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A powerful numerical technique for treating twelfth-order boundary value problems

  • Rohul Amin , Kamal Shah , Imran Khan , Muhammad Asif , Kholod M. Abualnaja , Emad E. Mahmoud and Abdel-Haleem Abdel-Aty EMAIL logo
Published/Copyright: December 24, 2020
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Open Physics
From the journal Open Physics Volume 18 Issue 1

Abstract

In this article, a fast algorithm is developed for the numerical solution of twelfth-order boundary value problems (BVPs). The Haar technique is applied to both linear and nonlinear BVPs. In Haar technique, the twelfth-order derivative in BVP is approximated using Haar functions, and the process of integration is used to obtain the expression of lower-order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking the convergence of the proposed technique. A comparison of the results obtained by the present technique with results obtained by other techniques reveals that the present method is more effective and efficient. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The convergence rate using different numbers of collocation points is also calculated, which is approximately equal to 2.

Keywords: boundary value problem; Gauss elimination method; collocation method; Haar wavelet

1 Introduction

Boundary value problems (BVPs) with higher order arises in the field of astrophysics, hydrodynamics and hydro-magnetic stability, fluid dynamics, astronomy, beam and long wave theory, and applied physics and engineering. Many researcher studied higher-order BVPs because of mathematical importance and their uses in various field of applied sciences [1,2,3,4,5]. These equations modeled physics of stability problems in hydrodynamics. In the presence of magnetic field and in the direction of gravity, when an infinite horizontal layer of fluid is heated from below, subject to the axis of rotation, instability takes place inside. When this phenomena of instability take place as ordinary convection, then it is modeled by tenth-order BVPs [1], and when the aforesaid instability take place as over stability, then this phenomena is modeled using twelfth-order BVPs [2]. Because of the widespread applications of higher-orders BVPs, many researchers showed more interest in the numerical solution of these equations. Bishop et al. [6] modeled the phenomena of torsional vibration of beams as eighth-order BVP. Siddiqi and Akram [7,8] introduced nonic spline and non-polynomial spline methods for the numerical solution of special linear eighth-order BVP models. In ref. [7,8], Siddiqi also proved that the convergence of the aforementioned techniques is of second order. Wazwaz [9] established an efficient technique using the Adomian decomposition technique to solve some special eighth-order BVPs numerically. Recently, a numerical technique based on polynomial splines of degree six was introduced by Siddiqi and Twizell in ref. [10] for the numerical solution of some special types of eighth-order BVPs. However, it was found that at the points adjacent to the boundary, the method diverges. Further, in ref. [11,12], Twizell et al. observed the same problems by solving some higher-order BVPs numerically. It was investigated that the results diverge because of the utilization of test functions with lower order in the aforesaid technique. In ref. [10], the authors applied differential quadrature methods (DQM) that use the higher-order test functions in the entire domain. However, the shortcoming of the DQM method is that it dealt only with second-order BVPs. For higher-order BVPs, a δ -point method should be used [11]. The details of improvement and applications of the DQM technique can be seen in ref. [10,11]. The current improvement in the DQM leads to the establishment of generalized DQM [13,14]. Theoretically, the generalized DQM can be used for any higher-order BVPs coupled with conventional δ -point method. In structure mechanics, the generalized DQM is used for the numerical solution of fourth- and sixth-order BVPs [14,15,16,17,18]. Furthermore, in fluid mechanics, it is used for the numerical solution of linear Onsager and nonlinear Blasius problems with third and sixth orders, respectively [13]. The initial value problems of second, third, and fourth orders are also solved using the aforesaid method [14]. The generalized DQM is extended by Liu et al. in ref. [19] for the numerical solution of eighth-order BVPs. In ref. [20], a reproducing kernel method is introduced by Geng et al. for the solution linear tenth-order BVPs. A variational iterative method is used by Siddiqi et al. for the numerical solution of tenth-order BVPs in ref. [21]. Akram et al. developed a numerical technique based on eleventh degree spline for linear special case of BVPs [22]. A tenth degree spline was used by Twizell et al. for the numerical solution of tenth-order BVPs and faced some problems in getting results near boundaries of the interval in ref. [23]. A new numerical technique was introduced by Boutayeb et al. for the solution of twelfth-order BVPs that arise in thermal instability [30]. Siddiqi et al. used tenth and twelfth degree spline for the numerical simulation of tenth- and twelfth-order BVPs [12,23]. Mustahsan et al. [24] used higher-order B-spline differential quadrature rule to approximate the generalized Rosenau-RLW equation. Harim et al. [25] gives a new scheme for positivity preserving interpolation using rational quartic spline of (quartic/quadratic) with three parameters. Tassaddiq et al. [26] found the numerical solution of the nonlinear differential equations arising in the study of hydrodynamics and hydro-magnetic stability problems using a new cubic B-spline scheme. Ghaffar et al. [27] proposed and analyzed a tensor product of nine-tic B-spline subdivision scheme to reduce the execution time needed to compute the subdivision process of quad meshes. Mustafa et al. [28] developed a numerical approach for solving second-order singularly perturbed BVPs. Khalid et al. [29] used the cubic B-splines to find the numerical solution of linear and nonlinear eighth-order BVPs. There is a meager literature available on the numerical simulation of twelfth-order BVPs and the related eigenvalue problems. Agarwal et al. worked on the existence and uniqueness of twelfth-order BVPs [30].

In this article, we developed a collocation method based on Haar wavelet for the numerical simulation of twelfth-order BVPs. The following nonlinear problem of twelfth order will be considered in this article:

(1) u ( 12 ) ( t ) = F ( t , u , u ( 1 ) , u ( 2 ) , u ( 3 ) , u ( 4 ) , u ( 5 ) , u ( 6 ) , u ( 7 ) , u ( 8 ) , u ( 9 ) , u ( 10 ) , u ( 11 ) ) 0 ≤ t ≤ 1 ,

where F is given function, whereas in case of linear, the following general form is considered:

(2) u ( 12 ) ( t ) + f ( t ) u ( t ) = g ( t ) , t ∈ [ 0 , 1 ] ,

with the following BCs:

(3) u ( 0 ) = α 0 , u ( 1 ) = β 0 u ( 1 ) ( 0 ) = α 1 , u ( 1 ) ( 1 ) = β 1 , u ( 2 ) ( 0 ) = α 2 , u ( 2 ) ( 1 ) = β 2 , u ( 3 ) ( 0 ) = α 3 , u ( 3 ) ( 1 ) = β 3 , u ( 4 ) ( 0 ) = α 4 , u ( 4 ) ( 1 ) = β 4 , u ( 5 ) ( 0 ) = α 5 , u ( 5 ) ( 1 ) = β 5 ,

where f ( t ) and g ( t ) are known functions.

