J. Nig. Soc. Phys. Sci. 4 (2022) 867

Journal of the
Nigerian Society

of Physical
Sciences

An Integral Transform-Weighted Residual Method for Solving
Second Order Linear Boundary Value Differential Equations

with Semi-Infinite Domain

E. I. Akinolaa,∗, R. A. Oderinub, S. Alaob, O. E. Opaleyeb

a Mathematics Programme, Bowen University, Iwo Osun State, Nigeria.
b Department of Pure and Applied Mathematics, Ladoke Akintola University, Ogbomoso. Oyo State. Nigeria

Abstract

Solving boundary value problem with semi-infinite domain in a conventional way or using some of the available approximate methods poses a
lot of challenges or better still seems almost impossible over the years, and some of the alternative ways for crossing this hurdle is by fixing
value for infinity that is in the domain. Here in this work, we present an Integral Transform (Aboodh Transform) - Weighted Residual Based
Method (AT-WRM) to address the afore-mentioned challenge. Aboodh Transform was used to transform and at the same time used to find the
inverse of the given differential equations while Weighted Residual via Collocation Method was used in order to avoid fixing value for infinity
as usual by introducing e−ix in trial function to decay the infinity that was part of the boundary condition. The accuracy of the presented method
was authenticated by solving three different problems. The excellent results obtained from the three solved problems validate the accuracy and
effectiveness of the method

DOI:10.46481/jnsps.2022.867

Keywords: Aboodh Transform, Weighted Residual, Partition Method, Boundary Value Problem, Semi-Infinite Domain.

Article History :
Received: 14 June 2022
Received in revised form: 20 September 2022
Accepted for publication: 21 September 2022
Published: 11 November 2022

c©2022 Journal of the Nigerian Society of Physical Sciences. All rights reserved.
Communicated by: W. A. Yahya

1. Introduction

Boundary conditions at infinity are commonly used to solve
linear and nonlinear phenomena that emerge in a wide range
of scientific and engineering sectors. For the solution of these
types of problems, a variety of mathematical techniques and
methodologies have been proposed. Oderinu and Aregbesola
[1], for example, employed the weighted residual approach to
address the challenge encountered in a semi-infinite domain

∗Corresponding author tel. no: +2348062723102.
Email address: emmanuel.idowu@bowen.edu.ng (E. I. Akinola )

by reducing the domain within (0,∞) utilizing the Gauss-
Laguerre integration formula with Galerkin and Moment
methods. Similarly, Odejide and Aregbesola [2] utilized
the same strategy to tackle the problems in a semi-infinite
domain, but the method of partition was employed. To handle
problems in an unbounded domain, several researchers have
employed a variety of strategies. Noor and Syed [3] used
Variational Iteration Method that was modified. Adewumi
et al. [4] used the combination of Laplace transformation
method and weighted residual method to provide a series
solution to problems in a semi-infinite domain, Peker et al. [5]
employed differential transform method and Pade approximant

1



E. I. Akinola et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 867 2

to solve Blasius problem, which appears in many areas of
science and engineering professions. Likewise, Sajid et al. [6]
presented the hybrid variational iteration algorithm method
based on the combination of variational iteration and shooting
methods to obtain the result of the same Blasius problem,
Akinola and Ogunlaran [7] presented solution to these types of
boundary-value problems in a semi-finite domain by Sumudu
transform decomposition method, several other methods have
also been employed to solve these type of problem and they
are: Adomian decomposition method [8], solution of the MHD
falner-Skan flow by Hankel-Pade method [9], variational itera-
tion approach [10] homotopy analysis decomposition method
for the solution of viscous boundary layer flow Due to a moving
sheet by Alao et al. [11], application of Laplace decomposition
method to boundary value equation in a semi-infinite domain
by Ogunsola et al. [12], a comparison study of numerical
techniques for solving ordinary differential equations defined
on a semi-infinite domain using rational Chebyshev functions
by Ramadan [13], an order four continuous numerical method
for solving general second order ordinary differential equations
by Obarhua [14], numerical algorithms for direct solution of
fourth order ordinary differential equations by Kuboye [15]
to mention a few and many other analytical and numerical
methods have also been employed to tackle problems of the
nature considered.

