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: [email protected]
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
FERENCES
M. Wazwazdifferential e
J.I. Ramos, equations, A
Eric B. BeckI, Prentice-H
Jungfeng Luvalue proble
M.I. Bhatti aComput. App
Md. Shafiqunumerical so33(2013) 53
Gear, C.W.Englewood
S.O. FatunlAcademic P
L. Lustman hypercube, C
] H.B. Thompfor second o
] Xiyon Chendifferential s
Graphical represe
hat the formula
Some linear a
ons whose an
he exact soluti
r order BVPs to
z, A new methequations, Appl.
Linearization Appl. Math. Comp
ker, Graham F, CHall, Inc (1981).
u, Variational Iteems, Computer M
and P. Bracken,pl. Math. 205 (2
ul Islam, Md. olutions of fourt3-64.
. Numerical iniCliffs, NJ (1971
a, Numerical Mress, Boston (19
B. Neta, W. GComputer Mathe
pson, C. Tisdell, order systems of
ng, Chengkui Zsystem, J. Math.
entation of exact
ations of this st
and nonlinear e
nalytical soluti
ions and we h
o get the desire
hod for solving Math. Comput.
techniques for put. 161 (2005)
Carey and J. Tin
eration Method fMathematics wit
Solutions of Di2007), 272-280.
Bellal Hossain,th order boundar
itial value prob).
Methods for Init988).
Gragg, Solution oematics with App
Systems of diffordinary differe
Zhong, Existen Anal. Appl. 312
and approximat
tudy are easy to
examples are t
ons are not av
have found a g
ed accuracy.
singular initial128 (2002) 45-5
singular initial 525-542.
nsley Oden, FIN
for Solving a noth Applications, 5
ifferential Equat
, On the use ory value problem
blems in ordin
tial Value Prob
of ordinary diffplications, 23 (1
ference equationential equations,
nce of positive 2 (2005) 14-23.
te solutions of v(
o understand a
tested to verify
vailable. The
good agreemen
l problems in t57.
value problem
NITE ELEMENTS
onlinear System 54(2007), 1133-
tions in a Bernst
of piecewise stams, GANIT Jn. B
nary differential
blems in Ordina
ferential initial v992), 65-72.
s associated withJ. Math. Anal. A
solutions for
Ru
(x) of equation (
and may be imp
y the effectiven
computed solu
nt. This metho
the second orde
ms of ordinary
TS An Introductio
of Second-order-1138.
tein Polynomials
andard polynomBangladesh Math
equations, Pre
ary Differential
value problems
h boundary valuAppl. 248 (2000)
a second orde
upa & Islam
13)
plemented
ness of the
utions are
od may be
er ordinary
differential
on, Volume
r Boundary
s Basis, Jn.
mials in the h. Soc. Vol.
entice-Hall,
Equations,
on an intel
ue problems ) 333-347.
er ordinary