:[email protected]

mailto:[email protected]

984 A. Khan et al. / Appl. Math. Comput. 171 (2005) 983–1003

1. Introduction

This paper is concerned with the development of non-polynomial spline

function approximation methods to obtain numerical solution of ordinary

and partial differential equations. The use of spline functions dates back at

least to the beginning of previous century. Piecewise linear functions had beenused in connection with the Peano�s existence proof for the solution to the

initial value problems of the ordinary differential equations, although these

functions were not called splines. Splines were first identified in the work of

Schoenberg, Sard and others. Usually a spline is a piecewise polynomial func-

tion defined in a region D, such that there exists a decomposition of D into sub-

regions in each of which the function is a polynomial of some degree m. Also

the function, as a rule, is continuous in D, together with its derivatives of order

upto (m � k) [32]. In other words spline function is a piecewise polynomial sat-isfying certain conditions of continuity of the function and its derivatives. The

applications of spline as approximating, interpolating and curve fitting func-

tions have been very successful [1], [18], [37],[34]). It is also interesting to note

that the cubic spline is a close mathematical approximation to the draughts-

man�s spline, which is a widely used manual curve-drawing tool. It has been

shown by Schoenberg [46] that a curve drawn by a mechanical spline to a first

order of approximation is a cubic spline function. Further, the solution of a

variety of problems of �best approximation� are the spline function approxima-tions. Later on, spline functions recieved a considerable amount of attention in

both theoretical and practical studies.

A number of authors have attempted polynomial and non-polynomial

spline approximation methods for the solution of differential equations; De

Boor [12–14], Ahlberg et al. [1], Loscalzo and Talbot [30,31], Bickley [7], Fyfe

[16,17], Albasiny and Hoskins [2], Sakai [43–45], Russell and Shampine [42],

Micula [33,34], Rubin and Khosla [41], Rubin and Graves [40], Daniel and

Swartz [11], Archer [3], Patricio [35,36], Tewarson [51,52], Usmani et al. [53–55], Jain and Aziz [22–24], Surla et al. [47–50], Iyengar and Jain [21], Chawla

and Subramanian [8–10], Irodotou- Ellina and Houstis [20], Rashidinia [39],

Fairweather and Meade [15] and others. Spline functions of maximum smooth-

ness were first considered in the numerical solution of initial value problems in

ordinary differential equations by Loscalzo and Talbot [30,31] and many inter-

esting connections with standard numerical integration techniques have been

established. For example, the trapezoidal rule and the Milne–Simpson predic-

tor–corrector method fall out as special cases of such spline approximations.The main reason why the above mentioned applications of spline functions

to the numerical integration of ordinary differential equations leads to unstable

methods is because the resulting numerical approximations are, in a certain

sense too smooth [56]). Loscalzo and Schoenberg [29] have shown that the

use of Hermite–Splines of lower order smoothness avoids completely the prob-

A. Khan et al. / Appl. Math. Comput. 171 (2005) 983–1003 985

lem of instability. The spline functions have been used by a number of authors

to solve both initial and boundary value problems of ordinary and partial dif-

ferential equations. The use of cubic splines for the solution of linear two point

boundary value problems was first suggested by Bickley [7]. His main idea was

to use the �condition of continuity� as a discretization equation for the linear

two point boundary value problems. Later, Fyfe [16] discussed the applicationof deferred corrections to the method suggested by Bickley by considering

again the case of (regular) linear boundary-value problems.

However, it is well known since then that the cubic spline method of Bickley

gives only O(h2) convergent approximations. But cubic spline itself is a fourth-

order process [37]. It was therefore natural to look for an alternative method

which would give fourth order approximations using cubic splines. We also

find that the applications of the spline functions to the solution of convec-

tion–diffusion problems has not been very encouraging. To be able to dealeffectively with such problems we introduce �spline functions� containing a

parameter x. These are �non-polynomial splines� defined through the solution

of a differential equation in each subinterval. The arbitrary constants being

chosen to satisfy certain smoothness conditions at the joints. These �splines� be-long to the class C2 and reduce into polynomial splines as parameter x ! 0.

The exact form of the spline depends upon the manner in which the parameter

is introduced. We have studied parametric spline functions: spline under com-

pression, spline under tension and adaptive spline. A number of spline relationshave been obtained for subsequent use.

The singular perturbation mathematical model plays an important role in

modelling fluid processes arising in applied mechanics. We have either the stiff

system of initial boundary value problems or the convection-diffusion problems.

It has been realized that when conventional methods are applied to obtain

numerical solution, the step size must be limited to extremely small values.

Any attempt to use a larger step size results in the calculations becoming unsta-

ble and producing completely erroneous results. In recent years, considerableattention has been devoted to the formulation and implementation of, in es-

sence, modified spline methods for the solution of certain classes of elliptic

boundary value problems on rectangles, see for example Houstis et al. [19]. It

is interesting to note that this approach was adopted by Irodotou-Ellina and

Houstis [20], in their quintic spline collocation methods for general linear fourth

order two point boundary value problems. Another quintic spline method

requiring a uniformmesh for a non-linear fourth order boundary value problem

is due to Chawla and Subramanian [10]. This method is based on Bickley�s idea[7] of using the continuity condition to construct a cubic spline approximation,

but here it is used only after some other method (e.g., a finite difference method)

has been used to obtain accurate nodal values. Fairweather andMeade [15] pro-

vide a comprehensive survey of both orthogonal and modified spline collocation

methods for solving ordinary and partial differential equations.


