A class of methods based on non-polynomial spline functions for the solution of a special...

Applied Mathematics and Computation 174 (2006) 1169–1180

www.elsevier.com/locate/amc

A class of methods based onnon-polynomial spline functions for the

solution of a special fourth-orderboundary-value problems

with engineering applications

Siraj-ul-Islam a,*, Ikram A. Tirmizi a, Saadat Ashraf b

a GIK Institute of Engineering Sciences and Technology, Topi (NWFP), Pakistanb University of Engineering and Technology, Peshawar (NWFP), Pakistan

Abstract

We use a quintic non-polynomial spline functions to develop a numerical method forcomputing approximations to the solution of a system of fourth-order boundary-valueproblems associated with plate deflection theory. We show that the present family ofmethods gives better approximations and generalize all the existing finite differenceand spline functions based methods up to order six. Convergence of the methods isshown through standard convergence analysis. Numerical exampls are given to illustratethe applicability and efficiency of the new method.� 2005 Elsevier Inc. All rights reserved.

Keywords: Quintic non-polynomial splines; Finite-difference methods; Plate deflection theory;Boundary-value problems

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.amc.2005.06.006

* Corresponding author.E-mail addresses: [email protected] (Siraj-ul-Islam), [email protected] (I.A. Tirmizi),

[email protected] (S. Ashraf).

mailto:[email protected]



1170 Siraj-ul-Islam et al. / Appl. Math. Comput. 174 (2006) 1169–1180

1. Introduction

We consider smooth approximation to the problem of bending a rectangularclamped beam of length l resting on elastic foundation. The vertical deflectionw of the beam satisfies the system

½Lþ ðk=DÞ�w ¼ D�1qðxÞ; L � d4=dx4;

wð0Þ ¼ wðlÞ ¼ w0ð0Þ ¼ w0ðlÞ ¼ 0;ð1:1Þ

where D is the flexural rigidity of the beam, and k is the spring constant of theelastic foundation, and the load q(x) acts vertically downwards per unit lengthof the beam. The details of the mechanical interpretation are given in [9].Mathematically, the system (1.1) belongs to a general class of boundary prob-lems of the form

½Lþ f ðxÞ�yðxÞ ¼ gðxÞ; a < x < b; ð1:2ÞyðaÞ ¼ A1; yðbÞ ¼ A2; y 0ðaÞ ¼ B1; y 0ðbÞ ¼ B2; ð1:3Þ

where f(x) and g(x) are continuous on [a,b] and Ai, Bi (i = 1,2) are finite realarbitrary constants. The analytical solution of (1.2) for arbitrary choices off(x) and g(x) cannot be determined. Faced with this difficulty we resort tonumerical method to find its solution. Usmani [14] has formulated a simplecondition that guarantees the uniqueness of the solution of the problem (1.2)and (1.3). This sufficient condition is given by

infa6x6b

f ðxÞ > � r

ðb� aÞ4;

whereby r = 500.5639 . . . An O(h3/2) method based on finite-difference schemeshas been developed in [4]. Usmani [14] has established and analyzed high orderfinite difference methods. The performance of these methods was comparedwith that of modified shooting techniques employing a fourth-order Runge–Kutta method. Noor and Tirmizi [8] have developed some some finite differ-ence method based on Pade�s approximants for solution of fourth-orderdifferential equation. In [12], Usmani used quintic and sixtic spline functionto establish smooth approximation of y(x). This approach has the advantagethat it provides approximation not only for y(x) but also for y(i)(x),i = 1,2, . . . , 4, at every point of range of integration. In recent papers Siraj-ul-Islam and Tirmizi [10,11] developed non-polynomial spline based methodsfor the solution of system of third-order obstacle problems and which providebasis for our method. In the present paper, we apply non-polynomial splinefunctions that have a polynomial and trigonometric part to develop a newnumerical method for obtaining smooth approximations to the solution ofspecial fourth-order differential equations of the type (1.1) and (1.2). Thespline function we propose in this paper have the form T n ¼ spanf1; x;

Siraj-ul-Islam et al. / Appl. Math. Comput. 174 (2006) 1169–1180 1171

x2; x3; cos kx; sin kxg where k is the frequency of the trigonometric part of thespline function which can be real or pure imaginary and which will be usedto raise the accuracy of the method [2,5,10,15]. Thus in each subintervalxi 6 x 6 xi+1, we have

spanf1; x; x2; x3; sin jkjx; cos jkjxg;or spanf1; x; x2; x3; sinh jkjx; cosh jkjxg;or spanf1; x; x2; x3; x4; x5g; ðwhen k ¼ 0Þ.