Here, we would employ the combination of an integral
transform proposed by Aboodh [16, 17] and Weighted Resid-
ual method by Crandall [18], Finlayson and Scriven [19],
Vichnevetsky[20] for finding the solution of problems with
semi infinite domain, inspired and motivated by the existing
research in this direction, the researchers decided to couple
weighted-residual method with Aboodh Transform (AT) in or-
der to compliment the drawback of Aboodh Transform that
cannot handle boundary value problems especially with semi-
infinite domain. The suggested technique uses the Aboodh
Transform method to generate a new form of trial function,
which is then substituted into the original equation to obtain
the residual function, then ultimately reduced by the proposed
method.

2. Basic Concept of the AT-WRM

2.1. Basic Idea of Aboodh Transform

Aboodh Transform is one out of many integral transforms
defined for function of exponential order [16, 17] by:

A =
{
f (t) : ∃M, k1, k2 > 0, | f (t)| < Me

−vt
}
, (1)

where k1, k2 may be finite or infinite in a given set M. The
Aboodh Transform of a function f (t) is defined as:

A[ f (t)] = K(v) =
1
v

∫
∞

0
f (t)e−vtdt, t ≥ 0, k1 ≤ v ≤ k2. (2)

Aboodh Transform of some of the functions are:

A(c) =
c
v2
,

where c is a constant

A[tn] =
n!

vn+2
, A[e±at] =

1
v2 ± av

,

A[sin(at)] =
a

v(v2 + a2)
, A[cos(at)] =

1
(v2 + a2)

.

Aboodh transform of first, second and nth derivatives are:

A[ f ′(t)] = vK(v) −
f (0)
v

A[ f ′′(0)] = v2 K(v) −
f ′(0)

v
− f (0)

A[ f n(0)] = vn K(v) −
n−1∑
k=0

f k(0)
v(2−n+k)

2.2. Basic Idea of Weighted Residual Method
Given a boundary value problem

M[u] = r(x), x ∈ D, Bµ[u] = γk, x ∈ ∂D. (3)

M[u] denotes a general differential operator, Bµ is the appro-
priate number of boundary conditions involved and D is the
domain with boundary ∂D.

The solution of (3) is found by assuming an approximate
solution called trial function to the dependent function u(x) in
the form:

Ω(x, a) = w0(x) +
n∑

i=1

aiwi(x), (4)

where ωi(x) are prescribed and satisfy the boundary conditions.
Upon substitution of (4) into (3) the residual R(x, a) in the
differential equation is obtained.

The objective is to minimize the residual such that it be-
comes smaller and smaller or tend to zero as the number n of
the function ωi(x) increased in the successive approximations.
Having obtained the residual, one of the methods, namely: Col-
location, Partition, Moment, least square and Galerkin of mini-
mizing the residual is employed.

3. Methodology- Aboodh Transform Weighted Residual
Based Method (AT-WRM)

Suppose we have a differential equation:

L[u(x)] = f (5)

in the domain ψ [1]

Bγ[u] = ψ on ∂ψ, (6)

where L[u(x)] denotes a general differential operator (linear or
non-linear) involving spatial derivatives of dependent variable
u, f is a known function of position, Bγ[u] represents the
appropriate number of boundary conditions, and ψ is the
domain (semi-infinite in this case) with the boundary ∂ψ.

The itemized steps below were followed in solving this type
of problem:

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E. I. Akinola et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 867 3

(i) We assumed a trial function that satisfies (6) in the form
[10]

n∑
i=0

die
−ikx

, the assumption above is due to the condition at infinity,
where di i = 0, 1, 2, ·, n are constants to be determined
and k is any constant value

(ii) Taking the Aboodh transform of (5) to have

A
[
L[u(x)]

]
= A[ f ] (7)

(iii) Substituting the trial function obtained in step 1 into (7)
to give

A
[
L
[ n∑

i=0

die
−ikx
− f

]]
= 0. (8)

(iv) Taking the Aboodh inverse transform of (8) to obtain a
new trial function given as

A−1
[
A
[
L
[ n∑

i=0

die
−ikx
− f

]]
= 0

]
(9)

(v) Imposing the boundary condition in finite domain on (9)
to give a set of equations

(vi) Likewise, substituting (9) into (5) to obtain the Resid-
ual function which is then minimized using collocation
method by making use of xk which is the zeros of the
Laguerre polynomial Ln(x) given [2, 21]

