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63 
J. mt. area res., Vol. 6, 2021 

   

         Journal of Mountain Area Research 
            

 

 

AN ENHANCED WAVELET BASED METHOD FOR NUMERICAL SOLUTION OF 

HIGH ORDER BOUNDARY VALUE PROBLEMS 
S. Haq1, and M. Sohaib2,3, * 

1. Faculty of Engineering Sciences, GIK Institute, Topi 23660, KPK, Pakistan. 

2. Department of Mathematics and Statistics, Bacha Khan University, Charsadda, 24461, Pakistan  

3. Department of Mathematics, COMSATS University, Islamabad 
 

ABSTRACT 

 The Legendre wavelet collocation method (LWCM) is suggested in this study for solving high-

order boundary value problems numerically. Eighth, tenth, and twelfth-order examples are used as 

test problems to ensure that the technique is efficient and accurate. In comparison to other 

approaches, the numerical results obtained using LWCM demonstrate that the method's accuracy 

is very good. The results indicate that the method requires less computational effort to achieve better 

results. 

KEYWORDS: Wavelet, Legendre wavelet, Legendre polynomials, Collocation points  

 

*Corresponding author: (Email: msohaib@bkuc.edu.pk, Phone: 0092-346-9648868)

 

1. INTRODUCTION 

The researchers are targeting ordinary 

differentials equation because of their 

importance in different areas of engineering, 

biomedical science, physics and 

mathematics. Due to their weight ordinary 

differential equation have been used in 

hydrodynamic and hydro magnetics stability, 

fluid mechanics[4], astronomy[1], introduction 

motors, beam and long wave theory [26]. An 

infinite layer of fluid with a certain incidence 

in rotation gives rise to instability/unsteadiness. 

The sixth order ordinary differential equation is 

modeled by the instability sets as an ordinary 

convection whereas eight order is in case of 

over stability. If heating of the same layer is 

under the influence magnetic field in direction 

of gravity the model is tenth order ordinary 

differential equation when instability sets as an 

ordinary convection and twelfth order 

ordinary differential equation in case of over 

stability [4, 1]. In literature different procedures 

have been used to achieve the solution of 

ordinary differential equation of different 

orders. These techniques include the method 

of Homotopy analysis (HAM) [16, 26, 43], 

Variational literation (VIM) [15, 20, 24], 

Homotopy Perturbation (HPM)[8, 16, 20], 

Modified Decomposition (MDM) [34], Optimal 

Homotopy Asymptotic (OHAM) [13] , Quintic 

B-spline (QBSM) [33] , Non-Polynomial Spline 

(NPSM) [28] , Eleven Degree Spline (EDSM) 

[29] , Finite difference (FDM) [3] , and Exp-

function (EFM) [36 ] etc. For the study of 8𝑡ℎ 

order boundary value problem (BVP) HPM was 

used by golbabai and javidi [8] , differential 

quadrature method (DQM) by liu and 

wu[18],OHAM by haq et al. [11], reproducing 

kernel space by Akram and Rehman [33, 2] . 

Siddiqi and Akram utilized NPSM and EDSM for 

the solution of tenth order ordinary differential 

equation [28, 29], Geng and li exercised VIM 

for the solution of tenth order boundary valve 

problems [6]. I. ullah et el. made use of new 

iterative method while Wazwaz used MDM for 

the solution od tenth order BVPs [34, 31]. The 

solution of twelfth order BVPs using DTM was 

Vol. 6, 2021 

https://doi.org/10.53874/jmar.v6i0.109 

Full length article 



 

 

Haq et al., J. mt. area res. 06 (2021)63-76 

64 
J. mt. area res., Vol. 6, 2021 

computed by islam et el. [13]. Approximate 

solution of twelfth order BVPs were obtained 

via splines and HPM in [30, 19] . Wavelet 

analysis has acquired popularity in recent 

years for numerical solutions of differential 

equations. The present study aims to formulate 

a Legendre wavelet collection method 

(LWCM)for high BVPs. The Legendre 

polynomials is used for the approximation of 

an unknown function and its derivatives. 

Legendre wavelet owes some important 

properties such as good interpolation, less 

computational cost, better accuracy with a 

smaller number of collection points. 

Motivated by these properties of the 

Legendre wavelet, we obtain approximate 

solutions of eight, tenth and twelfth order 

ordinary differential equation via LWCM in 

spectral mode was applied for the study of 

oscillatory type problems by Dizicheh. [5]. 

delay differential equation [32] , advection 

problems [10] , and lane-emden equation 

have been solved by LWCM. Authors in [25] 

have introduce operational matrix method 

using integration for Legendre wavelet. The 

same methodology has also been applied for 

solution of partial differential equation, 

integral and integro-differential equations, 

and fractional partial differential equations. 

Rest of the paper is organized as follows: in 

section 2 a short review of wavelet and 

Legendre wavelet is presented. Section 3 

gives a briefs description of the proposed 

LWCM. In section 4 error analysis is given. In 

section 5 the proposed method is used for the 

approximate solution of same high order BVPs. 

Section 6 is devoted to the conclusion.  

2. WAVELET 

    Wavelet arises and different applied 

sciences such as mathematics, quantum 

mechanics and engineering. Compact 

support, orthogonality, regularity, 

orthonormality, symmetry and good accuracy 

of approximation are the properties of the 

wavelets. A wavelet is a type of function that 

is made up of the dilation and translation of a 

single function known as the mother wavelet. 

