GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 37 (2017) 161-174

NUMERICAL SOLUTIONS OF SYSTEM OF SECOND ORDER BOUNDARY VALUE PROBLEMS USING

GALERKIN METHOD

Mahua Jahan Rupa1 and Md. Shafiqul Islam2

1Department of Mathematics, University of Barisal, Barisal-8200, Bangladesh 2Department of Applied Mathematics, University of Dhaka, Dhaka-1000, Bangladesh

2Corresponding author: email: mdshafiqul@du.ac.bd

Received 26.07.2017 Accepted 26.10.2017

ABSTRACT

In this paper we derive the formulation of one dimensional linear and nonlinear system of second

order boundary value problems (BVPs) for the pair of functions using Galerkin weighted residual

method. Here we use Bernstein and Legendre polynomials as basis functions. The proposed method

is tested on several examples and reasonable accuracy is found. Finally, the approximate solutions

are compared with the exact solutions and also with the solutions of the existing methods.

Keywords: Galerkin method, second order linear and nonlinear BVP, Bernstein and Legendre

polynomials.

1. Introduction

Ordinary differential systems have been focused in many studies due to their frequent appearance

in various applications in physics, engineering, biology and other fields. Wazwaz [1] applied the

Adomian decomposition method to solve singular initial value problems in the second order

ordinary differential equations, Ramos [2] proposed linearization techniques for solving singular

initial value problems (IVPs) of ordinary differential equations, and there are other papers [7 – 9]

for solving second order IVPs. However, many classical numerical methods used to solve second-

order IVPs that cannot be applied to second order BVPs. For a nonlinear system of second order

BVPs [3], there are few valid methods to obtain the numerical solutions. Many authors [10, 11]

discussed the existence of solutions to second order systems, including the approximation of

solutions via finite difference method. Lu [4] proposed the variational iteration method for solving

a nonlinear system of second order BVPs. Since piecewise polynomials can be differentiated and

integrated easily and can be approximated to any function of any accuracy desired. Hence

Bernstein polynomials have been used by many authors. Very recently, Bhatti and Bracken [5]

used Bernstein polynomials for solving two point second order BVP, but it is limited only to first

order nonlinear IVP. Besides spline functions and Bernstein polynomials, there are another type of

piecewise continuous polynomials, namely Legendre polynomials [6].

162 Rupa & Islam

Therefore, the purpose of this paper is devoted to use two kinds of piecewise polynomials:

Bernstein and Legendre polynomials widely for solving system of linear and nonlinear second

order BVP exploiting Galerkin weighted residual method.

2. Some Special Polynomials

In this section we give a short description on Bernstein [5] and Legendre [6] polynomials which

are used in this paper.

(a) Bernstein polynomials

The general form of the Bernstein polynomials of nth degree over the interval [ , ] is defined by

[6] , ( ) = ( − ) ( − )( − ) , ≤ ≤ = 0,1,2, ………… . , Note that each of these + 1 polynomials having degree satisfies the following properties:

i. , ( ) = 0, if < 0 or > ,

ii. ∑ , ( ) = 1

iii. , ( ) = , ( ) = 0, 1 ≤ ≤

The first 11 Bernstein polynomials of degree ten over the interval [0,1] , are given below:

i. , ( ) = (1 − )

ii. , ( ) = 10(1 − )

iii. , ( ) = 45(1 − )

iv. , ( ) = 120(1 − )

v. , ( ) = 210(1 − )

vi. , ( ) = 252(1 − )

vii. , ( ) = 210(1 − )

viii. , ( ) = 120(1 − )

ix. , ( ) = 45(1 − )

x. , ( ) = 10(1 − )

xi. , =

All these polynomial will satisfy the corresponding homogeneous form of the essential boundary

conditions in the Galerkin method to solve a BVP.

(b) Legendre Polynomials

The general form of the Legendre polynomials [6] over the interval [1, −1] is defined by

( ) = (−1) (2 − 2 )!2 ! ( − )! ( − 2 )!

Numerical Solutions of System of Second Order Boundary Value 163

where = for even and = for odd.