Ln(x) = e
x d

n

d xn
(e−x xn)

Ak =
(n!)2

xk[L′n(xk)]2
(10)

Table 1: Laguerre Polynomial and the Corresponding Weight Functions

n xk Ak
2 0.58578644 0.85355339

3.41421356 0.14644661
4 0.32254769 0.60315410

1.74576110 0.35741869
4.53662030 0.03888791
9.39507091 0.00053929

6 0.22284660 0.45896467
1.18893210 0.41700083
2.99273633 0.11337338
5.77514357 0.01039920
9.83746742 0.00026102

15.98287398 0.00000090

(vii) The number of equations obtained equalled the number
of constants to be determined and, so we solved the re-
sulting algebraic equations through Maple 18 to obtain
the constants di which are substituted back into the trial
function in step 1 to find the solution.

3.1. Numerical Illustration
Illustration 1:

8
d2u
d x2

+ 2
du
d x
− u = 8e−

x
4 (11)

subject to the boundary conditions

u(0) = 1, u(∞) = 0,

with exact solution given as:

uEXACT = 9e
−

1
2 x − 8e−

1
4 x.

Assuming the trial function

u =
n∑

i=0

die
−ikx

with n = 6 and k = 14 gives:

u = d0+d1e
−

1
4 x+d2e

−
1
2 x+d3e

−
3
4 x+d4e

−x
+d5e

−
5
4 x+d6e

−
3
2 x.(12)

Finding the Aboodh transform of (11) and substituting (12) into
the resulting equation gives:

A[u] = K[v] =
1
v2

[ u′(0)
v

+ u(0) + A
[
e−

x
4 +

1
8

u−
1
4

u′
]]
.(13)

Upon application of inverse Aboodh Transform on (13), we
have:

u = A−1
[ 1
v2

[ u′(0)
v

+ u(0) + A
[
e−

x
4 +

1
8

u −
1
4

u′
]]]
. (14)

Substituting (14) into (11) gives:

R = 3 +
5
16

d3e
−

3
4 x +

3
8

d4e
−x

+
7

16
d5e
−

5
4 x +

1
2

d6e
−

3
2 x +

1
8

d0

+
1
8

d1 − e
−

1
4 x −

1
2

x +
1
32

d0 −
1

128
d0(x

2
+ 16)

−
1
72

d3
(
4 − 3x + 5e−

3
4 x

)
−

1
64

d4
(
5 − 5x + 3e−x

)
−

1
400

d5
(
36 − 45x + 14e−

5
4

)
−

1
144

d6
(
14 − 21x + 4e−

3
2 x

)
−

1
8

e−
1
4 x(16 + 3d1) −

1
16

d1(x − 4)

+
1
36

d3
(
− 3 −

15
4

e−
3
4 x

)
+

1
32

d4(−5 − 3e
−x) +

1
200

d5(
− 45 −

35
2

e−
5
4 x

)
+

1
72

d6
(
− 21 − 6e−

3
2 x

)
. (15)

Subjecting (14) to the initial condition u(0) = 1, and collocate
(11) using the roots of Laguerre polynomial of six points
at x = 0.22284660, 1.18893210, 2.99273633, 5.77514357,
9.83746742, 15.98287398 to take care of the boundary con-
dition at infinity-u(∞) = 0. This gives rise to seven algebraic
equations with seven unknown altogether.

Solving the seven equations for the unknown constants
through Maple 18, we have:

d0 = 1.88748 × 10
−7, d1 = −8.000002108199, d2 = 9.000009965150,

d3 = −0.000023454290, d4 = 0.000028981727,
d5 = −0.000018037567, d6 = 0.000004464431. (16)

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E. I. Akinola et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 867 4

Substituting (16) into the assumed trial function (12) gives the
desired solution to (11) as:

uAT−WRM = 1.88748 × 10
−7
− 8.000002108199e

−1
4 x

+ 9.000009965150e
−1
2 x − 0.000023454290e

−3
4 x

+ 0.000028981727e−x − 0.000018037567e
−5
4 x

+ 0.000004464431e
−3
2 x (17)

 

 

 

 

Figure 1: Graphical Validation of the Initial Boundary Condition of Illustration
1

Illustration 2:

8
d2u
d x2

+ 2
du
d x
− u = 8e−

3x
4 , 0 ≤ x ≤∞, (18)

with the exact solution

uEXACT = −3e
−

1
2 x + 4e−

3
4 x.