This article is organized as follows: Haar functions are defined in Section 2. Numerical Haar wavelet collocation (HWC) technique for the solution of both nonlinear and linear twelfth-order BVPs is given in Section 3. In Section 4, some problems from literature are given for the validation of HWC method. Conclusion is given in the final Section 5.

2 Haar wavelet

The Haar functions are piecewise constant functions having three values 1, − 1 , and 0. The Haar scaling function on interval [ γ 1 , γ 2 ) is given by [31]:

(4) h 1 ( t ) = 1 for t ∈ [ γ 1 , γ 2 ) , 0 otherwise .

The mother wavelet for the HW functions on [ γ 1 , γ 2 ) is

(5) h 2 ( t ) = 1 for t ∈ γ 1 , γ 1 + γ 2 2 , − 1 for t ∈ γ 1 + γ 2 2 , γ 2 , 0 otherwise .

All the other terms in the HW series can be represented in t ∈ [ ρ 1 , ρ 2 ) except the scaling function

(6) h i ( t ) = 1 for t ∈ [ ρ 1 , ρ 2 ) , − 1 for t ∈ [ ρ 2 , ρ 3 ) , 0 otherwise ,

where

ρ 1 = γ 1 + ( γ 2 − γ 1 ) ζ m , ρ 2 = γ 1 + ( γ 2 − γ 1 ) ζ + 0.5 m , ρ 3 = γ 1 + ( γ 2 − γ 1 ) ζ + 1 m ,

where integer m = 2 r and r = 0 , 1 , … , r ′ , and let the integer ζ = 0 , 1 , … , m − 1 . The number i can be obtained as i = m + ζ + 1 . In the interval [ 0 , 1 ] , ρ 1 , ρ 2 , and ρ 3 are defined as:

(7) ρ 1 = ζ m , ρ 2 = ζ + 0.5 m , ρ 3 = ζ + 1 m .

Any function of L 2 [ 0 , 1 ) space of square integrable function is expressed as:

(8) u ( t ) = ∑ i = 1 ∞ λ i h i ( t ) .

This series is truncated at finite N terms for approximation purpose, i.e.

u ( t ) ≈ ∑ i = 1 N λ i h i ( t ) .

We use the symbol

(9) p i , 1 ( t ) = ∫ 0 t h i ( t ) d t ,

and the value of the above integral is calculated by definition of h i and is given by

(10) p i , 1 ( t ) = t − ρ 1 for t ∈ [ ρ 1 , ρ 2 ) , ρ 3 − t for t ∈ [ ρ 2 , ρ 3 ) , 0 elsewhere .

Thus value of p i , 2 is

p i , 2 ( t ) = ∫ 0 t p i , 1 ( s ) d s ,

and by simplifying this integral, we have

(11) p i , 2 ( t ) = 1 2 ( t − ρ 1 ) 2 if t ∈ [ ρ 1 , ρ 2 ) , 1 4 m 2 − 1 2 ( ρ 3 − t ) 2 if t ∈ [ ρ 2 , ρ 3 ) , 1 4 m 2 if t ∈ [ ρ 3 , 1 ) , 0 elsewhere .

In addition, the value of p i , 3 is given by

p i , 3 ( t ) = ∫ 0 t p i , 2 ( s ) d s ,

and by simplifying this integral, we obtain

(12) p i , 3 ( t ) = 1 6 ( t − ρ 1 ) 3 if t ∈ [ ρ 1 , ρ 2 ) , 1 4 m 2 ( t − ρ 2 ) − 1 6 ( ρ 3 − t ) 3 if t ∈ [ ρ 2 , ρ 3 ) , 1 4 m 2 ( t − ρ 2 ) if t ∈ [ ρ 3 , 1 ) 0 elsewhere .

Similarly, the value of p i , 4 is given by

p i , 4 ( t ) = ∫ 0 t p i , 3 ( s ) d s ,

and by simplifying this integral, we obtain

(13) p i , 4 ( t ) = 1 24 ( t − ρ 1 ) 4 if t ∈ [ ρ 1 , ρ 2 ) , 1 8 m 2 ( t − ρ 2 ) 2 − 1 24 ( ρ 3 − t ) 4 + 1 192 m 4 ( ρ 3 − t ) 3 if t ∈ [ ρ 2 , ρ 3 ) , 1 8 m 2 ( t − ρ 2 ) 2 + 1 192 m 4 if t ∈ [ ρ 3 , 1 ) 0 elsewhere .

Generally,

(14) p i , n ( t ) = ∫ 0 t p i , n − 1 ( t ) d t .

Thus p i , n ( t ) is obtained as follows [10]:

(15) p i , n ( t ) = 0 for t ∈ [ 0 , ρ 1 ) , ( t − ρ 1 ) n n ! for t ∈ [ ρ 1 , ρ 2 ) , [ ( t − ρ 1 ) n − 2 ( ρ 1 − ρ 2 ) n ] n ! for t ∈ [ ρ 2 , ρ 3 ) , 1 n ! [ ( t − ρ 1 ) n − 2 ( ρ 1 − ρ 2 ) n + ( t − ρ 3 ) n ] , for t ∈ [ ρ 3 , 1 ) . for n = 1 , 2 , 3 … , 12 .

We also introduce the following notation:

(16) C i , ν = ∫ 0 1 p i , ν ( t ) d t .

For HWC technique, the interval [ α , β ] is discretized using the formula:

(17) t m = α + ( β − α ) m − 1 / 2 2 M m = 1 , 2 , 3 , 4 , … , 2 M .

Equation (17) is known as collocation point (CP). Gauss points (GPs) are also known as integration points because numerical integration is carried out at these points. These points are represented as:

G j = h j − 1 2 + 3 − 3 6 , G j + 1 = h j − 1 2 + 3 + 3 6 , j = 1 , 2 , 3 , 4 , … , N − 1 .