As evident from Usmani�s [12] polynomial quintic and sextic spline functionproduces only suboptimal second and fourth-order accurate method whereasnon-polynomial spline functions produces optimal method of order two forarbitrary a and b such that c = 1 � 2a � 2b and of order four and six for spe-cific values of a and b [4,9,10]. This improvement is because of introduction ofparameter k in the trigonometric part of Tn.

The main idea is to use the condition of continuity to get recurrence relationfor (1.1). The advantage of our method is higher accuracy with the samecomputational effort. In comparison with the finite difference methods[1,2,4,7,8,13–15] spline solution has its own advantages. For example, oncethe solution has been computed, the information required for spline interpola-tion between mesh points is available. This is particularly important when thesolution of the boundary value problem is required at different locations in theinterval [a, b]. This approach has the added advantage that it not only providescontinuous approximations to y(x), but also for y(i)(x), i = 1,2, . . . , 4, at everypoint of the range of integration. Also, the C1-differentiability of the trigono-metric part of non-polynomial splines compensates for the loss of smoothnessinherited by polynomial splines.

2. Numerical method

In order to develop the numerical method for approximating solution of ofdifferential Eqs. (1.2) and (1.3), we define a grid of N + 1 equally spaced pointsxi = a + ih, i = 0,1, . . . ,N, whereby h ¼ b�a

Nþ1. For each segment, the polynomial

Pi(x) has the form

P iðxÞ ¼ ai sin kðx� xiÞ þ bi cos kðx� xiÞ þ ciðx� xiÞ3 þ diðx� xiÞ2

þ eiðx� xiÞ þ fi; i ¼ 0; 1; . . . ;N ; ð2:1Þ

where ai, bi, ci, di, and ei are constants and k is free parameter. The functionPi(x), which interpolates y(x) at the mesh points xi depends on k and reducesto quintic spline in [a, b] as k ! 0.

Let yi be an approximation to y(xi), obtained by the segment Pi(x) of themixed splines function passing through the points (xi,yi) and (xi+1,yi+1). To


obtain the necessary conditions for the coefficients introduced in (2.1), we donot only require that Pi(x) satisfies (1.1) at xi and xi+1 and that the boundaryconditions (1.2) are fulfilled, but also the continuity of first, second, and thirdderivatives at the common nodes (xi, yi).

To derive expression for the coefficients of (2.2) in terms of yi, yi+1, Di, Di+1,Si and Si+1, we first define

P iðxiÞ ¼ yi; P iðxiþ1Þ ¼ yiþ1; P 0iðxiÞ ¼ Di;

P 0iðxiþ1Þ ¼ Diþ1; P ð4Þ

i ðxiÞ ¼ Si; P ð4Þi ðxiþ1Þ ¼ Siþ1.

ð2:2Þ

From algebraic manipulation we get the following expressions:

ai ¼Siþ1 � Si cosðhÞ

k4 sinðhÞ; bi ¼ h4

Si

h4;

ci ¼ � 2ðyiþ1 � yiÞh3

� ðSiþ1 � SiÞð1þ cosðhÞÞh2k3 sinðhÞ

þ ðDiþ1 þ DiÞh

þ 2ðSiþ1 � SiÞh3k4

;

di ¼ � 2Di þ Diþ1

hþ 3ðyiþ1 � yiÞ

h2� 3ðSiþ1 � SiÞ

h2k4

þ ½Siþ1ð2þ cosðhÞÞ � Sið1þ 2 cosðhÞÞ�hk3 sinðhÞ

;

ei ¼ Di �ðSiþ1 � Si cosðhÞÞ

k3 sinðhÞ;

fi ¼ yi �ðSiþ1 � Si cosðhÞÞ

k3 sinðhÞ;

ð2:3Þwhereby h = kh and i = 0,1,. . .,N � 1.