To obtain the solution to (18), salient points or procedures item-
ized in section 3 were carried out one after the other with the
same trial function as (12), the same number of points (n = 6)
and k = 14 . Hence, the constant terms of (12) and the approxi-
mate result of (18) are obtained as:

d0 = −2.2517 × 10
−9, d1 = 2.5194 × 10

−8, d2 = −3.000000119291,

d3 = 4.000000281182, d4 = −3.47939 × 10
−7, d5 = 2.16854 × 10

−7,

d6 = −5.3749 × 10
−8 (19)

and

uAT−WRM = −2.2517 × 10
−9

+ 2.5194 × 10−8e−
1
4 x

− 3.000000119291e−
1
2 x + 4.000000281182e−

3
4 x

− 3.47939 × 10−7e−x + 2.16854 × 10−7e−
5
4 x

− 5.3749 × 10−8e−
3
2 x (20)

 

 

 

 

Figure 2: Graphical Validation of the Initial Boundary Condition of Illustration
2

Illustration 3: Consider the problem solved by Odejide [2]

d2u
d x2

+ 2
du
d x
− 2u = −e−2x, 0 ≤ x ≤∞ (21)

subject to the boundary conditions

u(0) = 1, u(∞) = 0.

The exact solution to (21) is:

uEXACT =
1
2

(
e(−1+

√
i)x

+ e−2x
)

This time around the new trial function

u =
n∑

i=0

die
(−1+

√
i)x

4



E. I. Akinola et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 867 5

Table 2: Comparison between AT-WRM and the Exact for Illustration 1

x uEXACT uAT−WRM |uEXACT − uAT−WRM|
0.0 1.000000000000 0.9999999999995 5.2791 × 10−13

0.1 0.758585524280 0.7585855246842 4.04668539 × 10−10

0.2 0.533701366318 0.5337013671228 8.04471306 × 10−10

0.3 0.324423897197 0.3244238983415 1.144766413 × 10−9

0.4 0.129877433414 0.1298774348212 1.406555289 × 10−9

0.5 -0.050768173034 -0.05076817144136 1.593182583 × 10−9

0.6 -0.218299825266 -0.2182998235440 1.721015535 × 10−9

0.7 -0.373463358686 -0.3734633568723 1.812797763 × 10−9

0.8 -0.516965610304 -0.5169656084104 1.893055570 × 10−9

0.9 -0.649476385479 -0.6494763834940 1.985061000 × 10−9

1.0 -0.771630327158 -0.7716303250490 2.1089594485 × 10−9

Table 3: Comparison between AT-WRM and the Exact for Illustration 2

x uEXACT uAT−WRM uMWR[2] |uEXACT − uAT−WRM| |uEXACT − uMWR|
0.0 1.000000000000 0.999999999999 0.9999999995 7.0 × 10−13 5.0 × 10−10

0.1 0.857285671812 0.8572856718066 0.8572856728 5.51848747 × 10−12 1.8 × 10−9

0.2 0.728319651592 0.7283196515825 0.7283196512 1.032911322 × 10−11 8.0 × 10−10

0.3 0.611940945763 0.6119409457479 0.6119409463 1.445554232 × 10−11 3.0 × 10−10

0.4 0.507080623493 0.5070806234760 0.5070806239 1.765087242 × 10−11 1.0 × 10−10

0.5 0.412754765950 0.4127547659297 0.4127547648 1.994249358 × 10−11 1.2 × 10−9

0.6 0.328057944442 0.3280579444209 0.3280579430 2.151874020 × 10−11 1.0 × 10−9

0.7 0.252157188311 0.2521571882890 0.2521571879 2.264794352 × 10−11 1.1 × 10−9

0.8 0.184286406269 0.1842864062451 0.1842864050 2.362236467 × 10−11 1.0 × 10−9

0.9 0.123741227565 0.1237412275399 0.1237412271 2.472101384 × 10−11 1.0 × 10−10

1.0 0.069874231826 0.06987423180009 0.06987423173 2.618660748 × 10−11 2.7 × 10−10

is assumed for (21) with n = 6. Following the same process as
done for the illustrations 1 and 2 we have:

d0 = −3.464061362 × 10
−11, d1 = 0.5000000099703,

d2 = −9.79486026690 × 10
−8, d3 = 0.5000003319506,

d4 = −5.104366705162 × 10
−7, d5 = 3.679792467715

−7,

d6 = −1.014800109354 × 10
−8 (22)

and

uAT−WRM = −3.464061362 × 10
−11e−x + 0.5000000099703e−2x

− 9.79486026690 × 10−8e−2414213562x

+ 0.5000003319506e−2732050808x

− 5.104366705162 × 10−7e−3.000000x

+ 3.679792467715−7e−3.236067977x

− 1.014800109354 × 10−8e−3.449489743x (23)

3.2. Discussion of Results
An Integral Transform -Weighted Residual based technique

has been successfully used in solving second order linear
boundary value problems with semi- infinite domain. In order
to examine the reliability of the proposed method ,three differ-
ent illustrations were considered. Tables 2, 3 and 4 show the

 

Figure 3: Graphical Validation of the Initial Boundary Condition of Illustration
3

results of the three illustrations respectively and in all the three
tables it was observed that absolute difference between exact

5



E. I. Akinola et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 867 6

Table 4: Comparison between AT-WRM and the Exact for Illustration 3

x uEXACT uAT−WRM uMWR[2] |uEXACT − uAT−WRM| |uEXACT − uMWR|
0.0 1.000000000000 1.000000000001 1.00000000 1.346801 × 10−13 0.0000000
0.1 0.7898337356130 0.7898337356130 0.7898312080 8.472091 × 10−14 2.5275 × 10−6

0.2 0.6246723675307 0.6246723675306 0.6246709988 9.183796 × 10−14 1.3692 × 10−6

0.3 0.4947063913440 0.4947063913439 0.4947069736 6.180149 × 10−14 5.826 × 10−6

0.4 0.3922992773092 0.3922992773093 0.3923013693 1.0272004 × 10−13 2.0923 × 10−6

0.5 0.3114991915313 0.3114991915314 0.3115024402 9.230426 × 10−14 3.2486 × 10−6

0.6 0.2476617911460 0.2476617911461 0.2476660586 1.4779032 × 10−13 4.2673 × 10−6

0.7 0.1971585649673 0.1971585649672 0.1971637757 9.0332704 × 10−14 5.2108 × 10−6

0.8 0.1571511081549 0.1571511081545 0.1571571826 3.30057946 × 10−13 6.0744 × 10−6

0.9 0.1254162556993 0.1254162556989 0.1254231315 4.38164242 × 10−13 6.8759 × 10−6

1.0 0.1002104788741 0.1002104788736 0.1002181358 5.38651959 × 10−13 7.6569 × 10−6

solution and Aboodh Transform-Weighted Residual Method is
very negligible. Figures 1, 2 and 3 also show graphical com-
parison between the exact solution and Aboodh Transform-
Weighted Residual Method for the three illustrations and it was
observed that there are no significant difference between the
two sets of solutions.

4. Conclusion

This research work has presented Aboodh Transform-
Weighted Residual Method (AT-WRM) as a means of
providing solution to three different second order linear
boundary value problems in a semi-infinite domain. The results
obtained so far as presented in the tables and figures showed
that the method is superb, provides brilliant result and at the
same time proves a vital tool to surmount the challenges of
seeking how to handle the condition at infinity. Moreover, it
has been shown that the combination of an integral transform-
Adoodh Transform and Weighted Residual Method demostrate
an excellent mathematical tool in obtaining near exact result if
not the real exact solution of any boundary value problem and
most especially the one with semi-infinite domain.

Motivation

The reason for compling the two methods is to be able to
solve boundary value problems within semi-infinite domain by
the process of Adoodh Transform method as Adoodh Trans-
form cannot independently deal with boundary condition at in-
finity naturally. In addition,to improve the efficiency of the
method of Weighted Residual Method.

Acknowledgements

The authors thank Dn. Stephen Taiwo Akinola who did the
editorial and the anonymous referees for their careful reading
of the manuscript and valuable comments for the improvement
of the quality of the paper.

References

[1] R. A. Oderinu & Y. Aregbesola, “Weighted Residual Method In A Semi
Infinite Domain Using an Un-Patitioned Method”, International Journal
of Applied Mathematics 25 (2012) 25.