[9]. Considered the following family of 

continuous wavelets 

 

 
ℱ (𝑥;  𝛼, 𝛽) = |𝛼|

−
1
2 ℱ(𝛼−1(𝑥 − 𝛽)), 

𝛼, 𝛽𝜖ℝ, 𝛼 ≠ 0, 
 (2.1) 

where α and β are positively dilation and 

translation parameter which very 

continuously. Let us assume  

𝛼 =  𝛼0
−𝛿  ,  𝛽 = 𝑚𝛽0𝛼0 

𝛿 , 𝛼0 > 1, 𝛽0 > 0, 

where 𝛿, 𝑚 belong to 𝕫+ then Eq. (2.1) takes 

the form 

ℱ (𝑥;  𝛿, 𝑚) = 𝛼
𝛿
2 ℱ(2𝛿 𝑥 − 𝑚𝛽0). 

In above equation ℱ (𝑥;  𝛿, 𝑚)  forms a 

wavelet basis for 𝐿2(ℝ). In orthonormal basis 

𝛼0 = 2 and 𝛽0 = 1 and is given by  

ℱ (𝑥;  𝛿, 𝑚) = 2
𝛿
2  ℱ(2𝛿 𝑥 − 𝑚). 

If  𝑛 = 0,1, . . ., 𝑁 − 1 and 𝑚 = 1, 2, . . ., 2𝛿−1 

then the Legendre wavelet are defined on 

the interval [0,1) as follows [24]: 

 

ℱ(𝑥; 𝑚, 𝑛)

= {
(𝑛 + 1)

1
22

𝛿
2 𝑄𝑛(𝑘)   for 2

1−𝛿 (𝑚 − 1) ≤ 𝑥 < 21−𝛿 𝑚

0,            otherwise
  
 

 

 

 

In the above definition 𝑘 = (2𝛿 𝑥 + 1 − 2𝑚), the 

coefficient (𝑛 +
1

2
)

1

2
 is for orthonormality and 

𝑄𝑛 (∗)  represent Legendre polynomial of 

degree 𝑛 which can be computed with help  

 

of the following recursion relation [37]  

𝑄0(𝑥) = 1,  𝑄1(𝑥) = 𝑥, 

(𝑛 + 1)𝑄𝑛+1(𝑥) − (2𝑛 + 1)𝑥𝑄𝑛 (𝑥) + 𝑛𝑄𝑛−1(𝑥) = 0,   𝑛 = 1,2, . . 



 

 
Haq et al., J. mt. area res. 06 (2021)63-76 

65 
J. mt. area res., Vol. 6, 2021 

3. DESCRIPTION OF THE TECHNIQUE 

    In this section we represent the procedure 

how the method is applied to ODE to find its 

approximate solution. It should be 

emphasized that the ODE can be transformed 

into a system of algebraic equations (linear or 

nonlinear) using collocation points, and the 

solution can then be found from these 

equations. For this purpose, let us consider the 

following nth order initial for boundary value 

problem. 

 

𝑈(𝑛)(𝑥) =

𝐹 (𝑥, 𝑈(𝑥), 𝑈′(𝑥), … , 𝑈(𝑛−1)(𝑥)) , 𝑎1 ≤ 𝑥 ≤

𝑎2, 

 (3.1) 

having the following initial (case i) or 

boundary (case ii, iii) conditions 

Case i 

 𝑈(𝐾−1)(𝑎1) = 𝛾𝑘 ,   𝑘 = 1, 2, 3, … , 𝑛  (3.2) 

Case ii 

 𝑈(𝐾−1)(𝑎𝑖 ) = 𝛾𝑘
𝑖 , 𝑖 = 1, 2, 𝑘 = 1, 2, 3, … ,

𝑛

2
.  (3.3) 

 Case iii 

 𝑈(𝐾−2)(𝑎𝑖 ) = X𝑘
𝑖 , 𝑖 = 1, 2, 𝑘 = 1, 2, 3, … ,

𝑛

2
.  (3.4) 

In the above equation 𝛾𝑘, 𝛾𝑘
𝑖 ,  X𝑘

𝑖 , are known 

constant and U is known function. Now the 

function 𝑈(𝑥)  can be approximated by the 

Legendre wavelet over interval [0, 1) [16] and 

is given as follows: 

 𝑈(𝑥) = ∑ ∑ 𝛽𝑖
𝑗

∞

𝑗=0

∞

𝑖=1

ℱ 𝑖
𝑗
(𝑥),  (3.5) 

where 𝛽𝑖
𝑗

 are unknown constants to be 

calculated. In actual computation Eq. (3.5) is 

written 

 𝑈(𝑥) = ∑ ∑ 𝛽𝑖
𝑗

𝑀−1

𝑗=0

2𝛿−1

𝑖=1

ℱ 𝑖
𝑗
(𝑥),  (3.6) 

which can be written as 

 𝑈(𝑥) = 𝐵ℱ (𝑥),  (3.7) 

where 𝐵 and ℱ are respectively row and 

Colum matrices defined as follow: 

𝐵 = [𝛽1
0,  𝛽1

1,  𝛽1
1, … , 𝛽1

𝑀−1,  𝛽2
0,  𝛽2

1, … , 𝛽2
𝑀−1, … , 𝛽

2𝑘−1
0 , 𝛽

2𝑘−1
1 , … , 𝛽

2𝑘−1
𝑀−1

], 

ℱ(𝑥) = [ℱ1
0(𝑥), ℱ1

1(𝑥), ℱ1
𝑀−1(𝑥), ℱ2

0(𝑥), … , ℱ2
𝑀−1(𝑥), … , ℱ

2𝑘−1
0 (𝑥), … , ℱ

2𝑘−1
𝑀−1(𝑥)]

𝑇
. 