The first ten Legendre polynomials are given below ( ) = ( ) = (3 − 1) ( ) = (5 − 3 ) ( ) = (35 − 30 + 3) ( ) = (63 − 70 + 15 ) ( ) = (231 − 315 + 105 − 5) ( ) = (429 − 693 + 315 − 35 ) ( ) = (6435 − 1201 + 6903 − 1260 − 35) ( ) = (12155 − 25740 + 18018 − 4620 + 315 ) ( ) = 1256 (46189 − 109395 + 90090 − 30030 + 3465 − 63)

3. System of Second Order Differential Equations

General linear system of two second-order differential equations in two unknowns functions ( ) and ( ), is a system of the form [2]

( ) + ( ) + ( ) + ( ) + ( ) + ( ) = ( )( ) + ( ) + ( ) + ( ) + ( ) + ( ) = ( ) where ( ), ( ), ( ), ( ) are given functions, and ( ), ( )are continuous,

= 1,2,3,4,5,6

And general nonlinear system of two second-order differential equations in two unknowns

functions ( ) and ( ), is a system of the form [4] ( ) + ( ) + ( ) + ( ) + ( ) + ( ) + , ) = ( )( ) + ( ) + ( ) + ( ) + ( ) + ( ) + ( , ) = ( )

where ( ), ( ), ( ), ( ) are given functions, , are nonlinear functions and ( ), ( )are continuous, = 1,2,3,4,5,6

164 Rupa & Islam

4. Formulation of Second Order BVP

Let us consider the one dimensional system of second order differential equations [3] − ( ) + ( ) ( ) + ( ) ( ) = ( )− ( ) + ( ) ( ) + ( ) ( ) = ( ) (1) for the pair of functions ( ) and ( ) in 0 < < 1. Since each equation is of second order, two

boundary conditions are required to specify each of the solution components ( ) and ( ) uniquely. For convenience, we assume homogeneous Dirichlet data at the ends as boundary

conditions

(0) = (1) = (0) = (1) = 0 (2) The data include the prescribed functions , , , , and , which are assumed to be bounded and

sufficiently smooth to ensure subsequent variational integrals are well defined and the problem is

“well posed”.

Let us consider two trial approximate solutions for the pair of functions ( ) and ( ) of system

(1) given by ( ) = ∑ ( ), ≥ 1( ) = ∑ ( ), ≥ 1 (3)

where and are parameter, ( ) are co-ordinate functions (here Bernstein and Legendre

polynomials) which satisfy boundary conditions (2).

Now apply Galerkin Method [1] in system (1) we get weighted residual system of equations

(− ( ) + ( ) ( ) + ( ) ( )) ( ) = ( ) ( )(− ( ) + ( ) ( ) + ( ) ( )) ( ) = ( ) ( ) (4)

Integrating by parts and setting ( ) = 0 at the boundary = 0 and = 1, then we obtain system

of weighted residual equations

( ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( )) = ( ) ( )( ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( )) = ( ) ( ) (5)

Now putting the representation (3) into (5) we get

( ′( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( ))= ( ) ( )

Numerical Solutions of System of Second Order Boundary Value 165

( ′( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( ))= ( ) ( )

We can write above equation as

( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( )= ( ) ( )

( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( )= ( ) ( )

= 1,2,3, … ,

Equivalently,

, + , =, + , = (6)

= 1,2,3, … ,

where, , = [ ( ) ( ) + ( ( ) ( ) ( ))]

, = ( ( ) ( ) ( ))

= ( ) ( )

, = [ ( ) ( ) + ( ( ) ( ) ( ))]

, = ( ( ) ( ) ( )) = ( ) ( )

, = 1,2,3, … ,

166 Rupa & Islam

For = 1, 2, … , we get system of linear equations, which involve parameter and which can

be obtained by solving system (6). System (6) can be assembled by element matrix contribution [3]. Since there is no direct method to solve nonlinear BVPs, so we describe the proposed method

for nonlinear BVPs through numerical examples in the next section.

5. Numerical Examples

In this study, we use three BVPs; two linear and one nonlinear, which are available in the existing

literature [4], the Dirichlet boundary conditions are considered to verify the effectiveness of the

derived formulations. For each case we find the approximate solutions using different number of

parameters with Bernstein and Legendre polynomials, and we compare these solutions with the

exact solutions, and graphically which are shown in the same diagram.

Example 1

Consider the following system of equations [4] ( ) + ( ) + ( ) = 2 ( ) + 2 ( ) + 2 ( ) = −2 (7) subject to the boundary conditions (0) = (1) = 0, (0) = (1) = 0(8) where 0 < < 1. The exact solution of (7) are ( ) = − and ( ) = − .