3 Haar collocation technique

In this section, HWC scheme is developed for solution of twelfth-order both linear and nonlinear BVPs. We developed HWC method for interval [ 0 , 1 ] . Let u ( 12 ) ( t ) ∈ L 2 [ 0 , 1 ) , then

(18) u ( 12 ) ( t ) = ∑ i = 1 N a i h i ( t ) .

Integrating equation (18) from 0 to t and using BCs, we obtain the values of u ( 11 ) , u ( 10 ) , u ( 9 ) , u ( 8 ) , u ( 7 ) , u ( 6 ) , u ( 5 ) , u ( 4 ) , u ( 3 ) , u ( 2 ) , u ( 1 ) , and u ( t ) as follows:

(19) u ( 11 ) ( t ) = u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 1 ( t ) ,

(20) u ( 10 ) ( t ) = u ( 10 ) ( 0 ) + t u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 2 ( t ) ,

(21) u ( 9 ) ( t ) = u ( 9 ) ( 0 ) + t u ( 10 ) ( 0 ) + t 2 2 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 3 ( t ) ,

(22) u ( 8 ) ( t ) = u ( 8 ) ( 0 ) + t u ( 9 ) ( 0 ) + t 2 2 u ( 10 ) ( 0 ) + t 3 6 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 4 ( t ) ,

(23) u ( 7 ) ( t ) = u ( 7 ) ( 0 ) + t u ( 8 ) ( 0 ) + t 2 2 u ( 9 ) ( 0 ) + t 3 6 u ( 10 ) ( 0 ) + t 4 24 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 5 ( t ) ,

(24) u ( 6 ) ( t ) = u ( 6 ) ( 0 ) + t u ( 7 ) ( 0 ) + t 2 2 u ( 8 ) ( 0 ) + t 3 6 u ( 9 ) ( 0 ) + t 4 24 u ( 10 ) ( 0 ) + t 5 120 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 6 ( t ) ,

(25) u ( 5 ) ( t ) = α 5 + t u ( 6 ) ( 0 ) + t 2 2 u ( 7 ) ( 0 ) + t 3 6 u ( 8 ) ( 0 ) + t 4 24 u ( 9 ) ( 0 ) + t 5 120 u ( 10 ) ( 0 ) + t 6 720 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 7 ( t ) ,

(26) u ( 4 ) ( t ) = α 4 + t α 5 + t 2 2 u ( 6 ) ( 0 ) + t 3 6 u ( 7 ) ( 0 ) + t 4 24 u ( 8 ) ( 0 ) + t 5 120 u ( 9 ) ( 0 ) + t 6 720 u ( 10 ) ( 0 ) + t 7 5040 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 8 ( t ) ,

(27) u ( 3 ) ( t ) = α 3 + t α 4 + t 2 2 α 5 + t 3 6 u ( 6 ) ( 0 ) + t 4 24 u ( 7 ) ( 0 ) + t 5 120 u ( 8 ) ( 0 ) + t 6 720 u ( 9 ) ( 0 ) + t 7 5040 u ( 10 ) ( 0 ) + t 8 40320 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 9 ( t ) ,

(28) u ( 2 ) ( t ) = α 2 + t α 3 + t 2 2 α 4 + t 3 6 α 5 + t 4 24 u ( 6 ) ( 0 ) + t 5 120 u ( 7 ) ( 0 ) + t 6 720 u ( 8 ) ( 0 ) + t 7 5040 u ( 9 ) ( 0 ) + t 8 40320 u ( 10 ) ( 0 ) + t 9 362880 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 10 ( t ) ,

(29) u ( 1 ) ( t ) = α 1 + t α 2 + t 2 2 α 3 + t 3 6 α 4 + t 4 24 α 5 + t 5 120 u ( 6 ) ( 0 ) + t 6 720 u ( 7 ) ( 0 ) + t 7 5040 u ( 8 ) ( 0 ) + t 8 40320 u ( 9 ) ( 0 ) + t 9 362880 u ( 10 ) ( 0 ) + t 10 3628800 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 11 ( t ) ,

(30) u ( t ) = α 0 + t α 1 + t 2 2 α 2 + t 3 6 α 3 + t 4 24 α 4 + t 5 120 α 5 + t 6 720 u ( 6 ) ( 0 ) + t 7 5040 u ( 7 ) ( 0 ) + t 8 40320 u ( 8 ) ( 0 ) + t 9 362880 u ( 9 ) ( 0 ) + t 10 3628800 u ( 10 ) ( 0 ) + t 11 39916800 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 12 ( t ) .

Now to find the unknown u ( 6 ) ( 0 ) , u ( 7 ) ( 0 ) , u ( 8 ) ( 0 ) , u ( 9 ) ( 0 ) , u ( 10 ) ( 0 ) , and u ( 11 ) ( 0 ) in equation (30), we integrate equations (24)–(29) each from 0 to 1, and using BCs we obtain the following expressions:

(31) β 5 − α 5 = u ( 6 ) ( 0 ) + 1 2 u ( 7 ) ( 0 ) + 1 6 u ( 8 ) ( 0 ) + 1 24 u ( 9 ) ( 0 ) + 1 120 u ( 10 ) ( 0 ) + 1 720 u ( 11 ) ( 0 ) + ∑ i = 1 N a i c i , 6 ,

(32) β 4 − α 4 − α 5 = 1 2 u ( 6 ) ( 0 ) + 1 6 u ( 7 ) ( 0 ) + 1 24 u ( 8 ) ( 0 ) + 1 120 u ( 9 ) ( 0 ) + 1 720 u ( 10 ) ( 0 ) + 1 5040 u ( 11 ) ( 0 ) + ∑ i = 1 N a i c i , 7 ,

(33) β 3 − α 3 − α 4 − 1 2 α 5 = 1 6 u ( 6 ) ( 0 ) + 1 24 u ( 7 ) ( 0 ) + 1 120 u ( 8 ) ( 0 ) + 1 720 u ( 9 ) ( 0 ) + 1 5040 u ( 10 ) ( 0 ) + 1 40320 u ( 11 ) ( 0 ) + ∑ i = 1 N a i c i , 8 ,