2. Spline functions

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

D:a = x0 < x1 < x2 < � � � < xN�1 < xN = b where hj = xj � xj�1, for j = 1(1)N.

Assume we are given the values fujgNj¼0 of a function u(x), with [a,b] as its do-

main of definition. A spline function of degree m with nodes at the points xj,j = 0,1,2, . . . ,N is a function SD(x) with the following properties:

(a) In each sub interval [xj,xj+1], j = 0,1, . . . ,N � 1, SD(x) is a polynomial of

degree m.

(b) SD(x) and its first (m � 1) derivatives are continuous on [a,b]. If the func-

tion SD(x) has only (m � k) continuous derivatives, then k is defined as

the deficiency of the polynomial spline and is usually denoted by SD

(m,k), see [41]. The cubic spline is a piecewise cubic polynomial of defi-ciency one, e.g. SD (3,1). The cubic spline procedure can be described

as follows.

Consider a function u(x) such that at the mesh points xi,u(xi) = ui. A cubic

polynomial is specified on the interval [xj�1,xj]. The four constants are related

to the function values uj�1, uj as well as certain spline derivatives mj�1, mj or

Mj�1, Mj. The quantities mj, Mj are the spline derivative approximations to

the function derivatives u 0(xj), u00(xj) respectively. Similarly considered on the

interval [xj,xj+1]. The continuity of the derivatives is then specified at xj. The

procedure results in equations for mj, Mj, j = 1,2, . . . ,N � 1. Boundary condi-

tions are required at j = 0 and j = N. The system is closed by the governing dif-

ferential equation for u(xj), where the derivatives are replaced by their spline

approximations mj, Mj.

In this paper, we first give some definitions and basic results on cubic and

quintic spline functions. The definition of the cubic and quintic spline functions

is extended to piecewise non-polynomial functions depending on a parameterx. For x! 0 these (non-polynomial) functions reduce to ordinary cubic or

quintic splines. Depending on the choice of parameter, the spline function is

known as cubic spline in compression, cubic spline in tension or adaptive cubic

spline. Similarly three of the splines derived from quintic spline are termed

�parametric quintic spline-I�, �parametric quintic spline-II� and �adaptive quinticspline�.

3. Cubic spline functions

Definition

A cubic spline function SD(x), of class C2 [a,b], interpolating to a function

u(x) defined on [a,b] is such that


(a) In each interval [xj�1,xj], SD(x) is a polynomial of degree at most three.

(b) The first and second derivatives of SD(x) are continuous on [a,b].

Thus we may write in [xj�1,xj],

S00DðxÞ ¼ Mj�1

xj � xhj

þMjx� xj�1

hj¼ �zMj�1 þ zMj; ð1Þ

where S00DðxjÞ ¼ Mj and hj = xj�xj�1, z = (x � xj�1)/hj, �z ¼ 1� z.

Integrating (1) twice, and setting x = xj and x = xj�1, the constants of inte-gration are determined by the conditions SD(xj�1) = uj�1, SD(xj) = uj. Finally,

we obtain

SDðxÞ ¼ zuj þ �zuj�1 þ h2j ½q3ðzÞMj þ q3ð�zÞMj�1�=3!; ð2Þ

where q3(z) = z3 � z, q3(0) = q3(±1) = 0, q3 is an odd function of z.

The function SD(x) in the interval [xj,xj+1] is obtained with j + 1 replacing j

in (2) so that

SDðxÞ ¼ zujþ1 þ �zuj þ h2jþ1½q3ðzÞMjþ1 þ q3ð�zÞMj�=3!; ð3Þ

where z = (x � xj)/hj+1.

Cubic spline relations

For simplicity we assume that the subintervals are of equal length so that

hj = hj+1 = h, for j = 1(1)N. The continuity of the first derivative of SD(x) at

x = xj requires S0Dðxj�Þ ¼ S0

DðxjþÞ. Since,

S0Dðxj�Þ ¼ 1

hðuj � uj�1Þ þ

h6ð2Mj þMj�1Þ; j ¼ 1ð1ÞN ;

S0DðxjþÞ ¼ 1

hðujþ1 � ujÞ �

h6ð2Mj þMjþ1Þ; j ¼ 0ð1ÞN � 1

ð4Þ

the following spline relation may be obtained:

ðiÞ Mjþ1 þ 4Mj þMj�1 ¼6

h2ðujþ1 � 2uj þ uj�1Þ; j ¼ 1ð1ÞN � 1:

Similarly we obtain

ðiiÞ mjþ1 þ 4mj þ mj�1 ¼3

hðujþ1 � uj�1Þ;

ð5Þ

where mj ¼ S0DðxjÞ.