Using the continuity condition of the second and third derivatives at (xi,yi),that is P ðnÞ

i�1ðxiÞ ¼ P ðnÞi ðxiÞ where n = 2 and 3, we get the following consistency

relations for i = 1,2, . . . ,N:

Di�1 þ 4Di þ Di�1 ¼3

hðyiþ1 � yi�1Þ �

3

k4hðSi�1 þ SiÞ

þ 1

k3 sinðhÞðSi�1 þ Siþ1Þð2þ cosðhÞÞ ð2:4Þ

and

Diþ1 � Di�1 ¼2

hðyiþ1 � 2yi þ yi�1Þ þ

h2

6k sinðhÞ ðSi�1 � 2Si cosðhÞ þ Siþ1Þ

� 2

hk4ðSi�1 � 2Si þ Siþ1Þ

þ 1

k3 sinðhÞðSi�1 � 2Si þ Siþ1Þð1þ cosðhÞÞ. ð2:5Þ


Using Eqs. (2.4) and (2.5), we get following scheme:

yi�2 þ yiþ2 � h4ðSi�2 þ Siþ2Þ�1

h3 sin hþ 1

6h sin hþ 1

h4

� �� 4yi�1 � 4yiþ1 � h4ðSi�1 þ Siþ1Þ

2ð1þ cos hÞh3 sin h

� cos h� 2

3h sin h� 4

h4

� �þ 6yi � h4Si � 2ð1þ 2 cos hÞ

h3 sin h� 4 cos h� 1

3h sin hþ 6

h4

� �¼ 0; ð2:6Þ

where Si = f(xi,yi), i = 0,1, . . . ,N + 1.For simplicity we rewrite Eq. (2.6)

yi�2 � 4yi�1 þ 6yi � 4yiþ1 þ yiþ2 ¼ h4½aSi�2 þ bSi�1 þ Sicþ bSiþ1 þ aSiþ2�;ð2:7Þ

where

a ¼ �1

h3 sin hþ 1

6h sin hþ 1

h4

� �; b ¼ 2ð1þ cos hÞ

h3 sin h� cos h� 2

3h sin h� 4

h4

� �;

c ¼ � 2ð1þ 2 cos hÞh3 sin h

� 4 cos h� 1

3h sin hþ 6

h4

� �and i = 2, . . . ,N � 1.

The local truncation errors ti, i = 2, . . . ,N � 1, associated with our scheme(2.7) is

ti ¼h4ð1�2ðaþbÞþcÞyð4Þi þh6 1

6�ð4aþbÞ

� �yð6Þi

þh8 180� 16

12aþ 1

12b

� �� yð8Þi þh10 17

302400� 32

180aþ 1

360b

� �� yð10Þi ðfiÞþOðh11Þ:

(ð2:8Þ

The recurrence relation (2.7) gives rise to a family of methods of differentorders as given below:

(i) Sixth-order method

For a ¼ �64320

; b ¼ 320� 16a and c = 1�2a � 2b (2.7) reduces to

yi�2 � 4yi�1 þ 6yi � 4yiþ1 þ yiþ2

¼ h4½aSi�2 þ bSi�1 þ Sicþ bSiþ1 þ aSiþ2� þ 3306881000000

h10yð10Þi . ð2:9Þ

(ii) Fourth-order method

For a ¼ �107199

; b ¼ 16� 4a and c = 1�2a � 2b (2.7) reduces to


¼ h4½aSi�2 þ bSi�1 þ Sicþ bSiþ1 þ aSiþ2� þ 1929281000000



(iii) Second-order method

For a ¼ �64319

; b ¼ 72400, and c = 1�2 a � 2b (2.7) gives


¼ h4½aSi�2 þ bSi�1 þ Sicþ bSiþ1 þ aSiþ2� � 7776510000000


Each of the above recurrence relations gives (n � 2) linear equations in n un-knowns yi, i = 1, . . . ,N. We need two more equations at each end of the rangeof integration for each of them.

For sixth-order method (2.9) these two equations are developed by methodof undetermined coefficients and are given by

� 11

2y0 þ 9y1 �

9

2y2 þ y3

¼ 3hy00 þ h4X5i¼0

biyðivÞi

!� h10

960yð10Þðn1Þ þOðh11Þ; i ¼ 1

and

yN�2 �9

2yN�1 þ 9yN � 11

2yNþ1

¼ �3hy 0Nþ1 þ h4X5i¼0

b5�iyðivÞN�4þi �

h10

960yð10ÞðnN Þ þOðh11Þ; i ¼ N ;

where

ðb0; b1; b2; b3; b4; b5Þ ¼1

60480ð4233; 43274; 5662; 3432;�1391; 230Þ.