[2] S. Odejide & Y. Aregbesola, “Applications of method of weighted resid-
uals to problems with semi-finite domain”, Rom. Journ. Phys. 56 (2011)
14.

[3] M.A.Noor & S.T. Mohyud-Din, “Modified variational iteration method
for a boundary layer problem in unbounded domain”, International Jour-
nal of Nonlinear Science 7 (2009) 426.

[4] A.Adewumia, et al., “Laplace-weighted residual method for problems
with semi-infinite domain”, Journal of Modern Methods in Numerical
Mathematics 7 (2016) 59.

[5] H.A.Peker, O. Karaolu & G. Oturanç, “The differential transformation
method and Pade approximant for a form of Blasius equation”, Mathe-
matical and Computational Applications 16 (2011) 507.

[6] M. Sajid, N. Ali, & T. Javed, “A Hybrid Variational Iteration Method for
Blasius Equation”, Applications and Applied Mathematics: An Interna-
tional Journal (AAM), 10 (2015) 15.

[7] E.Akinola, et al., “On the Application of Sumudu Transform Series De-
composition Method and Oscillation Equations”, Asian J. Mathematics,
2 (2017) 1.

[8] S. Khuri, “On the decomposition method for the approximate solution of
nonlinear ordinary differential equations”, International Journal of Math-
ematical Education in Science and Technology 32 (2001) 525.

[9] S.Abbasbandy, & T. Hayat, “Solution of the MHD FalknerSkan flow by
Hankel Pade method”, Physics Letters A 373 (2009) 731.

[10] A.M. Wazwaz, “The variational iteration method for solving two forms
of Blasius equation on a half-infinite domain”, Applied Mathematics and
Computation 188 (2007) 485.

[11] S. Alao, R. A. Oderinu, F. O. Akinpelu & E. I. Akinola, “Homotopy Anal-
ysis Decomposition Method for the Solution of Viscous Boundary Layer
Flow Due to a Moving Sheet”, Journal of Advances in Mathematics and
Computer Science 32 (2019) 1.

[12] A.W. Ogunsola, R.A. Oderinu, M. Taiwo & J.A. Owolabi, ”Application of
Laplace Decomposition Method to Boundary Value Equation in a Semi-
Infinite Domain”, International Journal of Difference Equations (IJDE)
17 (2022) 75.

[13] M.A. Ramadan, T. Radwan, M.A. Nassar & M.A. Abd El Salam, “A
Comparison Study of Numerical Techniques for Solving Ordinary Dif-
ferential Equations Defined on a Semi-Infinite Domain Using Rational
Chebyshev Functions by Ramadan”, Journal of Function Spaces 2021
(2021) 12 .

[14] F. O. Obarhua & O. J. Adegboro, “An order four continuous numerical
method for solving general second order ordinary differential equations”,
J. Nig. Soc. Phys. Sci. 3 (2021) 42.

[15] J.O. Kuboye, O. R. Elusakin & O.F. Quadri, “Numerical Algorithms for
Direct Solution of Fourth Order Ordinary Differential Equations”, J. Nig.
Soc. Phys. Sci., 2 (2020) 218.

6



E. I. Akinola et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 867 7

[16] K. S. Aboodh, “The New Integral Transform ’Aboodh Transform”,
Global Journal of Pure and Applied Mathematics 9 (2013) 35.

[17] K. S. Aboodh, “Application of new transform Aboodh transform to partial
differential equations”, Global Journal of Pure and Applied Mathematics,
10 (2014) 249.

[18] S.H. Crandall, “Engineering analysis: A survey of numerical proce-
dures”, McGraw-Hill, 1956.

[19] B. Finlayson & L. E. Scriven, “The method of weighted residual; a re-

view”, Appl. Mech. Rev. 19 (1966) 735.
[20] R. Vichnevetsky, “Use of functional approximation methods in the com-

puter solution of initial value partial differential equation problems”,
IEEE Transactions on Computers 100 (1969) 499.

[21] R. A.Oderinu & A. S.Aregbesola, “Using Laguerres Quadrature in
Weighted Residual Method for Problems with Semi Infinite Domain”, In-
ternational Journal of pure and Applied Mathematics 75 (2012) 371.

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