From Eq. (3.7) we can write  

 𝑈𝑖 (𝑥) = 𝐵ℱ 𝑖 (𝑥), 𝑖 = 1, 2, … , 𝑛.  (3.8) 

Using Eq. (3.7) and (3.8) in Eq. (3.1)-(3.4) 

implies that  

 𝐵ℱ 𝑛(𝑥) = ℱ (𝑥, 𝐵ℱ(𝑥), 𝐵ℱ ′(𝑥), … , 𝐵ℱ (𝑛−1)(𝑥)).   (3.9) 

 𝐵ℱ (𝑘−1)(𝑎1) = 𝛾𝑘, 𝑘 = 1, 2, 3, … , 𝑛. (3.10) 

 𝐵ℱ
(𝑘−1)(𝑎𝑖 ) = 𝛾𝑘

𝑖 , 𝑖 = 1, 2, 𝑘 = 1, 2, 3, … , 𝑛
2
.  (3.11)  

 

𝐵ℱ (𝑘−1)(𝑎𝑖 ) = X𝑘
𝑖 , 𝑖 = 1, 2,  

𝑘 = 1, 2, 3, … ,
𝑛

2
. 

 (3.12) 

From Eq. (3.6) there are 𝑀2𝑘−1  constants to 

be determined. The calculation of these 

constants needs 𝑀2𝑘−1  equations. Now 𝑛 

number of equations can be derived from Eq. 

(3.10) or (3.11) or (3.12) and rest of 𝑀2𝑘−1 − 𝑛 

equations can be obtained from Eq. (3.9) 

using the collocation points and are given by 

 𝐵ℱ 𝑛(𝑥𝑗 ) = ℱ (𝑥𝑗 , 𝐵ℱ(𝑥𝑗 ), 𝐵ℱ ′(𝑥𝑗 ), … , 𝐵ℱ 𝑛−1(𝑥𝑗 )),   (3.13) 

where 

𝑥𝑗 =
𝑗−0.5

2𝑘𝑀−𝑛
,  𝑗 = 1, 2, … , 2𝑘−1𝑀 − 𝑛.                     (3.14) 

Therefore Eq. (3.13) coupled with Eq. (3.10) or 

(3.11) or (3.12) give system of 2𝑘−1𝑀  

equations and one can find the value of the 

unknown constants 𝛽𝑖
𝑗
. Using the value of 𝛽𝑖

𝑗
 

in Eqs. (3.6) will give the required solution. 

4. ERROR ANALYSIS 

    Theorem (Sohaib and Haq [7]): Assume 

that 𝑈(𝑥)  and 𝑈∗(𝑥)  be the exact 

approximate solution of Eq. (3.1) respectively. 

Moreover, also assume that |𝑈′(𝑥)| ≤ 𝐵1  and 

|𝑈′′(𝑥)| ≤ 𝐵2,  where 𝐵1  and 𝐵2  are constant 

than  

‖𝐸‖2 ≤ ∑ ∑
3𝐵2

2

2𝑛5(2𝑚 − 3)4

∞

𝑚=𝑀

∞

𝑛=2𝑘−1

. 

5. NUMERICAL EXAMPLES 

In this section the proposed method (LWCM) is 

used to find the approximate solution of linear 



 

 

Haq et al., J. mt. area res. 06 (2021)63-76 

66 
J. mt. area res., Vol. 6, 2021 

and nonlinear eight, tenth, and twelfth order 

boundary value problems. To determine the 

technique's efficacy, the results are 

compared to those obtained using other 

methods available in the literature. 

5.1 EIGHT ORDER ODEs 

Example 1: Let us consider the following linear 

eight order boundary value problem [26] 

 
𝑈(8)(𝑥) −  𝑈(𝑥) =  −8𝑒 𝑥  ,     0 < 𝑥

< 1, 
 (5.1) 

with boundary condition 

 𝑈(𝑗)(0) =  1 − 𝑗,   𝑈(𝑘)(1) =  −𝑘𝑒,

𝑗 = 0, 1, … ,5  and    𝑘 = 1, 2. 

The exact solution of problem (5.1) is given by 

𝑈(𝑥) =  (1 − 𝑥)𝑒 𝑥 . 

Solution: Taking 𝛿 = 1 and 𝑀 =  17, Eq.  

(3.6) become 

 𝑈(𝑥) =  ∑ 𝛽1
𝑗

16

𝐽=0

ℱ1
𝑗
(𝑥) = 𝐵ℱ(𝑋),  (5.2) 

where 𝐵 and ℱ are 

 
𝐵 = [𝛽1 

0 , 𝛽1
1, … , 𝛽1

16]    

  ℱ(𝑥) = [ℱ1
0, ℱ1

1, … , ℱ1
16]𝑇 

  

Using Eq. (5.1) and boundary conditions we 

obtain 

 𝐵ℱ (8)(𝑥) −  𝐵ℱ(𝑥) =  −8𝑒 𝑥    (5.3) 

and  

 

𝐵ℱ (𝑗)(0) = 1 − 𝑗,   𝐵ℱ (𝑘)(1)

=  −𝐾𝑒,    

𝑗 = 0, 1, 2, … , 5   and    𝐾 = 1, 2.                