For fourth-order method (2.10) these equations are

�11

2y0þ9y1�

9

2y2þ y3

¼ 3hy00þh4

2808yð4Þ0 þ151yð4Þ1 þ52yð4Þ2 �yð4Þ3

� �þ h8

6720yð8Þðn1ÞþOðh9Þ; i¼ 1

and

yN�2 �9

2yN�1 þ 9yN � 11

2yNþ1

¼ �3hy 0Nþ1 þh4

280�yð4ÞN�2 þ 52yð4ÞN�1 þ 151yð4ÞN þ 8yð4ÞNþ1

� �þ h8

6720yð8ÞðnNÞ þOðh9Þ; i ¼ N .


The schemes at the boundaries for second-order method (2.11) are,

� 11

2y0 þ 9y1 �

9

2y2 þ y3

¼ 3hy00 �h4

20�3yð4Þ0 þ 18yð4Þ1

� �þ 7h6

40yð6Þðn1Þ þOðh7Þ; i ¼ 1;

yN�2 �9

2yN�1 þ 9yN � 11

2yNþ1

¼ �3hy0Nþ1 þh4

2018yð4ÞN � 3yð4ÞNþ1

� �þ 7

40h6yð6ÞðnN Þ þOðh7Þ; i ¼ N .

In the above methods a < n1 < x3; xN�2 < nN < b.

Remarks

(i) When a ¼ �1720

;b ¼ 124720, and c ¼ 474

720then our method (2.7) reduces to

Usmani�s sixth-order finite difference method [14].(ii) When a ¼ 0; b ¼ 1

6, and c ¼ 4

6then our method (2.7) reduces to Usmani�s

fourth-order finite difference method [14].

(iii) When a ¼ � 75000

; b ¼ � 3231875

, and c ¼ 49377500

then our method (2.7) reduces to

Noor and Tirmizi fourth-order finite difference method [8].(iv) When a ¼ 1

360;b ¼ 56

360, and c ¼ 246


Usmani�s second-order method [12] based on polynomial sextic splinefunctions.

(v) When a ¼ 148; b ¼ 12

48, and c ¼ 22

48then our method (2.7) reduces to Al-said

and Noor second-order method [3].(vi) When a ¼ 1

81; b ¼ 14

81, and c ¼ 51

81then our method (2.7) reduces to Noor

and Tirmizi second-order method [8].(vii) When a ¼ 1

120; b ¼ 26

120, and c ¼ 66

120then our method (2.7) reduces to D.J

Fyfe second-order method [5].(viii) When a ¼ 1

120;b ¼ 26

120, and c ¼ 66


Usmani�s second-order method [12] based on polynomial quintic splinefunctions and which resembles case (vi).

3. Convergence analysis

The method is described in matrix form in the following way. LetA ¼ ðaijÞNi;j¼1 denote the five banded matrix. Clearly the system (2.9), (2.10)

and (2.11) along with boundaries can be expressed in matrix form as


AY ¼ Cþ T; ð3:1ÞAeY ¼ C; ð3:2ÞAE ¼ T; ð3:3Þ

where Y ¼ ðyiÞ; eY ¼ ð~yiÞ; C ¼ ðciÞ; T ¼ ðtiÞ; E ¼ ðeiÞ ¼ ðyi � ~yiÞ be N-dimen-sional column vectors and A = A0 + Q,Q = h4BG,G = diag(gi), i = 1,2, . . . ,NA0 = P2 is five band matrix of order n, where P = (pij) is a tridiagonal matrixdefined by

pij ¼2 i ¼ j ¼ 1; 2; . . . ; n;

�1 ji� jj ¼ 1;

0 otherwise;

8><>: ð3:4Þ

B ¼

�b1 �b2 �b3 �b4 �b5

�b �c �b �a

�a �b �c �b �a

�a �b �c �b �a

�a �b �c �b �a

. . . . . . .

. . . .