The relations (5) are called the continuity or consistency relations of thecubic spline. The following relations may also be easily obtained


ðiÞ mj ¼ � h6ðMjþ1 þ 2MjÞ þ

ðujþ1 � ujÞh

;

ðiiÞ mjþ1 ¼h6ðMj þ 2Mjþ1Þ þ

ðujþ1 � ujÞh

;

ðiiiÞ mjþ1 � mj ¼h2ðMjþ1 þMjÞ:

ð6Þ

Eq. (5) lead to a system of N equations for the N + 2 unknowns Mj or mj

respectively. The additional two equations are obtained from boundary condi-

tions on M or m. The resulting tridiagonal system for Mj or mj is diagonally

dominant and may be solved by an efficient algorithm, [1]. Therefore, giventhe values uj the Eq. (5) with appropriate boundary conditions form a closed

system for Mj or mj and with (2) or (3) the values SD(x) can be found at all

intermediate locations.

3.1. Parametric cubic spline functions

Definition

A function SD (x,s) of class C2 [a,b] which interpolates u(x) at the meshpoints {xj}, depends on a parameter s, and reduces to cubic spline SD(x), in

[a,b] as s ! 0, is termed a parametric cubic spline function. Since the para-

meter s can occur in SD(x,s) in many ways, such a spline in not unique. The

three parametric cubic splines derived from cubic spline by introducing the

parameter in three different ways are termed as �cubic spline in compres-

sion�,�cubic spline in tension� and �adaptive cubic spline�.

3.1.1. Cubic spline in compression

If SD(x,s) is a parametric cubic spline satisfying the differential equation

S00Dðx; sÞ þ sSDðx; sÞ ¼ ½S00

Dðxj�1; sÞ þ sSDðxj�1; sÞ�ðxj � xÞ

hj

þ ½S00ðxj; sÞ þ sSDðxj; sÞ�ðx� xj�1Þ

hj; ð7Þ

where x 2 [xj�1,xj],SD(xj,s) = uj,hj = xj � xj�1 and s > 0 then it is termed cubic

spline in compression (see [4,5,25,27,38]).

Solving the differential Eq. (7) and using interpolatory conditions at xjand xj�1 to determine the constants of integration, we get after writing

x ¼ hjffiffiffis

p,

SDðx; sÞ ¼ zuj þ �zuj�1 þ h2j ½q1ðzÞMj þ q1ð�zÞMj�1�=x2; ð8Þ

where z = (x � xj�1)/hj, �z ¼ 1� z, q1ðzÞ ¼ z� sinxzsinx , q1(0) = q1(±1) = 0, q1(z) is

an odd function of z.


Similarly in interval [xj,xj+1] we obtain with j + 1 replacing j in (8)

SDðxÞ ¼ zujþ1 þ �zuj þ h2jþ1½q1ðzÞMjþ1 þ q1ð�zÞMj�=x2: ð9Þ

Spline relations

For uniform mesh i.e. h = hj = hj+1, spline relations corresponding to (5) (i),

(ii) may be obtained as:

ðiÞ aMjþ1 þ 2bMj þ aMj�1 ¼1

h2ðujþ1 � 2uj þ uj�1Þ;

ðiiÞ amjþ1 þ 2bmj þ amj�1 ¼ðaþ bÞ


ð10Þ

where

mj ¼ S0Dðxj; sÞ; Mj ¼ S00

Dðxj; sÞ; a ¼ 1

x2ðx cos ecx� 1Þ;

b ¼ 1

x2ð1� x cotxÞ:

ð11Þ

The consistency relation for (10)(i) leads to the equation 2a + 2b = 1, whichmay also be expressed as x=2 ¼ tanðx=2Þ. This equation has a zero root and

an infinite no. of non-zero roots, the smallest positive being

x = 8.986818916. . . For x2¼ tan x

26¼ 0, a = b = 1/4. In this case spline relations

(10) reduce to

ðiÞ h2

4ðMjþ1 þ 2Mj þMj�1Þ ¼ d2uj;

ðiiÞ h2ðmjþ1 þ 2mj þ mj�1 ¼ ðujþ1 � uj�1Þ:

ð12Þ

For x ! 0, (a,b) ! (1/6,1/3) and q1(z)/x2 ! q0(z)/6 so that the Eq. (10) re-

duces to the corresponding cubic spline relation (5).

For (a,b) ! (1/12,5/12) the Eq. (10) reduces to the well-known fourth order

Numerov�s method and quartic spline method of Usmani [54].

3.1.2. Cubic spline in tension

If the function SD(x,s) is a parametric cubic spline satisfying the following

differential equation in sub interval [xj�1,xj],

S00Dðx; sÞ � sSDðx; sÞ ¼ ½S00

Dðxj�1; sÞ � sSDðxj�1; sÞ�ðxj � xÞ

hj

þ ½S00ðxj; sÞ � sSDðxj; sÞ�ðx� xj�1Þ

hj; ð13Þ


where s> 0 is a tension factor, SD(xj) = uj, S0DðxjÞ ¼ mj, S00

DðxjÞ ¼ Mj, then

SD(x,s) is termed as �cubic spline in tension� (see [4,40]).