(34) β 2 − α 2 − α 3 − 1 2 α 4 − 1 6 α 5 = 1 24 u ( 6 ) ( 0 ) + 1 120 u ( 7 ) ( 0 ) + 1 720 u ( 8 ) ( 0 ) + 1 5040 u ( 9 ) ( 0 ) + 1 40320 u ( 10 ) ( 0 ) + 1 362880 u ( 11 ) ( 0 ) + ∑ i = 1 N a i c i , 9 ,

(35) β 1 − α 1 − α 2 − 1 2 α 3 − 1 6 α 4 − 1 24 α 5 = 1 120 u ( 6 ) ( 0 ) + 1 720 u ( 7 ) ( 0 ) + 1 5040 u ( 8 ) ( 0 ) + 1 40320 u ( 9 ) ( 0 ) + 1 362880 u ( 10 ) ( 0 ) + 1 3628800 u ( 11 ) ( 0 ) + ∑ i = 1 N a i c i , 10 ,

(36) β 0 − α 0 − α 1 − 1 2 α 2 − 1 6 α 3 − 1 24 α 4 − 1 120 α 5 = 1 720 u ( 6 ) ( 0 ) + 1 5040 u ( 7 ) ( 0 ) + 1 40320 u ( 8 ) ( 0 ) + 1 362880 u ( 9 ) ( 0 ) + 1 3628800 u ( 10 ) ( 0 ) + 1 39916800 u ( 11 ) ( 0 ) + ∑ i = 1 N a i c i , 11 .

Finally solving these equations simultaneously from equations (31)–(36), we get the below unknown u ( 6 ) ( 0 ) , u ( 7 ) ( 0 ) , u ( 8 ) ( 0 ) , u ( 9 ) ( 0 ) , u ( 10 ) ( 0 ) , and u ( 11 ) ( 0 ) , respectively:

(37) u ( 6 ) ( 0 ) = − 6470201353035691 6757837294824 β 5 + 118516580759761 32010808238640 α 5 + 2151202903913401 3201080823864 β 4 − 27713230633076743 7602566956677 α 4 − 270012512586410 98734635801 β 3 − 19984894766555 76793605623 α 3 − 128319829296376810 7602566956677 β 2 − 210931438483313885 2172161987622 α 2 − 577747094319923377549975 70107941048334922089 β 1 − 2207122273538488435 15205133913354 α 0 − 35091125295843245 400135102983 α 1 − 2207122273538488435 15205133913354 β 0 − 42990435239 215108139000 ∑ i = 1 N a i c i , 6 − 2343267753345241 1600540411932 ∑ i = 1 N a i c i , 7 − 566218149728848055 15205133913354 ∑ i = 1 N a i c i , 8 + 290305544850841105 15205133913354 ∑ i = 1 N a i c i , 9 + 78079399025187 1792567825 ∑ i = 1 N a i c i , 10 + 20731003199424 358513565 ∑ i = 1 N a i c i , 11 ,

(38) u ( 7 ) ( 0 ) = 216602183598847 12702701682 β 5 − 1340551706356039 114324315138 β 4 + 3689401654484092 57162157569 α 4 + 7331969120981 1143243151380 α 5 + 3746059237983320 230380816869 β 3 + 5756575050500 577395531 α 3 + 17760563974515640 57162157569 β 2 + 101959974465237070 57162157569 α 2 + 950775109242156354562900 23369313682778307363 β 1 + 151757773725749150 57162157569 α 0 + 93164723291813900 57162157569 α 1 + 151757773725749150 57162157569 β 0 + 91157358413 2830370250 ∑ i = 1 N a i c i , 6 + 1498319261246479 57162157569 ∑ i = 1 N a i c i , 7 + 72589568721481930 1086080993811 ∑ i = 1 N a i c i , 8 − 20268761916539210 57162157569 ∑ i = 1 N a i c i , 9 − 92127552012888 358513565 ∑ i = 1 N a i c i , 10 − 69772030495776 71702713 ∑ i = 1 N a i c i , 11 ,

(39) u ( 8 ) ( 0 ) = − 82753633919459575 563153107902 β 5 + 26280691607835565 266756735322 β 4 − 6441191282440120 230380816869 β 3 − 1403002294599812260 2534188985559 α 4 − 323345554427839 533513470644 α 5 − 2416572966468620 25597868541 α 3 − 6884103002774936200 2534188985559 β 2 − 5629521122144896450 362026997937 α 2 + 294793943471530806860500 23369313682778307363 β 1 − 58631564241173563550 2534188985559 α 0 − 1905938466108421700 133378367661 α 1 − 58631564241173563550 2534188985559 β 0 − 1774331045959 3585135650 ∑ i = 1 N a i c i , 6 − 29977939959398485 133378367661 ∑ i = 1 N a i c i , 7 + 137394719262534170 2534188985559 ∑ i = 1 N a i c i , 8 + 7926586457424639050 2534188985559 ∑ i = 1 N a i c i , 9 + 166738121772516 358513565 ∑ i = 1 N a i c i , 10 + 676798603391232 71702713 ∑ i = 1 N a i c i , 11 ,

(40) u ( 9 ) ( 0 ) = 190386159422417555 281576553951 β 5 − 58990309819975385 133378367661 β 4 − 8674505759448400 230380816869 β 3 + 31867182012494208400 2534188985559 β 2 + 6432400181871939640 2534188985559 α 4 + 1243141312426859 266756735322 α 5 + 10439997376280600 25597868541 α 3 + 25990558745826853300 362026997937 α 2 − 1946079602919231779603000 3338473383254043909 β 1 + 270957847562001413900 2534188985559 α 0 + 8824481642824250600 133378367661 α 1 + 270957847562001413900 2534188985559 β 0 + 5707992885539 1792567825 ∑ i = 1 N a i c i , 6 + 136802974884271090 133378367661 ∑ i = 1 N a i c i , 7 − 442502191489463300 2534188985559 ∑ i = 1 N a i c i , 8 − 36967696475892148100 2534188985559 ∑ i = 1 N a i c i , 9 + 112480197536952 358513565 ∑ i = 1 N a i c i , 10 − 3577685847232896 71702713 ∑ i = 1 N a i c i , 11 ,