Solutions using Bernstein polynomials:

We use Bernstein polynomials as trial solution to solve the system (7). Consider trial approximate

solutions be

( ) = , ( )( ) = , ( ) (9)

where and are parameter and , ( ) are co-ordinate functions of Bernstein polynomials

which satisfy conditions (8).

Using the method illustrated in section 4, finally we get,

− , ( ) , ( ) + , ( ) , ( ) + , ( ) , ( )= ( ) , ( ) 10( )

Numerical Solutions of System of Second Order Boundary Value 167

− , ( ) , ( ) + 2 , ( ) , ( ) + 2 , ( ) , ( )= ( ) , ( ) 10( ) = 1,2, … ,

The above equations are equivalent to the matrix form , + , =

, + , =

where, , = [ − , ( ) , ( ) + ( , ( ) , ( ))]

, = ( , ( ) , ( ))

= 2( ) , ( )

, = [ − , ( ) , ( ) + (2 , ( ) , ( ))]

, = (2 , ( ) , ( )) = −2 , ( ) , = 1,2, … ,

Similarly, we can derive the equation (10) using Legendre polynomials.

Table 1: Results of ŭ(x) of equation (7) in Example 1

Exact value

Approximate Solution ( ) Absolute error Approximate

Solution ( ) Absolute error Absolute error [4]

Legendre polynomial n=2 Bernstein polynomial n=9 . 0.0000000000 0.0000000000 0.0000000000 0.000000000 0.0000000000 . . −0.0900000000 −0.0900000000 6.9388939 × 10 −0.09000000 2.8008740130 × 10 . . −0.1600000000 −0.1600000000 2.7755575 × 10 −0.16000000 3.6821549280 × 10 . . −0.2100000000 −0.2100000000 0.0000000000 −0.21000000 6.2091685960 × 10 . . −0.2400000000 −0.2400000000 0.0000000000 −0.24000000 1.0877099220 × 10 . . −0.2500000000 −0.2500000000 0.0000000000 −0.25000000 1.3281251100 × 10 . . −0.2400000000 −0.2400000000 0.0000000000 −0.24000000 1.0877088120 × 10 . . −0.2100000000 −0.2100000000 0.0000000000 −0.16000000 6.2091748410 × 10 . . −0.1600000000 −0.1600000000 0.0000000000 −0.16000000 3.6821518060 × 10 . . −0.0900000000 −0.0900000000 0.0000000000 −0.09000000 2.8008692590 × 10 . . 0.0000000000 0.0000000000 0.0000000000 0.000000000 0.0000000000 .

168

Tab

x

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Exa

Con

Figure 1: G

ble 2: Results of v

Exact value

0 0.000000000

1 0.090000000

2 0.160000000

3 0.210000000

4 0.240000000

5 0.250000000

6 0.240000000

7 0.210000000

8 0.160000000

9 0.090000000

0 0.000000000

Figure 2: G

ample 2:

nsider the follo

Graphical represe

v(x) of equation

ApproximatSolution v(x

Legendre

0 0.000000000

0 0.090000000

0 0.160000000

0 0.210000000

0 0.240000000

0 0.250000000

0 0.240000000

0 0.210000000

0 0.160000000

0 0.090000000

0 0.000000000

Graphical represe

owing system o

entation of exact

n (7) in example

te x)

Absolute er

e polynomial n=2

00 0.00000000

00 6.9388939×

00 2.7755575×

00 0.00000000

00 0.00000000

00 0.00000000

00 0.00000000

00 0.00000000

00 0.00000000

00 0.00000000

00 0.00000000

entation of exact

of equations [4

t and approxima

e 1

rror ApproximSolution v

Bern

000 0.0000000

10–18 0.0900000

10–17 0.1600000

000 0.2100000

000 0.2400000

000 0.2500000

000 0.2400000

000 0.1600000

000 0.1600000

000 0.0900000

000 0.0000000

and approximat

]

te solutions of u

mate v(x)