Solving (13) and using interpolatory conditions at xj and xj�1, we obtain

SDðx; sÞ ¼ zuj þ �zuj�1 þ h2j ½q2ðzÞMj þ q2ð�zÞMj�1�=x2; ð14Þ

where

z ¼ ðx� xj�1Þ=hj; �z ¼ 1� z; x ¼ffiffiffis

phj; q2ðzÞ ¼

sinhxzsinhx

� z;

q2ð0Þ ¼ q2ð�1Þ ¼ 0 and hj ¼ xj � xj�1:

Similarly in interval [xj,xj+1] we obtain

SDðx; sÞ ¼ zujþ1 þ �zuj þ h2jþ1½q2ðzÞMjþ1 þ q2ð�zÞMj�=x2 ð15Þ

where z = (x � xj)/hj, �z ¼ 1� z;x ¼ffiffiffis

phjþ1, hj+1 = xj+1 � xj.

Spline relations

For uniform mesh i.e., h = hj = hj+1, the following spline relations are

obtained:

ðiÞ aMjþ1 þ 2bMj þ aMj�1 ¼1

h2d2uj;

ðiiÞ amjþ1 þ 2bmj þ amj�1 ¼ðaþ bÞ


ð16Þ

where a ¼ 1x2 ð1� x cos echxÞ, b ¼ 1

x2 ðx cothx� 1Þ, x ¼ hffiffiffis

pwhen x! 0,

(a,b) ! (1/6,1/3), and q2(z)/x2 ! q0(z)/6, so that Eqs. (14)–(16) reduces to

the cubic spline relations.

3.1.3. Adaptive cubic spline

If the function SD(x,x) is a parametric cubic spline satisfying the following

differential equation

aS00Dðx;xÞ � bS 0

Dðx;xÞ ¼x� xj�1

hjðaMj � bmjÞ þ

xj � xhj

ðaMj�1 � bmj�1Þ;

ð17Þ

where xj�1 6 x 6 xj, a and b are constants,

S0Dðxj;xÞ ¼ mj; S

00Dðxj;xÞ ¼ Mj; hj ¼ xj � xj�1; and x > 0; then SDðx;xÞ

is termed as �adaptive cubic spline�.Solving (17) and using the interpolatory constants SD(xj�1,x) = uj�1,

SD(xj,x) = uj, we have


SDðx;xÞ ¼ Aj þ Bjexz �h2jx3

p2ðxzÞðMj �xhjmjÞ

� �

þh2jx3

p2ð�x�zÞ Mj�1 �xhjmj�1

� �� ; ð18Þ

where

z ¼ x� xj�1

hj; x ¼ bhj

a; �z ¼ 1� z; p2ðzÞ ¼ 1þ zþ 1

2z2;

Ajðex � 1Þ ¼ �uj þ uj�1ex �h2jx3

ðp2ðxÞ � exÞ Mj �xhjmj

� ��

�h2jx3

ðp2ð�xÞex � 1Þ Mj�1 �xhjmj�1

� �� ;

Bjðex � 1Þ ¼ uj � uj�1

þh2jx2

x2þ 1

� �Mj �

xhjmj

� �þ x

2� 1

� �Mj�1 �

xhjmj�1

� �� :

The function SD(x,x) on the interval [xj,xj+1] is obtained with j + 1 replacing j

in Eq. (18).

Spline relations

For uniform mesh i.e hj = hj+1 = h, the condition of continuity of first or sec-

ond derivative of SD(x,x) at xj yields the following equation:

Mjþ1 �xhmjþ1

� �½e�xp2ðxÞ � 1� þ Mj �

xhmj

� �½e�xfðp2ð�xÞex � 1Þ

� ðp2ðxÞ � exÞg� � Mj�1 �xhmj�1

� �e�x½p2ð�xÞex � 1�

¼ �x3

h2½e�xujþ1 � ð1þ e�xÞuj þ uj�1�: ð19Þ

It may be noted that the continuity of first derivative of SD(x,x) implies that of

second derivative and vice-versa. We obtain the following relations:

ðiÞ mj�1 ¼ �hðA1Mj�1 þ A2MjÞ þ ðuj � uj�1Þ=h;ðiiÞ mj ¼ hðA3Mj�1 þ A4MjÞ þ ðuj � uj�1Þ=h;

ðiiiÞ ah2x

Mj�1 ¼ �ðA4mj�1 þ A2mjÞ þ B1ðuj � uj�1Þ=h;

ðivÞ ah2x

Mj ¼ ðA3mj�1 þ A1mjÞ þ B2ðuj � uj�1Þ=h;

ð20Þ


where

A1 ¼1

4ð1þ aÞ þ a

2x; A2 ¼

1

4ð1� aÞ � a

2x;

A3 ¼1

4ð1þ aÞ � a

2x; A4 ¼

1

4ð1� aÞ þ a

2x;