 At 𝑥 =  𝑥𝑗  (see Eq.  (3.14) the above 

equation becomes 

 
𝐵ℱ 8(𝑥𝑗 ) −  𝐵ℱ(𝑥𝑗 ) =  −8𝑒

𝑥𝑗  ,  

   𝑗 = 1, 2, … , 9 
 (5.4) 

Where 𝑥𝑗  are the collection points given by 

Eq. (3.14). Eq. (5.4) together with the boundary 

conditions give system of seventeen 

equations in seventeen unknowns. Solving the 

system so obtained the unknown is given as 

follow: 

𝛽1
0 = 7.1828 × 10−1, 𝛽1

1 = −2.682 × 10−1,   𝛽1
2

= −9.6049 × 10−2, 

𝛽1
3 = −1.3310 × 10−2,   𝛽1

4 = −1.1655 ×  10−3,  

  𝛽1
5 = −7.5008 × 10−5, 𝛽1

6 = −3.82352 × 10−6,  

  𝛽1
7 = −1.61514 × 10−7,   𝛽1

8 = −5.82777 × 10−9, 

𝛽1
9 = −1.83571 × 10−10, 𝛽1

10 = −5.13164 × 1012,  

𝛽1
11 = −1.28957 × 10−13, 𝛽1

12 = −2.94351 × 1015,    

𝛽1
13 = −6.15476 × 10−17,   𝛽1

14 = −1.18679 × 10−18, 

              𝛽1
15 = −2.11898 × 10−20,   𝛽1

16

= −3.90587 × 10−22. 

Putting value in 𝛽1
𝑗
 in Eq. (5.2), the solution of 

the problem (5.1) is  

𝑈(𝑥) = 1 − 2.22 × 10−16𝑥 − 0.5𝑥2 − 0.33𝑥 3 − 0.125𝑥 4 − 0.03𝑥5 

−0.007𝑥6  − 0.001𝑥7 − 0.0002𝑥 8 − 0.00002𝑥 9 − 2.84 × 10−6𝑥10 

−2.51 × 10−7𝑥11 − 2.3 × 10−8𝑥12 − 1.91 × 10−9𝑥13 

−1.54 × 1010𝑥14 − 8.57 × 10−12𝑥15 − 1.23 × 10−12𝑥16. 

The result obtained and comparison with the 

method in [2, 8, 12] are given in table 1. From 

the table it can be observed that the result 

got using LWCM are more accurate than that 

other method. As we increase the value of the 

M the accuracy become better. This show 

accuracy of the present technique.  

Example 2: Let us consider another eight order 

BVP [33] given as follow: 

 
𝑈(8)(𝑥) + 𝑥𝑈(𝑥) = −𝑒 𝑥 (48 +  15𝑥 + 𝑥 3),

0 < 𝑥 < 1,    
 (5.5) 

with the following boundary conditions 

 
𝑈𝑘 (0) = 𝑘 − (2 − 𝑘)  

𝑈𝑘 (1) = −𝑘2𝑒  for  𝑘 = 0, 1, 2, 3. 
 (5.6) 

The theoretical solution of problem (5.5) -(5.6) 

is  

 𝑈(𝑥) = 𝑥(1 − 𝑥)𝑒 𝑥 .  (5.7) 

Solution: As in example 1 when 𝛿  = 1 and 

𝑀 =  17 then Eq. (3.6) transform to Eq. (5.2) 

the use of Eu. (5.2) convert the Eq. (5.5) and 

(5.6) to the following equations 

 𝐵ℱ 8(𝑥) + 𝑥𝐵ℱ(𝑥) = −𝑒 𝑥 (48 + 15 + 𝑥 3),   (5.8) 

and 

 
𝐵ℱ (𝑘)(0) = 𝑘(2 − 𝑘)  

 𝐵ℱ (𝑘)(1) = −𝐾2𝑒,   𝑘 = 0,1,3.  
 (5.9) 

At the collection points 𝑥𝑗  (see Eq. (3.14)), the 

Eq. (5.8) yields 



 

 
Haq et al., J. mt. area res. 06 (2021)63-76 

67 
J. mt. area res., Vol. 6, 2021 

𝐵ℱ (8)(𝑋𝐽 ) + 𝑥𝐵ℱ(𝑥𝑗 ) = −𝑒
𝑥𝑗 (48 + 15𝑥𝑗 + 𝑥𝑗

3), 𝑗 = 1,2, … ,9 (5.10) 