Absolu

nstein polynomial n

0000 0.00000

0000 2.8008740

0000 3.6821549

0000 6.2091685

0000 1.0877099

0000 1.3281251

0000 1.0877088

0000 6.2091748

0000 3.6821518

0000 2.8008692

0000 0.00000

te solutions of ν

Ru

u(x) of equation (

ute error

Abso

n=9

000000 0.00

0130×10–11 0.00

9280×10–11 0.00

5960×10–11 0.00

9220×10–11 0.00

1100×10–11 0.00

8120×10–11 0.00

8410×10–11 0.00

8060×10–11 0.00

2590×10–11 0.00

000000 0.00

ν (x) of equation

upa & Islam

(7)

olute error

[4]

00000000

00000000

00000000

00000000

00000000

00000000

00000000

00000000

00000000

00000000

00000000

(7)

Num

(

(

v

u

subj

u(0

whe

Usi

Tab

x

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

merical Solution

2)()

()12()(

xxux

uxx

ject to the bou

) = u(1) = 0, v

ere 0 < x < 1. T

ing the same pr

ble 3: Results of ŭ

Exact value

0 0.000000000

1 0.3090169944

2 0.587785252

3 0.8090169944

4 0.951056516

5 1.000000000

6 0.951056516

7 0.8090169944

8 0.587785252

9 0.3090169944

0 0.000000000

Figure 3: G

s of System of S

)sin(

)cos()(

xx

vxx

undary conditio

v(0) = v(1) = 0

The exact solut

rocedure of exa

ŭ(x) of equation

ApproximatSolution v(x

Legendre

0 0.000000000

4 0.308763936

3 0.809559744

4 0.809559744

3 0.950671680

0 0.999124000

3 0.95067316

4 0.809562432

3 0.588521344

4 0.308766240

0 0.000000000

raphical represe

Second Order Bo

sin()( 2x

ons

tions of (11) ar

ample 1 we get

(11)

te x)

Absolute er

e polynomial n=3

00 0.00000000

60 2.53058374×

40 5.42749625×

40 5.42749625×

00 3.84836395×

00 8.76000000×

60 3.83300295×

20 5.45437625×

40 7.36091707×

00 2.50754374×

00 0.00000000

ntation of exact

oundary Value

)12() xx

re u(x) = sin(x

t the following

rror ApproximSolution v

Berns

000 0.0000000

×10–4 0.3215460

×10–4 0.8230835

×10–4 .08230835

×10–4 0.9507983

×10–4 0.9691424

×10–4 0.9539908

×10–4 0.8134388

×10–4 0.5536130

×10–4 0.3147098

000 0.0000000

and approximat

)cos( x

x) and v(x) = x2

g Table 3 and g

mate v(x)

Absolute

stein polynomial n=

000 0.000000

957 1.25291013

958 1.40666014

958 1.40666014

300 2.5186306

805 3.08575195

751 2.9343588

915 4.4218971

521 3.41722001

746 5.69288021

000 0.000000

te solutions of u(

2 – x.

graphs.

e error Absol

=8

00000 0.000

36×10–2 3.0000

47×10–2 2.5000

47×10–2 7.8000

6×10–4 1.6600

53×10–2 2.7700

2×10–3 3.8700

5×10–3 4.5900

16×10–2 4.4900

16×10–3 3.0900

00000 0.000

(x) of equation (

169

(11)

(12)

lute error

[4]

00000000

0000×10–4

0×00010–3

0000×10–3

0000×10–2

0000×10–2

0000×10–2

0000×10–2

0000×10–2

0000×10–2

00000000

11)

170

Tab

x

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Exa

Con

(

(

v

u

subj

u(0

whe

We

solu

ble 4: Results of v

Exact value

0 0.000000000

1 0.3090169944

2 0.587785252

3 0.8090169944

4 0.951056516

5 1.000000000

6 0.951056516

7 0.8090169944

8 0.587785252

9 0.3090169944

0 0.000000000

Figure 4: G

ample 3 :

nsider the follo

)()

c)()(

uxuxx

xuxx

ject to the bou

) = u(1) = 0,

ere 0 < x < 1. T

e use Legendre

ution be of the

v(x) of equation

ApproximatSolution v(x

Legendre

0 0.000000000

4 0.308763936

3 0.809559744

4 0.809559744

3 0.950671680

0 0.95067316

3 0.95067316

4 0.809562432

3 0.588521344

4 0.308766240

0 0.000000000

Graphical represe

owing equation

s2)(

s)()os(2 xx

xvx

undary conditio

v(0) = v(1)