(41) u ( 10 ) ( 0 ) = − 54329321975175680 40225221993 β 5 + 16470508603713200 19054052523 β 4 + 18436417520000 215108139 β 3 − 9112831100007568000 362026997937 β 2 + 2244515359000000 / 1927317413 β 1 − 77417447408684518400 / 362026997937 α 0 − 2521304667636953600 19054052523 α 1 − 1830556686934688800 362026997937 α 4 − 225581421589120 19054052523 α 5 − 2613911678271200 3656838363 α 3 − 51908694786208086400 362026997937 α 2 − 77417447408684518400 362026997937 β 0 − 2885469830656 358513565 ∑ i = 1 N a i c i , 6 − 38702962690381600 19054052523 ∑ i = 1 N a i c i , 7 + 89949647479520000 362026997937 ∑ i = 1 N a i c i , 8 + 10635309524455760000 362026997937 ∑ i = 1 N a i c i , 9 − 243552012850944 358513565 ∑ i = 1 N a i c i , 10 + 8047769347763712 71702713 ∑ i = 1 N a i c i , 11 ,

(42) u ( 11 ) ( 0 ) = 328017836400 71702713 β 5 − 33777580400 11321481 β 4 + 18436417520000 215108139 β 3 + 13068852928000 215108139 β 2 + 3393234313600 215108139 α 4 − 3377758040 11321481 α 5 − 1415792400000 71702713 α 3 + 71878691080000 215108139 α 2 + 3095254640000 11321481 β 1 + 117619676360000 215108139 α 0 + 3095254640000 11321481 α 1 + 117619676360000 215108139 β 0 − 6482565936 358513565 ∑ i = 1 N a i c i , 6 + 67555160800 11321481 ∑ i = 1 N a i c i , 7 − 1633606616000 215108139 ∑ i = 1 N a i c i , 8 − 16102693784000 215108139 ∑ i = 1 N a i c i , 9 + 816803307936 358513565 ∑ i = 1 N a i c i , 10 − 39206558780928 71702713 ∑ i = 1 N a i c i , 11 .

3.1 Linear case

For linear case substituting the values from equations (18) and (30) to equation (2), simplification leads the following system of equations:

∑ i = 1 N a i h i ( t j ) + f ( t j ) t j 6 720 − 42990435239 215108139000 c i , 6 − 2343267753345241 1600540411932 c i , 7 − 566218149728848055 15205133913354 c i , 8 + 290305544850841105 15205133913354 c i , 9 + 78079399025187 1792567825 c i , 10 + 20731003199424 358513565 c i , 11 + t j 7 5040 91157358413 2830370250 c i , 6 + 1498319261246479 57162157569 c i , 7 + 72589568721481930 1086080993811 c i , 8 − 20268761916539210 57162157569 c i , 9 − 92127552012888 358513565 − 69772030495776 71702713 c i , 11 + t j 8 40320 − 1774331045959 3585135650 c i , 6 − 29977939959398485 133378367661 c i , 7 + 137394719262534170 2534188985559 c i , 8 + 7926586457424639050 2534188985559 c i , 9 + 166738121772516 358513565 c i , 10 + 676798603391232 71702713 c i , 11 + t j 9 362880 5707992885539 1792567825 c i , 6 + 136802974884271090 133378367661 c i , 7 − 442502191489463300 2534188985559 c i , 8 − 36967696475892148100 2534188985559 c i , 9 + 112480197536952 358513565 c i , 10 − 3577685847232896 71702713 c i , 11 + t j 10 3628800 − 2885469830656 358513565 c i , 6 − 38702962690381600 19054052523 c i , 7 + 89949647479520000 362026997937 c i , 8 + 10635309524455760000 362026997937 c i , 9 − 243552012850944 358513565 c i , 10 + 8047769347763712 71702713 c i , 11 + t j 11 39916800 − 6482565936 358513565 c i , 6 + 67555160800 11321481 c i , 7 − 1633606616000 215108139 c i , 8 − 16102693784000 215108139 c i , 9 + 816803307936 358513565 c i , 10 − 39206558780928 71702713 c i , 11 + a i p i , 12 ( t j ) .

= h ( t j ) − f ( t j ) α 0 + t j α 1 + t j 2 2 α 2 + t j 3 6 α 3 + t j 4 24 α 4 + t j 5 120 α 5 + t j 6 720 − 6470201353035691 6757837294824 β 5 + 118516580759761 32010808238640 α 5 + 2151202903913401 3201080823864 β 4 − 27713230633076743 7602566956677 α 4 − 270012512586410 98734635801 β 3 − 19984894766555 76793605623 α 3 − 128319829296376810 7602566956677 β 2 − 210931438483313885 2172161987622 α 2 − 577747094319923377549975 70107941048334922089 β 1 − 2207122273538488435 15205133913354 α 0 − 35091125295843245 400135102983 α 1 − 2207122273538488435 15205133913354 β 0 + t j 7 5040 216602183598847 12702701682 β 5 − 1340551706356039 114324315138 β 4 + 3689401654484092 57162157569 α 4 + 7331969120981 1143243151380 α 5 + 3746059237983320 230380816869 β 3 + 5756575050500 577395531 α 3 + 17760563974515640 57162157569 β 2 + 101959974465237070 57162157569 α 2 + 950775109242156354562900 23369313682778307363 β 1

+ 151757773725749150 57162157569 α 0 + 93164723291813900 57162157569 α 1 + 151757773725749150 57162157569 β 0 + t j 8 40320 − 82753633919459575 563153107902 β 5 + 26280691607835565 266756735322 β 4 − 6441191282440120 230380816869 β 3 − 1403002294599812260 2534188985559 α 4 − 323345554427839 533513470644 α 5 − 2416572966468620 25597868541 α 3 − 6884103002774936200 2534188985559 β 2 − 5629521122144896450 362026997937 α 2 + 294793943471530806860500 23369313682778307363 β 1 − 58631564241173563550 2534188985559 α 0 − 1905938466108421700 133378367661 α 1 − 58631564241173563550 2534188985559 β 0 + t j 9 362880 190386159422417555 281576553951 β 5 − 58990309819975385 133378367661 β 4 − 8674505759448400 230380816869 β 3 + 31867182012494208400 2534188985559 β 2 + 6432400181871939640 2534188985559 α 4 + 1243141312426859 266756735322 α 5