Eq. (5.10) together with Eq. (5.9) gives 

seventeen equations. Solving these equations 

gives the constant given by  

 𝛽1
0 = 2.817 × 10−1, 𝛽1

1 = 4.84503 × 10−2, 

   𝛽1
2 = −1.20648 × 10−1, 𝛽1

3 = −3.13013 × 10−2,   

 𝛽1
4 = −3.95534 ×  10−3,   𝛽1

5 = −3.31312 × 10−4, 

 𝛽1
6 = −2.07696 × 10−5,   𝛽1

7 = −1.04055 × 10−6,  

  𝛽1
8 = −4.34174 × 10−8, 𝛽1

9 = −1.55226 × 10−9,   

 𝛽1
10 = −4.85482 × 1011,   𝛽1

11 = −1.34945 × 10−12, 

 𝛽1
12 = −3.37545 × 1014,   𝛽1

13 = −7.67494 × 10−16,    

𝛽1
14 = −1.59956 × 10−17, 𝛽1

15 = −3.08249 × 10−19,  

  𝛽1
16 = −5.50016 × 10−21, 

Substituting the value of the unknown 𝛽1
𝑗
 in 

Eq. (5.2) the solution of the problem (5.8) -(5.9) 

can be obtained. In table 2 the absolute error 

using the proposed method is compared with 

those of the method in [2, 27, 14] available in 

the literature. The results obtained using the 

proposed method are clearly superior to those 

obtained using the other methods, as shown 

in the table. The accuracy is increase as we 

increase the value of 𝑀 . This show that the 

LWCM can be applied to the differential 

equation for better results. 

Example 3: Let us consider another eight-order 

boundary value problem taken from [33] and 

is given below: 

𝑈8(𝑥) +  𝑈7(𝑥) + 2𝑈(6)(𝑥) + 2𝑈(5)(𝑥) +  2𝑈(4)(𝑥) +  2𝑈(3)(𝑥) 

   +2𝑈"(𝑥) + 𝑈′(𝑥) +  𝑈(𝑥)

= 14 cos −16 sin −4𝑥 sin 𝑥,   0 < 𝑥

<  1 (5.10) 

with the boundary condition 

 

𝑈(𝑘)(0) =
2𝑘3 − 3𝑘2 − 2𝑘

3
 

𝑈(𝑘)(1) = (𝑘 + 2𝑘2 − 𝑘3) sin 1 − (4𝑘

− 5𝑘2 + 𝑘3) cos 1,   

𝑘 = 0,1,2,3. 

 (5.11) 

The exact solution of the problem (5.10) -

(5.11) is 

𝑈(𝑥) = (𝑥 2 − 1) sin 𝑥. 

Using 𝛿 = 1 and 𝑀 =  17 Eq. (3.6) takes the 

form given by Eq. (5.4) and plugging the value 

from Eq. (5.4) into Eqs. (5.10) -(5.11) the 

following can be obtained 

𝐵{ℱ (8)(𝑥) + ℱ (7)(𝑥) + 2ℱ (6)(𝑥) +  2ℱ (5)(𝑥) + 2ℱ (4)(𝑥)

+ 2ℱ (3)(𝑥) + 2ℱ (2)(𝑥) + ℱ ′(𝑥) + ℱ(𝑥)} 

= 14 cos −16 sin −4𝑥 sin 𝑥,                                                            (5.12) 

 

𝐵ℱ (𝑘)(0) =
2𝐾3 − 3𝐾2 − 2𝐾

3
,  

𝐵ℱ (𝑘)(1) = (𝑘 + 2𝑘2 − 𝑘3) sin 1 − (4𝑘

− 5𝑘2

+ 𝑘3) cos 1,

𝑘 = 0,1,2,3. 

 (5.13) 

Utilizing the collection points 𝑥𝑘  (given by Eq.  

(3.14)) in Eq. (5.12) we arrived at 

𝐵{ℱ (8)(𝑥𝑘 ) + ℱ
(7)(𝑥𝑘 ) + 2ℱ

(6)(𝑥𝑘 ) + 2ℱ
(5)(𝑥𝑘 ) + 2ℱ

(4)(𝑥𝑘 )

+ 2ℱ (3)(𝑥𝑘 ) + 2ℱ
(2)(𝑥𝑘 ) + ℱ

(′)(𝑥𝑘 ) + ℱ(𝑥𝑘 )}

= 14 cos 𝑥𝑘 − 16 sin 𝑥𝑘 − 4𝑘𝑥 sin 𝑥𝑘, 

𝑘 = 0,1,2, … ,8.                                     (5.14) 

Eq. (5.14) and (5.13) generates a system of 

seventeen equation in seventeen unknowns 

which can be easily solved for the unknown 

𝐵1,
𝐽

  J = 0, 1, 2,…,16. The value of these 

unknowns is computed, and the required 

solution has been obtained from (5.2). The 

absolute error has been computed with those 

of quintic B-spline method (QBSM) [33] 

available in the literature and are tabulated in 

table 3. From the table it is straight forward 

that the result of the present method is very 

good as compared to other technique. 

Example 4: Let us consider the following 

nonlinear eight order BVP [11] 

 𝑈(8)(𝑥) = 𝑒−𝑥 𝑈2(𝑥),    0 < 𝑥 < 1  (5.15) 

Subject to the boundary condition 

 
𝑈(𝑘)(0) = 1  𝑈(𝑘) (1) = 𝑒𝑥,    𝑘

= 0,2,4,6. 
 (5.16) 

Exact solution of the problem (5.15) -(5.16) is  



 

 

Haq et al., J. mt. area res. 06 (2021)63-76 

68 
J. mt. area res., Vol. 6, 2021 

𝑈(𝑥) = 𝑒 𝑥 . 