The exact solut

polynomials a

form

n (11)

te x)

Absolute er

e polynomial n=3

00 0.00000000

60 2.53058374×

40 5.42749625×

40 5.42749625×

00 3.84836395×

60 8.76000000×

60 3.83300295×

20 5.45437625×

40 7.36091707×

00 2.50754374×

00 0.00000000

entation of exact

ns [4]

)1()sin(

()sin( 2

xxx

xxx

on

= 0

tions of (13) ar

as trial approxim

rror ApproximSolution v

Bern

000 0.0000000

×10–4 0.3215460

×10–4 0.8230835

×10–4 0.8230835

×10–4 0.9507983

×10–4 0.9691424

×10–4 0.9539908

×10–4 0.8134388

×10–4 0.5536130

×10–4 0.3147098

000 0.0000000

and approximat

()(sin)

()cos()2222 xx

xx

re u(x) = (x – 1

mate solution t

mate v(x)

Absolu

nstein polynomial n

0000 0.00000

0957 1.252910

5958 1.406660

5958 1.406660

3300 2.51863

4805 3.085751

8751 2.934358

8915 4.421897

0521 3.417220

8746 5.692880

0000 0.00000

te solutions of v(

).cos()

)cos()21(2 xx

xx

) sin(x) and v(x

to solve the sy

Ru

ute error

Abso

n=4

000000 0.00

0136×10–2 3.0000

0147×10–2 2.5000

0147×10–2 7.8000

06×10–4 1.6600

953×10–2 2.7700

882×10–3 3.8700

715×10–3 4.5900

0016×10–2 4.4900

0216×10–3 3.0900

000000 0.00

(x) of equation (

)

x) = x – x2.

stem (13). Con

upa & Islam

olute error

[4]

00000000

00000×10–4

00×00010–3

00000×10–3

00000×10–2

00000×10–2

00000×10–2

00000×10–2

00000×10–2

00000×10–2

00000000

11)

(13)

(14)

nsider trial

Numerical Solutions of System of Second Order Boundary Value 171

( ) = ( )( ) = ( ) (15)

where and are parameter ( ) are co-ordinate functions of Legendre polynomials which

satisfy conditions (14).

Using the method illustrated in section 4, finally we get

− ( ) ( ) + ( ( ) ′ ( ) + cos( ) ′ ( ) ( ) = ( ) ( ) 16( )

− ( ) ( ) + ( ) ( ) + ( ) ( )= ( ) ( ) , = 1,2, , ……… , 16( ) The above equations are equivalent to the matrix form , + , = 17( ) , + ( , + , ) = 17( ) where, , = − ( ) ( ) + (( ′ ( ) ( ))

, = (cos( ) ′ ( ) ( ))

= (sin( ) + ( − + 2) cos( ) + (1 − 2 ) cos( )) ( )

, = − ( ) ( )

, = ( ( ) ′ ( ))

, = ( ( ) ( ))

172 Rupa & Islam

= (−2 + sin( ) + ( − 1) sin ( ) + ( − ) cos( )) ( )

, = 1,2, ……… , . The initial values of these coefficients are obtained by applying Galerkin method to the BVP

neglecting the nonlinear term in ( ). That is, to find initial coefficients we will solve the

system

, + , = , + , = 17( )

whose matrices are constructed from , = − ( ) ( ) + (( ( ) ( ))

, = cos( ) ( ) ( )

= (sin( ) + ( − + 2) cos( ) + (1 − 2 ) cos( )) ( )

Table 5: Results of ŭ(x) of equation (13)

x

Exact value

Approximate Solution v(x)

Absolute error Approximate Solution v(x)

Absolute error

Absolute error

[4]

Legendre polynomial n=2 Bernstein polynomial n=9

0.0 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000

0.1 –0.0898500750 0.0000000000 1.421902218×10–3 -0.0898555699 5.494900000×10–6 3.00000×10–4

0.2 –0.1589354646 –0.0912719772 7.683689640×10–4 –0.1589352157 2.489000000×10–7 2.50000×10–3

0.3 –0.206641447 –0.1597038336 6.095282628×10–4 –0.2068641998 5.51000000×10–8 7.80000×10–3

0.4 –0.2336510054 –02062546164 1.767632585×10–3 –0.2336500057 9.9970000×10–7 1.66000×10–2