+ 10439997376280600 25597868541 α 3 + 25990558745826853300 362026997937 α 2 − 1946079602919231779603000 3338473383254043909 β 1 + 270957847562001413900 2534188985559 α 0 + 8824481642824250600 133378367661 α 1 + 270957847562001413900 2534188985559 β 0 + t j 10 3628800 − 54329321975175680 40225221993 β 5 + 16470508603713200 19054052523 β 4 + 18436417520000 215108139 β 3 − 9112831100007568000 362026997937 β 2 + 2244515359000000 / 1927317413 β 1 − 77417447408684518400 / 362026997937 α 0 − 2521304667636953600 19054052523 α 1 − 1830556686934688800 362026997937 α 4 − 225581421589120 19054052523 α 5 − 2613911678271200 3656838363 α 3

(43) − 51908694786208086400 362026997937 α 2 − 77417447408684518400 362026997937 β 0 + t j 11 39916800 328017836400 71702713 β 5 − 33777580400 11321481 β 4 + 18436417520000 215108139 β 3 + 13068852928000 215108139 β 2 + 3393234313600 215108139 α 4 − 3377758040 11321481 α 5 − 1415792400000 71702713 α 3 + 71878691080000 215108139 α 2 + 3095254640000 11321481 β 1 + 117619676360000 215108139 α 0 + 3095254640000 11321481 α 1 + 117619676360000 215108139 β 0 .

Gauss elimination technique is used for the solution of this N × N linear system. The solution gives the values of unknown coefficients a i ’s. The solution at discrete CPs is obtained using these coefficients a i ’s in equation (30).

3.2 Nonlinear case

For nonlinear twelfth-order BVP, substituting the values of u ( t j ) , u ( 1 ) ( t j ) , u ( 2 ) ( t j ) , u ( 3 ) ( t j ) , u ( 4 ) ( t j ) , u ( 5 ) ( t j ) , u ( 6 ) ( t j ) , u ( 7 ) ( t j ) , u ( 8 ) ( t j ) , u ( 9 ) ( t j ) , u ( 10 ) ( t j ) , u ( 11 ) ( t j ) , and u ( 12 ) ( t j ) in (1) and the CPs, simplification leads to the following nonlinear system:

F ( t j , u , u ( 1 ) , u ( 2 ) , u ( 3 ) , u ( 4 ) , u ( 5 ) , u ( 6 ) , u ( 7 ) , u ( 8 ) , u ( 9 ) , u ( 10 ) , u ( 11 ) ) = ∑ i = 1 N a i h i ( t j ) − f t j , α 0 + t j α 1 + t j 2 2 α 2 + t j 3 6 α 3 + t j 4 24 α 4 + t j 5 120 α 5 + t 6 720 u ( 6 ) ( 0 ) + t j 7 5040 u ( 7 ) ( 0 ) + t j 8 40320 u ( 8 ) ( 0 ) + t j 9 362880 u ( 9 ) ( 0 ) + t j 10 3628800 u ( 10 ) ( 0 ) + t j 11 39916800 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 12 ( t j ) , α 1 + t j α 2 + t j 2 2 α 3 + t j 3 6 α 4 + t j 4 24 α 5 + t j 5 120 u ( 6 ) ( 0 ) + t j 6 720 u ( 7 ) ( 0 ) + t j 7 5040 u ( 8 ) ( 0 ) + t j 8 40320 u ( 9 ) ( 0 ) + t j 9 362880 u ( 10 ) ( 0 ) + t j 10 3628800 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 11 ( t j ) , α 2 + t j α 3 + t j 2 2 α 4 + t j 3 6 α 5 + t j 4 24 u ( 6 ) ( 0 ) + t j 5 120 u ( 7 ) ( 0 ) + t j 6 720 u ( 8 ) ( 0 ) + t j 7 5040 u ( 9 ) ( 0 ) + t j 8 40320 u ( 10 ) ( 0 ) + t j 9 362880 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 10 ( t j ) , α 3 + t j α 4 + t j 2 2 α 5 + t j 3 6 u ( 6 ) ( 0 ) + t j 4 24 u ( 7 ) ( 0 ) + t j 5 120 u ( 8 ) ( 0 ) + t j 6 720 u ( 9 ) ( 0 ) + t j 7 5040 u ( 10 ) ( 0 ) + t j 8 40320 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 9 ( t j ) , α 4 + t j α 5 + t j 2 2 u ( 6 ) ( 0 ) + t j 3 6 u ( 7 ) ( 0 ) + t j 4 24 u ( 8 ) ( 0 ) + t j 5 120 u ( 9 ) ( 0 ) + t j 6 720 u ( 10 ) ( 0 ) + t j 7 5040 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 8 ( t j ) , α 5 + t j u ( 6 ) ( 0 ) + t j 2 2 u ( 7 ) ( 0 ) + t j 3 6 u ( 8 ) ( 0 ) + t j 4 24 u ( 9 ) ( 0 ) + t j 5 120 u ( 10 ) ( 0 ) + t j 6 720 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 7 ( t j ) , u ( 6 ) ( 0 ) + t j u ( 7 ) ( 0 ) + t j 2 2 u ( 8 ) ( 0 ) + t j 3 6 u ( 9 ) ( 0 ) + t j 4 24 u ( 10 ) ( 0 ) + t j 5 120 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 6 ( t j ) , + t j u ( 8 ) ( 0 ) + t j 2 2 u ( 9 ) ( 0 ) + t j 3 6 u ( 10 ) ( 0 ) + t j 4 24 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 5 ( t j ) , u ( 8 ) ( 0 ) + t j u ( 9 ) ( 0 ) + t j 2 2 u ( 10 ) ( 0 ) + t j 3 6 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 4 ( t j ) , u ( 9 ) ( 0 ) + t j u ( 10 ) ( 0 ) + t j 2 2 u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 3 ( t j ) , u ( 10 ) ( 0 ) + t j u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 2 ( t j ) , u ( 11 ) ( 0 ) + ∑ i = 1 N a i p i , 1 ( t j ) , ∑ i = 1 N a i h i ( t j ) .