. . . .

�a �b �c �b

�b5 �b4 �b3 �b2 �b1

0BBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCA

;

ð3:5Þ

ci ¼

2A1 � h2B1 þ h4½b0ðf0A1 þ g0Þ þ b1g1

þb2g2 þ b3g3 þ b4g4 þ b5g5�; i ¼ 1;

�A1 þ h4½aðf0A1 þ g0 þ g4Þ þ bðg1 þ g3Þ þ cg2�; i ¼ 2;

h4ðaðgi�2 þ giþ2Þ þ bðgi�1 þ giþ1Þ þ cgiÞ; 3 6 i 6 n� 2;

�A1 þ h4½aðfnþ1A1 þ gnþ1 þ gn�3Þ

þbðgn�2 þ gnÞ þ cgn�1�; i ¼ n� 1;

2A1 � h2B1 þ h4½b5gn�4 þ b4gn�3 þ b3gn�2 þ b2gn�1

þb1gn þ b0ðgnþ1 þ A1fnþ1Þ�; i ¼ n.

8>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>:

ð3:6Þ


Our main purpose is to derive a bound on kEk. From Eq. (3.3), we have

E ¼ A�1T

¼ ðA0 þ QÞ�1T

¼ ðI þ A�10 QÞ�1A0T;

kEk 6 ðI � A�10 QÞ�1

�� A�10

�� kTk;ð3:7Þ

where k Æ k represents the 1-norm in matrix vector. Using the result kIk = 1,and k(I + A)�1k 6 (I � kAk)�1 we get the following expression

kEk 6A�10

�� kTk1� A�1

0

�� kQk ð3:8Þ

provided that kA�10 k � kQk < 1.

Now from (2.7) we consider the following cases

Case (i) When a ¼ �64319

; b ¼ 72400

and c = 1 � 2a � 2b then kT 1k ¼77765

10000000h6 M6; M6 ¼ max jyð6ÞðxÞj, our method (2.11) gives second-

ordered method.Case (ii) When a ¼ �10

7199; b ¼ 1

6� 4a and c = 1 � 2a � 2b then kT 2k ¼

19292810000000

h8 M8; M8 ¼ max jyð8ÞðxÞj, our method (2.10) gives fourth-ordered method.

Case (iii) When a ¼ �64320

; b ¼ 320� 16a and c = 1 � 2a � 2b, then kT 3k ¼

33068810000000

h10M10,

M10 ¼max jyð10ÞðxÞj our method (2.9) gives sixth-ordered method: ð3:9Þ

So for different choices of a, b and c, we get second, fourth and sixth-ordermethods respectively.

According to [14], the matrix A0 is non-singular and its inverse satisfies theinequality

A�10

�� 65ðd � cÞ4h�4 þ 4ðd � cÞ2h2

364. ð3:10Þ

Thus, using (3.8)–(3.10) and the fact that kBk is finite number, and kGk 6

jg(x)j, we get

kEk 6kA�1

0 kkM5kT 1k1� h4kA�1

0 kkkBkgMffi Oðh2Þ;

kEk 6kA�1



kEk 6kA�1



ð3:11Þ

where k = 5(d � c)4 + 4(d � c)2h2jg(x)j 6 gM provided h4kA0 � 1kkkBkgM < 1.


4. Numerical results and discussion

In this section we illustrate the numerical techniques discussed in the previ-ous sections by the following two boundary-value problems of the (1.2) and(1.3).

Problem 1

yðivÞ þ 4y ¼ 1; yð�1Þ ¼ yð�1Þ ¼ 0;

y 0ð�1Þ ¼ �y 0ð1Þ ¼ sinh 2� sin 2

4ðcosh 2þ cos 2Þ . ð4:1Þ

The analytical solution of (4.1) is

yðxÞ ¼ 0.25½1� 2½sin 1 sinh 1 sin x sinh xþ cos 1 cosh 1 cos x cosh x�=ðcos 2þ cosh 2Þ�.

We summarize the experimental results in Table 1 for our second, fourth andsixth-order methods for problem 1 for brevity.