0.5 –0.2397127693 –0.2318833728 2.163619305×10–3 –0.2397123941 3.75200000×10–7 2.77000×10–2

0.6 –0.2258569894 –0.2375491500 1.645994158×10–3 –0.2258523662 4.62320000×10–6 3.87000×10–2

0.7 –0.1932653062 –0.2242109952 4.373505713×10–4 –0.1932605023 4.80390000×10–6 4.59000×10–2

0.8 –1434712182 –0.1928279556 8.878602201×10–4 –0.1434700211 1.19710000×10–6 4.49000×10–2

0.9 –0.0783326910 –0.1443590784 1.430719837×10–3 –0.0783311456 1.54540000×10–6 3.09000×10–2

1.0 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000 0.0000000000

, = − ( ) ( )

Num

Tab

x

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Onc

sub

con

fina

Con

We

resi

of t

fun

resu

merical Solution

10, (D ji

10 (Dj

i,j = 1, 2,

Figure 5: G

ble 6: Results of ν

Exact value

0 0.0000000000

1 0.0900000000

2 0.1600000000

3 0.2100000000

4 0.2400000000

5 0.2500000000

6 0.2400000000

7 0.2100000000

8 0.1600000000

9 0.0900000000

0 0.0000000000

ce the initial

stituted into eq

ntinues until th

al values of the

nclusions

e have derived,

idual method. T

the problem. I

ctions in the a

ults but also on

s of System of S

))()(( dxLxxL ij

)sin(2 xx

, ..., n.

raphical represe

ν (x) of equatio

Approximate Solution v(x)

Legendr

0 0.0000000000

0 0.0891388543

0 1.1593721549

0 0.2103612466

0 0.2417674742

0 0.2532521825

0 0.2444767162

0 0.2151024200

0 0.1647906387

0 0.0932027171

0 0.0000000000

values of the

quation 17(b) t

he converged v

e parameters in

in details, the

This method en

In this method

approximation.

n the formulatio

Second Order Bo

dx

22 (sin)1(xx

ntation of exact

on (13) using

Absolute erro

re polynomial n=2

0 0.000000000

3 8.61145740×1

9 6.27845120×1

6 3.61246620×1

2 1.76747424×1

5 3.25218250×1

2 4.47671616×1

0 5.10241998×1

7 4.79063872×1

1 3.20271714×1

0 0.000000000

e coefficients

to obtain new

values of the

nto (14), we obt

formulation of

nables us to ap

d, we have use

. The concentr

ons.

oundary Value

2 co)()( xxx

and approximat

or ApproximSolution v

Ber

00 0.0000000

10–4 0.0899952

10–4 0.1600002

10–4 0.21009999

10–2 0.23999999

10–3 0.2511119

10–2 0.2400001

10–2 0.2099999

10–3 0.1599879

10–3 0.0900501

00 0.0000000

ai are obtain

estimates for t

unknown para

tain an approxi

f system of sec

pproximate the

ed Legendre a

ration has give

)())os( dxxLx j

te solutions of u(

mate v(x)

Absolu

rnstein polynomial n=

0000 0.00000

2168 4.783200

2894 2.894000

9877 9.998770

9998 2.000000

9953 9.953000

1511 1.511000

9899 1.010000

9652 1.203480

1123 5.011230

0000 0.00000

ned from the

the values of a

ameters are ob

imate solution

cond order BVP

solutions at ev

and Bernstein p

en not only on

(x) of equation (

ute error

Abs

=9

000000 0.00

000×10–6 0.00

000×10–7 0.00

000×10–5 0.00

00×10–10 0.00

000×10–7 0.00

000×10–7 0.00

000×10–8 0.00

000×10–5 0.00

000×10–5 0.00

000000 0.00

system 17(c),

ai. This iteratio

btained. Substi

of the BVP (1

Ps by Galerkin

very point of th

polynomials a

n the performan

173

13)

solute error

[4]

000000000

000000000

000000000

000000000

000000000

000000000

000000000

000000000

000000000

000000000

000000000

they are

on process

ituting the

3).

n weighted

he domain

as the trial

nce of the

174

We

to s

des

com

app

RE

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

Figure 6: G

e may notice th

solve for BVP.

ired formulati

mpared with th

plied for higher

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