B1 ¼1

2ð1� aÞ; B2 ¼ � 1

2ð1þ aÞ; a ¼ coth

x2� 2

x:

We also obtain

h2½A2Mjþ1 þ ðA1 þ A4ÞMj þ A3Mj�1� ¼ d2uj; ð21Þ

B1ujþ1 � ðB1 þ B2Þuj þ B2uj�1 ¼ h½A2mjþ1 þ ðA1 þ A4Þmj þ A3mj�1�: ð22Þ

For x! 0 (i.e. bh/a ! 0), then we have a = 0, a/x = 1/6, A1 = A4 = 1/3,

A2 = A3 = 1/6, B1 = �B2 = 1/2, and the spline function given by (18) reduces

into cubic spline.

4. Quintic spline functions

DefinitionA quintic spline function SD(x), interpolating to a function u(x) defined on

[a,b] is such that

(i) In each subinterval [xj�1,xj], SD(x) is a polynomial of degree at most five.

(ii) The first, second, third and fourth derivatives of SD(x) are continuous on

[a,b].

Thus, we may write in [xj�1,xj],

Sð4ÞD ðxÞ ¼ xj � x

hF j�1 þ

x� xj�1

hF j ¼ �zF j�1 þ zF j; ð23Þ

where F j ¼ Sð4ÞD ðxjÞ.

Integrating (23) four times with respect to x and using the conditions

SDðxiÞ ¼ ui; S00DðxiÞ ¼ Mi; i ¼ j; j� 1;

we obtain

SDðxÞ ¼ zuj þ �zuj�1 þh2

3!½q3ðzÞMj þ q3ð�zÞMj�1� þ

h4

5!½q5ðzÞF j þ q5ð�zÞF j�1�;

ð24Þ


where

z ¼ x� xj�1

h; �z ¼ xj � x

hor �z ¼ 1� z; q3ðzÞ ¼ z3 � z;

q5ðzÞ ¼ z5 � ð10=3Þz3 þ ð7=3Þz; q3ð0Þ ¼ q5ð0Þ ¼ 0;

q3ð�1Þ ¼ q5ð�1Þ ¼ 0; q5ð�ffiffiffi7

p=3Þ ¼ 0:

The function SD(x) in interval [xj,xj+1] is obtained with j + 1 replacing j in (24)

as

SDðxÞ ¼ uj�zþ ujþ1zþh2

3!½q3ð�zÞMj þ q3ðzÞMjþ1� þ

h4

5!½q5ð�zÞF j þ q5ðzÞF jþ1�;

ð25Þwhere z ¼ x�xj

h , �z ¼ 1� z.

Spline relations

The continuity of the first, second, third and fourth derivatives at x = xjrequires

S0Dðxj�Þ ¼ S0

DðxjþÞ; S00Dðxj�Þ ¼ S00

DðxjþÞ;Sð3ÞD ðxj�Þ ¼ Sð3Þ

D ðxjþÞ and Sð4ÞD ðxj�Þ ¼ Sð4Þ

D ðxjþÞ:

Differentiating successively Eqs. (24) and (25) and putting x = xj, we obtain thefollowing one sided derivatives at x = xj.

ð1Þ S0Dðxj�Þ ¼ 1

hðuj � uj�1Þ þ

h6ð2Mj þMj�1Þ �

h3

360ð8F j þ 7F j�1Þ;

j ¼ 1ð1ÞN ;

ð2Þ S0DðxjþÞ ¼ 1

hðujþ1 � ujÞ �

h6ð2Mj þMjþ1Þ þ

h3

360ð8F j þ 7F jþ1Þ;

j ¼ 0ð1ÞN � 1;

ð3Þ S00Dðxj�Þ ¼ Mj ¼ S00

DðxjþÞ;

ð4Þ Sð3ÞD ðxj�Þ ¼ 1

hðMj �Mj�1Þ þ

h6ðF j�1 þ 2F jÞ;

ð5Þ Sð3ÞD ðxjþÞ ¼ 1

hðMjþ1 �MjÞ �

h6ðF jþ1 þ 2F jÞ;

ð6Þ Sð4ÞD ðxj�Þ ¼ F j ¼ Sð4Þ

D ðxjþÞ: ð26Þ

The continuity of first derivative implies

Mjþ1 þ 4Mj þMj�1 ¼6

h2ðujþ1 � 2uj þ uj�1Þ þ

h2

60ð7F jþ1 þ 16F j þ 7F j�1Þ;

j ¼ 1ð1ÞN � 1 ð27Þ


and the continuity of the third derivative implies,

Mjþ1 � 2Mj þMj�1 ¼h2

6ðF jþ1 þ 4F j þ F j�1Þ; j ¼ 1ð1ÞN � 1 ð28Þ

subtracting (28) from (27) and dividing by 6 we obtain

Mj ¼1

h2ðujþ1 � 2uj þ uj�1Þ �

h2

120ðF jþ1 þ 8F j þ F j�1Þ: ð29Þ

Elimination ofMj�s between (28) and (29) leads to the following useful relation:

F jþ2 þ 26F jþ1 þ 66F j þ 26F j�1 þ F j�2

¼ 120

h4ðujþ2 � 4ujþ1 þ 6uj � 4uj�1 þ uj�2Þ; j ¼ 2ð1ÞN � 2: ð30Þ

The following additional spline relations may be obtained

ðiÞ mj ¼1

6hð2ujþ1 þ 3uj � 6uj�1 þ uj�2Þ �

h3

720ð2F jþ1 þ 33F j þ 24F j�1 þ F j�2Þ;

ðiiÞ T j ¼1

h3ðujþ1 � 3uj þ 3uj�1 � uj�2Þ �

h120

ðF jþ1 � 33F j � 27F j�1 � F j�2Þ:

ð31ÞWe define K by KWj � Wj+2 + 26Wj+1 + 66Wj + 26Wj�1 + Wj�2 for any func-

tion W evaluated at the mesh points. Then we have the following relations con-

necting u and its derivatives (see [6, 17, 28, 55]).

ðiÞ Kmj ¼5

hðujþ2 þ 10ujþ1 � 10uj�1 � uj�2Þ;

ðiiÞ KMj ¼20

h2ðujþ2 þ 2ujþ1 � 6uj þ 2uj�1 þ uj�2Þ;

ðiiiÞ KT j ¼60

h3ðujþ2 � 2ujþ1 þ 2uj�1 � uj�2Þ;

ðivÞ KF j ¼120

h4ðujþ2 � 4ujþ1 þ 6uj � 4uj�1 þ uj�2Þ;

ð32Þ

where mj ¼ S0DðxjÞ, Mj ¼ S00

DðxjÞ, T j ¼ Sð3ÞD ðxjÞ and F j ¼ Sð4Þ

D ðxjÞ.

4.1. Parametric quintic spline functions

DefinitionA function SD(x,s) of class C4 [a,b], which interpolates u(x) at the mesh

points xj, j = 1(1)N, depends on a parameter s, reduces to ordinary quintic

spline SD(x) in [a,b] as s ! 0 is termed a parametric quintic spline function.

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

unique.The three parametric quintic splines derived from quintic spline by


introducing the parameter in three different ways are termed as �parametric

quintic spline-I�, �parametric quintic spline-II� and �adaptive quintic spline�.

4.1.1. Parametric quintic spline-I

If SD(x,s) = SD(x) is a parametric quintic spline satisfying the following dif-

ferential equation in the interval [xj�1,xj],

Sð4ÞD ðxÞ þ s2Sð2Þ

D ðxÞ ¼ ðF j þ s2MjÞx� xj�1

hþ ðF j�1 þ s2Mj�1Þ

xj � xh

¼ Qjzþ Qj�1�z; ð33Þ

where Qj = Fj + s2Mj, S00DðxjÞ ¼ Mj, S

ð4ÞD ðxjÞ ¼ F j and s > 0 then it is termed

�parametric quintic spline-I� (see [25,26,39]).

Solving the differential Eq. (33) and determining the four constants of inte-

gration from the interpolatory conditions at xj and xj�1, we obtain


3!½q3ðzÞMj þ q3ð�zÞMj�1�

þ hx

� �4 x2

3!q3ðzÞ � q1ðzÞ

� �F j þ

hx

� �4 x2

3!q3ð�zÞ � q1ð�zÞ

� �F j�1;

ð34Þ

where q3(z) and q1(z) are defined in Sections (3) and (3.1.1) respectively.In the same manner in [xj,xj+1] we obtain

SDðxÞ ¼ �zuj þ zujþ1 þh2

3!½q3ð�zÞMj þ q3ðzÞMjþ1�

þ hx

� �4 x2

3!q3ðzÞ � q1ðzÞ

� �F jþ1 þ

hx

� �4 x2

3!q3ð�zÞ � q1ð�zÞ

� �F j:

ð35Þ

Spline relations

Continuity of the first and second derivatives implies that


h2ðujþ1 � 2uj þ uj�1Þ

� 6h2ða1F jþ1 þ 2b1F j þ a1F j�1Þ;

ðiiÞ Mjþ1 � 2Mj þMj�1 ¼ h2ðaF jþ1 þ 2bF j þ aF j�1Þ:

ð36Þ

The consistency relation for (36)(ii) leads to the equation x2¼ tan x

2(see Section

3.1.1).