Using 𝛿 = 1 and 𝑀 =  17 in Eq. (3.6) which 

becomes 

 𝑈(𝑥) = ∑ 𝛽1
𝑗
ℱ1

𝑗
(𝑥) = 𝐵ℱ(𝑥)

16

𝑗=𝑜

  (5.17) 

The use of Eq. (5.17) in Eq. (5.15) and (5.16) 

gives 

 𝐵ℱ (8)(𝑥) = 𝑒 −𝑥 𝐵2ℱ 2(𝑥),   (5.18) 

 

 
𝐵ℱ (𝑘)(0) = 1,   𝐵ℱ (𝑘)(1) = 𝑒    

𝑘 = 0, 2, 4, 6. 
 (5.19) 

Eq. (5.18) at 𝑥 = 𝑥𝑗 (see Eq. (3.14)) gives the 

following system 

 
𝐵ℱ (8)(𝑥𝑗 ) = 𝑒

−𝑥𝑗 𝐵2ℱ (2)(𝑥𝑗 ),   

 𝑗 = 1, 2, … ,9. 
 (5.20) 

Value of seventeen unknown 𝛽1
𝑗 
 ̕s has been 

computed from the system of seventeen 

equations given by Eqs. (5.19) and (5.20). The 

solution of the problem has been obtained 

from the substitution of 𝛽1
𝑗 
 ̕in Eq. (5.17). The 

results, absolute error and comparison with 

the method in [2, 21, 12] are given in table 4. 

From the table the result of the present 

method is very good as compared to the 

other method of literature. The increase in the 

value of 𝑀 gives us more accurate results. 

5.2 TENTH ORDER ODEs 

In this section the examples of tenth order 

ODEs have been chosen for study. The 

method under consideration is applied to 

check the applicability and the accuracy of 

the technique. 

Example 5: Taking the linear tenth order ODEs 

[23] 

 

𝑈(10)(𝑥) + 5𝑈(𝑥) = 10 cos 𝑥 + 4(1 − 𝑥) sin 𝑥 , 0 < 𝑥 < 1, (5.21) 

with the boundary conditions 

𝑈(2𝑘)(𝜂) = (−1)𝐾+12𝑘 cos 𝜂 ,

𝜂 = 0, 1  𝑘 = 0, 1, 2, 3, 4. (5.22) 

Exact solution of the problem is given below  

𝑈(𝑥) = (𝑥 − 1) sin 𝑥. 

The problem has been solved using the same 

methodology as discussed in the previous 

examples. Absolute errors for differential 

values of 𝑀 have been calculated and are 

shown in table 5. From table the results 

obtained are very close to exact solution. 

Furthermore, accuracy can be enhanced by 

increasing the value of 𝑀.  

Example 6: Taking nonlinear BVP of order ten 

[31] 

𝑈(10)(𝑥) = 𝑒 −𝑥 𝑈2(𝑥),   0 < 𝑥 < 1,                     

(5.21) 

subject to the following boundary condition 

𝑈2𝑘 (𝜂) = 𝑒𝜂 ,   𝜂 = 0,1,   𝑘 = 0, 1, 2, 3, 4. 

The exact solution is 𝑈(𝑥) = 𝑒 𝑥. For the solution 

of this problem the proposed technique is 

applied when δ =  1, M =  19, 17, 15, 13, 11.  

The obtained results are tabulated in table 6 

and are compared with those of new iterative 

method (NIM) [31]. From the table the result of 

LWCM is better than that of NIM. 

5.3 TWELFTH ORDER ODEs 

Example 7: Consider twelfth order ODE [17] 

𝑈12(𝑥) + 𝑥𝑈(𝑥) = −(120 + 23𝑥 + 𝑥 3)𝑒 𝑥 ,   0 < 𝑥 < 1,   

(5.22) 

coupled with the following boundary 

condition  

𝑈(𝑘)(𝜂) = 𝑘(2(1 − 𝜂) − 𝑘)𝑒𝜂 ,   𝜂 = 0,1,   𝑘 = 0, 1, 2, 3, 4, 5. 

Exact solution is 𝑈(𝑥) = 𝑥(1 − 𝑥)𝑒 𝑥 .  The same 

procedure as discussed in previous example 

with 𝛿 =  1  and 𝑀 =  18, 16, 14  is 



 

 
Haq et al., J. mt. area res. 06 (2021)63-76 

69 
J. mt. area res., Vol. 6, 2021 

implemted here also. The obtained numerical 

results and comparison of the error with DTM 

[13] is tabulated in table 7. Here again the 

results are more accurate than those of DTM.  

Example 8: Another example of twelfth order 

BVP of the form [22] 

𝑈(12)(𝑥) = 2𝑒 𝑥 𝑈2(𝑥) + 𝑈(3)(𝑥),   0 < 𝑥 < 1,            

(5.23) 

subject to the boundary conditions 

𝑈(2𝑘)(𝜂) = 𝑒 −𝜂 ,   𝜂 = 0,1,   𝑘 = 0, 1, 2, 3, 4, 5. 

The exact solution of the problems is 𝑈(𝑥) =

𝑒 −𝑥 .  The solution of the problems has been 

obtained with the help of the same method 

taking 𝛿 = 1  and 𝑀 =  18, 16, 14 . The results 

so obtained are posted in the table 8 and are 

compared with the results of DTM [13] which 

are available in literature. Again, the results 

with LWCM are better. 