The unknown values of u ( 6 ) ( 0 ) , u ( 7 ) ( 0 ) , u ( 8 ) ( 0 ) , u ( 9 ) ( 0 ) , u ( 10 ) ( 0 ) , and u ( 11 ) ( 0 ) , which are used in the above nonlinear system, are already calculated above in equations (37)–(42). In addition, we can easily calculate the Jacobian of the above very long-lasting equation by taking their partial derivatives with respect to a i . Broyden’s technique is used for solution of this N × N nonlinear system of algebraic equations. The solution gives the values of unknown Haar coefficients a i ’s. The approximate solution at discrete CPs can be easily obtained using these unknown coefficients a i ’s in equation (30).

4 Numerical examples

In this section, some examples are given to show the performance of the HWC technique. Three linear and two nonlinear twelfth-order BVPs are tested using the proposed HWC algorithm. If u ap represents the approximate and u ex represents the exact solution at CPs and GPs, respectively, then the maximum absolute errors E cp and E gp are defined as:

E cp = max | u exc − u apc | ,

E gp = max | u exg − u apg | .

The root mean square root errors M cp and M gp at CPs and GPs are defined as:

M cp = 1 N ∑ i = 1 N | u exc − u apc | 2 ,

M gp = 1 N ∑ i = 1 N | u exg − u apg | 2 .

The convergence rate at CPs is denoted by R cp and defined as [32,33]:

(44) R cp = log [ u apc ( N / 2 ) / u apc ( N ) ] log 2 .

Problem 1. Consider the twelfth-order BVP [34]

(45) u ( 12 ) ( t ) + t u ( t ) = − ( 120 + 23 t + t 3 ) e t , t ∈ [ 0 , 1 ] , u ( 0 ) = 0 , u ( 1 ) = 0 , u ( 1 ) ( 0 ) = 1 , u ( 1 ) ( 1 ) = − e , u ( 2 ) ( 0 ) = 0 , u ( 2 ) ( 1 ) = − 4 e , u ( 3 ) ( 0 ) = − 3 , u ( 3 ) ( 1 ) = − 9 e , u ( 4 ) ( 0 ) = − 8 , u ( 4 ) ( 1 ) = − 16 e , u ( 5 ) ( 0 ) = − 15 , u ( 5 ) ( 1 ) = − 25 e .

The analytical solution is

(46) u ( t ) = t ( 1 − t ) e t .

The maximum absolute errors are shown in Table 1. It is observed that the errors in absolute values are better than those of Siddiqi and Akram [34] and Siddiqi and Twizell [35] as shown in Table 2. From Table 2, it is observed that HWC results are better than other methods. The convergence rate is also calculated, which is approximately equal to two. The comparison of approximate and exact solution is given in Figure 1.

Table 1

E cp , R cp , E gp , M cp , and M gp for Problem 1

J N = 2 J + 1 E cp R cp E gp M cp M gp
1 4 4.080066 × 10−09 — 1.890321 × 10−12 2.040527 × 10−09 9.452123 × 10−13
2 8 2.429096 × 10−09 0.7482 1.218525 × 10−12 8.771540 × 10−10 4.361947 × 10−13
3 16 8.924874 × 10−10 1.4445 4.515832 × 10−13 2.532597 × 10−10 1.256058 × 10−13
4 32 2.675610 × 10−10 1.7380 1.339484 × 10−13 6.567113 × 10−11 3.228519 × 10−14
5 64 7.306394 × 10−11 1.8726 3.624878 × 10−14 1.656898 × 10−11 8.121789 × 10−15
6 128 1.907864 × 10−11 1.9372 9.325873 × 10−15 4.151756 × 10−12 2.031743 × 10−15
7 256 4.873948 × 10−12 1.9688 2.386979 × 10−15 1.038536 × 10−12 5.138567 × 10−16
8 512 1.231670 × 10−12 1.9845 6.661338 × 10−16 2.596703 × 10−13 1.335675 × 10−16
9 1024 3.096424 × 10−13 1.9919 2.775557 × 10−16 6.492086 × 10−14 4.701457 × 10−17
Table 2

Comparison of maximum absolute errors of present method with other methods

Present method Siddiqi and Akram [34] Siddiqi and Twizell [35]
3.10 × 10−13 7.38 × 10−9 2.07 × 10−3
Figure 1 Comparison of numerical and analytical solutions for 32 CPs of Problem 1.
Figure 1

Comparison of numerical and analytical solutions for 32 CPs of Problem 1.

Problem 2. Next, we have twelfth-order BVP [34]

(47) u ( 12 ) ( t ) − u ( t ) = − 12 ( 2 t cos t + 11 sin t ) , − 1 ≤ t ≤ 1 , u ( − 1 ) = u ( 1 ) = 0 , u ( 1 ) ( − 1 ) = u ( 1 ) ( 1 ) = 2 sin 1 , u ( 2 ) ( − 1 ) = − u ( 2 ) ( 1 ) = − 4 cos 1 − 2 sin 1 , u ( 3 ) ( − 1 ) = u ( 3 ) ( 1 ) = 6 cos 1 − 6 sin 1 , u ( 4 ) ( − 1 ) = − u ( 4 ) ( 1 ) = 8 cos 1 + 12 sin 1 , u ( 5 ) ( − 1 ) = u ( 5 ) ( 1 ) = − 20 cos 1 + 10 sin 1 .

The analytical solution is

(48) u ( t ) = ( t 2 − 1 ) sin t .

The comparison of approximate and exact solution is given in Figure 2. It is observed that the errors in absolute values are better than those of Siddiqi and Akram [34] and Siddiqi and Twizell [35] as shown in Table 3. From Table 4, it is observed that HWC results are better than the other methods. The convergence rate is also calculated, which is approximately equal to two (Table 5). The comparison of approximate and exact solution is given in Figure 3 for 32 number of CPs.

Figure 2 Comparison of numerical and analytical solutions for 32 CPs of Problem 2.
Figure 2

Comparison of numerical and analytical solutions for 32 CPs of Problem 2.