From (36) we obtain

Mj ¼1


� h2 a1 þa6

� �F jþ1 þ 2 b1 þ

b6

� �F j þ a1 þ

a6

� �F j�1

� �: ð37Þ

Substituting for Mj+1, Mj and Mj�1 from (37) into (36) we arrive at the follow-

ing useful relation:

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

h4d4uj; ð38Þ

we can also write (38) as KF j ¼ 1h4d4uj, where the operator K is defined by

Kwj ¼ pðwjþ2 þ wj�2Þ þ qðwjþ1 þ wj�1Þ þ swj:

Multiplying (36)(i) by a and (36)(ii) by 6a1 and adding, we get

F j ¼1

12h2ða1b� b1aÞ

�ðaþ 6a1ÞMjþ1 þ 4ða� 3a1ÞMj þ ðaþ 6a1ÞMj�1

� 6a


�

using (36)(ii) we obtain

pMjþ2 þ qMjþ1 þ sMj þ qMj�1 þ pMj�2

¼ 1

h2aðujþ2 þ uj�2Þ þ 2ðb� aÞðujþ1 þ uj�1Þ þ ð2a� 4bÞuj�

; ð39Þ

where p ¼ a1 þ a6, q ¼ 2 1

6ð2aþ bÞ � ða1 � b1Þ

� , s ¼ 2½1

6ðaþ 4bÞ þ ða1 � 2b1Þ�,

a ¼ 1

x2ðx cos ecx� 1Þ; b ¼ 1

x2ð1� x cotxÞ; a1 ¼

1

x2

1

6� a

� �;

b1 ¼1

x2

1

3� b

� �:

Some other spline relations are:

ð1Þ mj ¼ � 1

6hðujþ2 � 6ujþ1 þ 3uj þ 2uj�1Þ þ

h3

6½pF jþ2 þ ðp0 þ 2pÞF jþ1

þ ðp þ 2p0ÞF j þ 2pF j�1�;

ð2Þ T j ¼1

h3ðujþ2 � 3ujþ1 þ 3uj � uj�1Þ � h½pF jþ2 þ ðp0 � p þ aÞF jþ1

þ ðp � p0 þ bÞF j � pF j�1�:ð40Þ

where S0DðxjÞ ¼ mj, S

ð3ÞD ðxjÞ ¼ T j and p0 ¼ 2 b1 þ b

6

�.


As s ! 0 that is x! 0, then ða; b; a1; b1Þ ! 16; 13; �7360

; �8360

�and

ðp0; p; q; sÞ ! 1120

ð8; 1; 26; 66Þ the spline defined by (34) reduces into quintic

spline (24) and the above spline relations reduce to corresponding ordinary

quintic spline relations.

For ðp; q; sÞ ! 1360

(1,56,246), the relation (38) reduces to the sextic spline

method.

4.1.2. Parametric quintic spline-II

If the function SD(x,s) = SD(x) is a parametric quintic spline satisfying the

following differential equation in the interval [xj�1,xj],

Sð4ÞD ðxÞ � s2Sð2Þ

D ðxÞ ¼ ðF j � s2MjÞx� xj�1

hþ ðF j�1 � s2Mj�1Þ

xj � xh

; ð41Þ

where s > 0, Qj = Fj � s2Mj, then SD(x) is termed as �parametric quintic spline-II�. Solving and using interpolatory conditions for finding constants of integra-

tion we obtain


3!½q3ðzÞMj þ q3ð�zÞMj�1�

þ hx

� �4

q2ðzÞ �x2

3!q3ðzÞ

� �F j þ

hx

� �4

q2ð�zÞ �x2

3!q3ð�zÞ

� �F j�1;

ð42Þ

where x = sh, q3 (z) and q2 (z) are defined in Sections (3) and (3.1.2)respectively.

Spline relations

Continuity of the first and third derivatives yields the following spline

relations



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

ð43Þ

Using (43) we obtain

Mj ¼1

h2ðujþ1 � 2uj þ uj�1Þ � h2 pF jþ1 þ p0F j þ pF j�1

� : ð44Þ

Substituting for Mj+1, Mj and Mj�1 from (44) into (43) we arrive at the follow-

ing useful relations


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

h4d4uj: ð45Þ

Multiplying (43)(i) by a and (43)(ii) by 6a1 and adding, we get

F j ¼1

12h2ða1b� b1aÞ

�ðaþ 6a1ÞMjþ1 þ 4ða� 3a1ÞMj þ ðaþ 6a1ÞMj�1

� 6a


�

using (43)(ii) we obtain

pMjþ2 þ qMjþ1 þ sMj þ qMj�1 þ pMj�2

¼ 1

h2aðujþ2 þ uj�2Þ þ 2ðb� aÞðujþ1 þ uj�1Þ þ ð2a� 4bÞuj�

: ð46Þ

Some other spline relations are:

ð1Þ mj ¼ � 1

6hðujþ2 � 6ujþ1 þ 3uj þ 2uj�1Þ þ

h3

6½pF jþ2 þ ðp0 þ 2pÞF jþ1

þ ðp þ 2p0ÞF j þ 2pF j�1�;

ð2Þ T j ¼1

h3ðujþ2 � 3ujþ1 þ 3uj � uj�1Þ � h½pF jþ2 þ ðp0 � p þ aÞF jþ1

þ ðp � p0 þ bÞF j � pF j�1�;

ð47Þ

where

p ¼ a1 þa6; q ¼ 2

1

6ð2aþ bÞ � ða1 � b1Þ

� �;

s ¼ 21

6ðaþ 4bÞ þ ða1 � 2b1Þ

� �;

p0 ¼ 2 b1 þb6

� �; a ¼ 1

x2ð1� x cos echxÞ; b ¼ 1

x2ð�1þ x cothxÞ;

a1 ¼1

x2� 1

6þ a

� �; b1 ¼

1

x2� 1

3þ b

� �:

As s ! 0 that is x ! 0, then ða; b; a1; b1Þ ! 16; 13; �7360

; �8360

�and

ðp0; p; q; sÞ ! 1120

ð8; 1; 26; 66Þ the spline defined by (42) reduces into quintic

spline (24) and the above spline relations reduce to corresponding ordinaryquintic spline relations.