 

 

Table 1: Absolute error comparison of LWCM when k=1 

𝑥𝑖  
LWCM   RKS [2]  HPM [8] VIM [12] 

 𝑀 = 13 𝑀 = 15 𝑀 = 17 𝑛 = 7 𝑛 = 30 𝑁 = 7  

0.25 1.38E-13 3.33E-16 2.22E-16 3.03E-10 7.50E-12 2.16E-09 4.58E-09 

0.50 4.78E-12 6.77E-15 0.00E-00 7.73E-09 2.35E-10 1.16E-07 9.84E-09 

0.75 1.91E-11 2.55E-14 2.22E-16 3.12E-08 1.08E-09 1.05E-06 1.10E-05 

1.0 2.63E-11 3.48E-14 4.57E-16 4.40E-08 1.59E-09 4.22E-06 1.86E-04 

 

Table 2: Absolute error comparison of LWCM when k=1 

𝑥𝑖 LWCM         RKS [2] Method in 

[27] 

Method in 

[14] 

  𝑀 = 17 𝑀 = 15 𝑀 = 13 𝑀 = 11 𝑀 = 9 𝑁 = 10     

0.1 1.25E-16 8.33E-17 8.13 E-14 8.72 E-11 5.57 E-09 1.63E-10 5.62 E-10 3.73 E-09 

0.2 1.67E-16 3.61 E-16 4.33 E-13 6.70 E-10 3.57 E-08 1.63 E-09 4.88 E-09 6.61 E-09 

0.3 5.55E-17 4.44 E-16 4.90 E-13 1.24 E-09 4.56 E-08 4.90 E-09 1.37 E-08 2.33 E-08 

0.4 5.55E-17 2.78 E-16 1.19 E-13 6.53 E-10 2.79 E-08 8.46 E-09 2.29 E-08 5.17 E-08 

0.5 0.00000 0.000000 2.55 E-13 1.16 E-09 1.60 E-07 1.01 E-09 2.71 E-08 9.76 E-08 



 

 

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70 
J. mt. area res., Vol. 6, 2021 

0.6 2.22E-16 2.78 E-16 5.89 E-13 2.47 E-09 2.47 E-07 8.68 E-09 2.38 E-08 1.78 E-06 

0.7 3.33E-16 3.33 E-16 8.34 E-13 2.31 E-09 2.12 E-07 5.15 E-09 2.38 E-08 4.12 E-06 

0.8 3.33E-16 2.78 E-16 5.89 E-13 1.03 E-09 9.44 E-08 1.76 E-09 5.54 E-09 1.83 E-04 

 

Table 3: Absolute error comparison of LWCM when k=1 

𝑥𝑖 LWCM         QBSM [33] 

  𝑀 = 17 𝑀 = 15 𝑀 = 13 𝑀 = 11 𝑀 = 9  

0.1 2.25E-15 7.77E-16 4.37E-14 4.94E-11 3.99E-09 2.68E-07 

0.2 1.97E-15 1.55E-15 2.31E-13 3.84E-10 2.86E-08 9.24E-07 

0.3 1.33E-15 2.11 E-15 2.64E-13 7.45E-10 5.22E-08 1.94E-06 

0.4 2.78E-16 2.33E-15 7.77E-14 5.20E-10 3.40E-08 3.52E-06 

0.5 1.05E-15 2.44E-15 1.08E-13 3.06E-10 2.37E-08 4.44E-06 

0.6 2.28E-15 2.55E-15 2.73E-13 1.02E-09 7.33E-08 4.68E-06 

0.7 2.94E-15 3.22E-15 3.94E-13 1.01E-09 7.36E-08 4.26E-06 

0.8 3.33E-15 4.11E-15 2.78E-13 4.57E-10 3.49E-08 2.86E-06 

0.9 3.16E-15 5.30E-15 4.77E-14 5.46E-11 4.48E-09 1.28E-06 

 

Table 4: Absolute error comparison of LWCM when k=1 

𝑥𝑖 LWCM         RKS [2]  Method in [21] Method in [14] 

  𝑀 = 17 𝑀 = 15 𝑀 = 13 𝑀 = 11 𝑀 = 9 𝑛 = 10 𝑛 = 20     

0.1 0.000000 0.000000 2.22E-16 3.77E-15 1.58E-13 7.53E-08 1.61E-08 1.27E-05 1.91E-07 

0.2 0.000000 2.22E-16 5.11E-15 8.84E-13 3.97E-11 1.43E-07 3.07 E-08 2.43E-05 1.25E-07 

0.3 2.22E-16 2.22E-16 1.12E-13 2.04E-11 10.00E-10 1.96E-07 4.23 E-08 3.35E-05 7.25E-08 

0.4 0.000000 2.22E-15 9.45E-13 1.84E-10 9.79E-09 2.31E-07 4.97 E-08 3.94E-05 4.85E-08 

0.5 0.000000 1.15E-14 4.82E-12 9.93E-10 5.72E-08 2.42E-07 5.23 E-08 4.16E-05 2.91E-07 

0.6 4.44E-16 4.31E-14 1.79E-11 3.86E-09 2.41E-07 2.30E-07 4.98 E-08 3.96E-05 7.80E-08 

0.7 4.44E-16 1.28E-13 5.40E-11 1.20E-08 8.08E-07 1.95E-07 4.24 E-08 3.38E-05 1.11E-07 

0.8 4.44E-16 3.29E-13 1.39E-10 3.17E-08 2.30E-06 1.42E-07 3.08 E-08 2.45E-05 1.71E-07 



 

 
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71 
J. mt. area res., Vol. 6, 2021 

0.9 4.44E-16 7.54E-13 3.20E-10 7.41E-08 5.75E-06 7.47E-08 1.62 E-08 1.29E-05 7.93E-08 

 