Problem 2

yðivÞ þ xy ¼ �ð8þ 7xþ x3Þex; 0 < x < 1;

yð0Þ ¼ yð1Þ ¼ 0; y 0ð0Þ ¼ 1; y 0ð1Þ ¼ �e;ð4:2Þ

with y(x) = x(1 � x)ex as its analytical solution. For the sake of comparison,we summarize the experimental results of our methods and the existing splineand finite difference methods of the same order corresponding to problem (4.2)in Tables 2–5.

It is verified from the Tables 1–5 that on reducing the step size form h to h/2,the maximum error observed error kEk is approximately reduced by a factor 1/2p, where p is the order of the method. Tables 1–5 show that our class ofmethods out performs the existing methods of the same order. The numericalcalculations were done on P-IV computer at the simulation LaboratoryGhulam Ishaq Khan Institute of Engineering Sciences Topi Pakistan.

Table 1The observed maximum errors kek corresponding to problem (4.1)

h Sixth-order method(2.9) with a = �6/4319,b = 4175/25914 andc = 1 � 2a � 2b

Fourth-order method(2.10) with a = � 10/7199,b = 3/20 � 16a andc = 1 � 2a � 2b

Second-order method(2.11) with a = �6/4319,b = 72/400 andc = 1 � 2a � 2b

1/8 9.9 · 10�9 1.56 · 10�8 1.83 · 10�5

1/16 1.89 · 10�10 1.58 · 10�9 4.67 · 10�6

1/32 2.82 · 10�12 3.51 · 10�10 1.01 · 10�6

Table 2The observed maximum errors kEk of y, y 0, y00, y000 and y(iv) for our sixth-order method (2.9)

h maxijy(xi)�yij maxijy 0(xi)�y 0ij maxijy00(xi)�y00 ij maxijy000(xi)�y000j maxijy(iv)(xi)�y(iv)j1/8 3.70(�10) 4.91(�6) 3.42(�4) 7.20(�3) 1.85(�10)1/16 7.51(�12) 1.99(�7) 2.24(�5) 1.20(�3) 4.16(�12)1/32 6.99(�14) 6.21(�9) 1.85(�6) 1.78(�4) 3.90(�14)

Table 3The observed maximum errors kEk for sixth-order methods

h 1/8 1/16 1/32

Our method (2.9) with a = �6/4319,b = 4175/25914 and c = 1�2a � 2b

3.70(�10) 7.51(�12) 6.99(�14)

Quintic Spline collocation [6] 3.31(�8) 2.13(�10) 3.34(�12)

Table 4The observed maximum errors kEk for fourth-order methods

h 1/8 1/16 1/32

Our method (2.10) a = �1/698,b = 3/20�16a, c = 1�2a � 2b

2.86(�9) 1.62(�10) 1.03(�12)

Usmani [14] 7.83(�8) 5.16(�9) 3.25(�10)Usmani [12] Sextic Spline 2.36(�7) 1.15(�8) 9.75(�10)Runge–Kutta fourth-order methodfor modified shooting method

1.14 · 10�6 8.79 · 10�8 –

Table 5The observed maximum errors kEk for second-order methods

h 1/8 1/16 1/32

Our method (2.11) a = �6/4319,b = 72/400, c = 1�2a � 2b

1.51(�5) 3.96(�6) 3.54(�8)

Usmani [14] 3.49(�4) 8.59(�5) 2.14(�5)Noor and Timizi [8] 1.27(�4) 3.31(�5) 8.32(�6)Al-Said and Noor Quartic Spline [3] 3.38(�4) 9.98(�5) 2.50(�5)Usmani [12] Quintic Spline 1.72(�4) 4.29(�5) 1.07(�5)Quintic Spline Collocation [6] 1.72(�4) 4.29(�5) 1.07(�5)


5. Conclusion

In this paper, we have developed a new numerical method for solvingfourth-order boundary-value problems based on non-polynomial splines. Thepresent methods enables us to approximate the solution at every point of therange of integration. The parameter dependency of the non-polynomial quintic


spline functions produces optimal sixth, fourth and second-order accuratemethods whereas polynomial spline fuctions produces suboptimal methodsof order two and six. The results obtained are very encouraging and our meth-ods performs better than the existing spline, collocation and finite differencemethods.

Acknowledgment

The first author is grateful to Higher Education Commission Pakistanfor granting scholarship for Ph.D. studies and University of Engg & TechPeshawar Pakistan for study leave.

References

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