4.1.3. Adaptive quintic spline

If the function SD(x,x) is a parametric quintic spline satisfying the following

differential equation in the interval [xj�1,xj],

aSð4ÞD ðx;xÞ � bSð3Þ

D ðx;xÞ ¼ ðaF j � bT jÞx� xj�1

hþ ðaF j�1 � bT j�1Þ

xj � xh

;

ð48Þwhere x 2 [xj�1,xj], Sð3Þ

D ðxj;xÞ ¼ T j, Sð4ÞD ðxj;xÞ ¼ F j;x ¼ bh=a > 0, a, b are

constants, then Sð3ÞD ðx;xÞ is termed as �adaptive quintic spline�.

Solving (48) and using the interpolatory conditions we obtain

SDðx;xÞ ¼ ujzþ uj�1�zþ k1½/1ðzÞ þ /2ðzÞ�Mj � k1½/1ðzÞex þ /2ðzÞ�Mj�1

þ k2qj½k/1ðzÞðp2ðxÞ � exÞ þ kðp2ðxÞ � 1Þ/2ðzÞþ zp4ðxÞ � p4ðxzÞ þ �z� þ k2qj�1½k/1ðzÞðp2ð�xÞex � 1Þþ k2ðp2ð�xÞ � 1Þ/2ðzÞ � �zp4ð�xÞ þ p4ð�x�zÞ � z�; ð49Þ

where

z ¼ x� xj�1

h; �z ¼ 1� z; Qj ¼ F j �

baT j; x ¼ bh

a; Mj ¼ S00

Dðxj;xÞ;

k1 ¼ kh2=x2; k2 ¼ h4=x5; /1ðzÞ ¼ x2z�z=2; /2ðzÞ ¼ exz � zex � �z;

k ¼ ðex � 1Þ�1and pN ðtÞ ¼ 1þ t þ t2

2!þ��þ tN

N !:

The function SD(x,x) on the interval [xj,xj+1] is obtained with j + 1 replacing j

in (49).

Spline relations

The condition of continuity of first and third derivative of SD(x,x) at xjyields the following equations:

ðiÞ ðujþ1 � 2uj þ uj�1Þ ¼ k1Mjþ1 þ k2Mj þ k3Mj�1 þ k4qjþ1 þ k5qj þ k6qj�1;

ðiiÞ Mjþ1 � ð1þ exÞMj þ exMj�1 ¼ k2f½1� kðp2ðxÞ � 1Þ�qjþ1

þ ½kðp2ðxÞex � p2ðxÞÞ � 3�qjþ ½kðp2ð�xÞ � 1Þex þ 1�qj�1g=k1;

ð50Þ

where

k1 ¼ ðex � p2ðxÞk1Þ; k2 ¼ ½exðp2ðxÞ � 3Þ þ ð3� p2ð�xÞÞ�k1;k3 ¼ ½exp2ð�xÞ � 1�k1;


k4 ¼ �k2kx2

2ðp2ðxÞ � exÞ þ kðp2ðxÞ � 1Þð1þx� exÞ þ p4ðxÞ �x� 1

� �;

k5 ¼ k2 �2xþ x3

3þ kx2 1� x2

4

� �ð1þ exÞ

� �and

k6 ¼ k2 � kx2

2ðp2ð�xÞex � 1Þ þ kðp2ð�xÞ � 1Þ½ðx� 1Þex þ 1�

�

þp4ð�xÞ þ x� 1

�:

Since the condition for the continuity of the fourth and higher derivatives is

same as that of the third derivative it follows that derivatives of all order for

the adaptive quintic spline are continuous.The spline function approximation methods have the following advantages:

In this approach u(x) is approximated by a different polynomial in each sub-

interval, thus providing approximations not only for u(x) but also for the deri-

vative of u(x) at every point of the interval [a,b]. We recall that by a finite

difference method we approximate u(x) at a finite set of grid points only in

[a,b]. Derivative boundary conditions are imposed directly without incurring

large local discretization errors. Unlike finite difference or finite element tech-

niques, with a spline approximation there appears to be no particular advan-tage gained with the divergence form for the equation. Unlike finite element

procedure, there are no quadratures to evaluate.

5. Conclusion

It is well known fact that if any discretization technique is applied to a

parameter dependent problem, then the behaviour of the discretization de-pends on a parameter.The reader�s are encouraged therefore, to use their intu-

itions to attempt to design an appropriate even order parametric spline

relations.

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A survey on parametric spline function approximation

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