Table 5: Absolute error comparison of LWCM when k=1 

𝑥𝑖 LWCM         

  𝑀 = 19 𝑀 = 17 𝑀 = 15 𝑀 = 13 𝑀 = 11 

0.1 2.36E-16 1.03E-15 8.77E-13 7.32E-10 1.05E-06 

0.2 1.94E-16 1.50E-15 1.68E-12 1.40E-09 2.00E-06 

0.3 2.22E-16 1.92E-15 2.34E-12 1.94E-09 2.76E-06 

0.4 1.94E-16 2.08E-15 2.78E-12 2.30E-09 3.26E-06 

0.5 1.94E-16 2.25E-15 2.97E-12 2.45E-09 3.43E-06 

0.6 1.39E-16 2.14E-15 2.86E-12 2.35E-09 3.27E-06 

0.7 1.11E-16 1.80E-15 2.46E-12 2.02E-09 2.79E-06 

0.8 5.55E-17 1.22E-15 1.81E-12 1.48E-09 2.03E-06 

0.9 1.39E-17 4.86E-16 9.57E-13 7.81E-10 1.07E-06 

 

Table 6: Absolute error comparison of LWCM when k=1 

𝑥𝑖 LWCM         NIM [31] 

  𝑀 = 19 𝑀 = 17 𝑀 = 15 𝑀 = 13 𝑀 = 11  

0.1 6.66E-16 6.66E-16 9.50E-13 9.09E-11 1.62E-07 4.11E-09 

0.2 1.78E-15 1.55E-15 1.84E-12 1.75E-10 3.09E-07 7.81E-09 

0.3 2.22E-15 1.77E-15 2.59E-12 2.45E-10 4.28E-07 1.08E-08 

0.4 3.11E-15 2.89E-15 3.13E-12 2.94E-10 5.06E-07 1.27E-08 

0.5 3.11E-15 2.66E-15 3.39E-12 3.17E-10 5.35E-07 1.33E-08 

0.6 3.11E-15 2.66E-15 3.32E-12 3.08E-10 5.12E-07 1.27E-08 

0.7 2.66E-15 2.22E-15 2.90E-12 2.68E-10 4.38E-07 1.08E-08 

0.8 2.66E-15 1.78E-15 2.15E-12 1.97E-10 3.20E-07 7.85E-09 

0.9 1.33E-15 4.44E-16 1.15E-13 1.05E-11 1.69E-07 4.12E-09 

 



 

 

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72 
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Table 7: Absolute error comparison of LWCM when k=1 

𝑥𝑖 LWCM     DTM [13] 

  𝑀 = 18 𝑀 = 16 𝑀 = 14  

0.1 1.25E-16 9.71E-17 7.49E-15 7.51E-14 

0.2 1.11E-16 0.000000 2.37E-13 2.77E-12 

0.3 5.55E-17 7.77E-16 1.23E-12 1.73E-11 

0.4 0.000000 2.05E-15 2.76E-12 5.02E-11 

0.5 5.55E-17 2.72E-15 3.54E-12 9.34E-11 

0.6 5.55E-17 2.00E-15 2.77E-12 1.28E-10 

0.7 5.55E-17 7.77E-16 1.24E-12 1.39E-10 

0.8 1.11E-16 1.11E-16 2.39E-13 1.23E-10 

0.9 1.67E-16 4.44E-16 8.22E-13 7.50E-11 

 

Table 8: Absolute error comparison of LWCM when k=1 

𝑥𝑖 LWCM     DTM [13] 

  𝑀 = 18 𝑀 = 16 𝑀 = 14  

0.1 3.55E-15 3.05E-12 1.94E-09 1.61E-07 

0.2 6.77E-15 5.80E-12 3.70E-09 3.07E-07 

0.3 9.21E-15 7.99E-12 5.09E-09 4.22E-07 

0.4 1.09E-14 9.39E-12 5.98E-09 4.97E-07 

0.5 1.15E-14 9.87E-12 6.29E-09 5.22E-07 

0.6 1.08E-14 9.39E-12 5.98E-09 4.96E-07 

0.7 9.27E-15 7.99E-12 5.09E-09 4.22E-07 

0.8 6.77E-15 5.80E-12 3.70E-09 3.07E-07 

0.9 3.72E-15 3.05E-12 1.94E-09 1.61E-07 



73 
J. mt. area res., Vol. 6, 2021 

6. CONCLUSION 

The Legendre wavelet collocation method is 

used to solve linear and nonlinear boundary 

value problems of orders eighth, tenth, and 

twelve in this work. The results obtained are 

compared with the results of QBSM, NIM, DTM 

and other method in [2, 8, 12, 27] from the 

available literature. It has been observed that 

the results of the proposed method are 

efficient, accurate, easy to apply, and needs 

less computational cost. The technique can 

be applied to partial [38], integral [39], and 

fractional differential equations [40] easily.   

 

DECLARATIONS 

Funding: Not applicable 

Conflicts of interest/Competing interests: 

The authors declare no conflict of interest  

Data availability: Not applicable 

Code availability: Not applicable 

Authors’ contributions:  

Siraj ul Haq: Original draft write up and Supervision 

Muhammad Sohaib: Conceptualization, 

Methodology, Reviewing and Editing.  

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 This work is licensed under a Creative Commons Attribution 4.0 International License. 

Received: 26 Aug. 2021. Revised/Accepted: 13 Dec. 2021. 

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