Table 3

E cp , R cp , E gp , M cp , and M gp for Problem 2

J N = 2 J + 1 E cp R cp E gp M cp M gp
1 4 4.087801 × 10−9 — 1.887768 × 10−12 2.044393 × 10−9 9.439349 × 10−13
2 8 2.358883 × 10−9 0.7932 1.162847 × 10−12 8.515840 × 10−10 4.162510 × 10−13
3 16 8.609238 × 10−10 1.4541 4.243827 × 10−13 2.440498 × 10−10 1.180074 × 10−13
4 32 2.578136 × 10−10 1.7396 1.253442 × 10−13 6.316870 × 10−11 3.020509 × 10−14
5 64 7.040099 × 10−11 1.8727 3.402834 × 10−14 1.593048 × 10−11 7.602254 × 10−15
6 128 1.838557 × 10−11 1.9370 8.881784 × 10−15 8.881784 × 10−15 1.906115 × 10−15
7 256 4.697249 × 10−12 1.9687 2.220446 × 10−15 9.983748 × 10−13 4.762571 × 10−16
8 512 1.187088 × 10−12 1.9844 6.106227 × 10−16 2.496279 × 10−13 1.254793 × 10−16
9 1,024 2.983625 × 10−13 1.9923 2.775558 × 10−16 6.240902 × 10−14 5.778539 × 10−17
Table 4

Comparison of maximum absolute errors of present method with other methods

Present method Siddiqi and Akram [34] Siddiqi and Twizell [35]
2.98 × 10−13 4.69 × 10−5 2.07 × 10−3
Table 5

E cp , R cp , E gp , M cp , and M gp for Problem 3

J N = 2 J + 1 E cp R cp E gp M cp M gp
1 4 4.791599 × 10−5 — 2.233135 × 10−8 2.396389 × 10−5 1.116627 × 10−8
2 8 2.464102 × 10−5 0.9594 1.196611 × 10−8 8.894615 × 10−6 4.283299 × 10−9
3 16 8.712958 × 10−6 1.4998 4.095127 × 10−9 2.467474 × 10−6 1.137446 × 10−9
4 32 2.589810 × 10−6 1.7503 1.188665 × 10−9 6.334443 × 10−7 2.857365 × 10−10
5 64 7.060396 × 10−7 1.8750 3.199413 × 10−10 1.594197 × 10−7 7.144727 × 10−11
6 128 1.843290 × 10−7 1.9375 8.298362 × 10−11 3.992147 × 10−8 1.786133 × 10−11
7 256 4.709231 × 10−8 1.9687 2.113087 × 10−11 9.984531 × 10−9 4.465273 × 10−12
8 512 1.190143 × 10−8 1.9844 5.331291 × 10−12 2.496393 × 10−9 1.116319 × 10−12
9 1,024 2.991534 × 10−9 1.9922 1.338707 × 10−12 6.241146 × 10−10 2.790845 × 10−13
Figure 3 Comparison of numerical and analytical solutions for 32 CPs of Problem 3.
Figure 3

Comparison of numerical and analytical solutions for 32 CPs of Problem 3.

Problem 3. Consider the linear twelfth-order BVP [34]

(49) u ( 12 ) ( t ) − u ( t ) = 1 4096 e sin t ( − 1823912 + 3439408 cos ( 2 t ) + 527012068 cos ( 4 t ) − 1060549512 cos ( 6 t ) + 59278120 cos ( 8 t ) − 313256 cos ( 10 t ) + 156 cos ( 12 t ) + 8338252 sin t + 171393617 sin ( 3 t ) + 1451213951 sin ( 5 t ) − 346542218 sin ( 7 t ) + 5688330 sin ( 9 t ) − 9713 sin ( 11 t ) + sin ( 13 t ) ) , t ∈ [ 0 , 1 ] , u ( 0 ) = 0 , u ( 1 ) = e sin ( 1 ) sin ( 1 ) , u ( 1 ) ( 0 ) = 1 , u ( 1 ) ( 1 ) = e sin ( 1 ) cos ( 1 ) + e sin ( 1 ) cos ( 1 ) sin ( 1 ) , u ( 2 ) ( 0 ) = 2 , u ( 2 ) ( 1 ) = 2 e sin ( 1 ) cos 2 ( 1 ) − e sin ( 1 ) sin ( 1 ) + e sin ( 1 ) cos 2 ( 1 ) sin ( 1 ) − e sin ( 1 ) sin 2 ( 1 ) , u ( 3 ) ( 0 ) = 2 , u ( 3 ) ( 1 ) = − e sin 1 cos 1 + 3 e sin 1 cos 3 1 − 7 e sin 1 cos 1 sin 1 + e sin 1 cos 3 1 sin 1 − 3 e sin 1 cos 1 sin 2 1 , u ( 4 ) ( 0 ) = − 4 , u ( 4 ) ( 1 ) = − 8 e sin ( 1 ) cos 2 ( 1 ) + 4 e sin ( 1 ) cos 4 ( 1 ) + e sin ( 1 ) sin ( 1 ) − 22 e sin ( 1 ) cos 2 ( 1 ) sin ( 1 ) + e sin ( 1 ) cos 4 ( 1 ) sin ( 1 ) + 7 e sin ( 1 ) sin 2 ( 1 ) − 6 e sin ( 1 ) cos 2 ( 1 ) sin 2 ( 1 ) + 3 e sin ( 1 ) sin 3 ( 1 ) , u ( 5 ) ( 0 ) = − 24 , u ( 5 ) ( 1 ) = e sin ( 1 ) cos ( 1 ) − 30 e sin ( 1 ) cos 3 ( 1 ) + 5 e sin ( 1 ) cos 5 ( 1 ) + 31 e sin ( 1 ) cos ( 1 ) sin ( 1 ) − 50 e sin ( 1 ) cos 3 ( 1 ) sin ( 1 ) + e sin ( 1 ) cos 5 ( 1 ) sin ( 1 ) + 60 e sin ( 1 ) cos ( 1 ) sin 2 ( 1 ) − 10 e sin ( 1 ) cos 3 ( 1 ) sin 2 ( 1 ) + 15 e sin ( 1 ) cos ( 1 ) sin 3 ( 1 ) .

The exact solution is

(50) u ( t ) = e sin t sin t .

Gauss elimination technique is used for solution of this linear problem. The E cp , R cp , E gp , M cp , and M gp are given in Table 3.

Problem 4. Consider the nonlinear twelfth-order BVP [36]

(51) u ( 12 ) ( t ) = 2 e t u 2 ( t ) + u ( 3 ) ( t ) , t ∈ [ 0 , 1 ] , u ( 2 k ) ( 0 ) = 1 , u ( 2 k ) ( 1 ) = e − 1 , k = 0 , 1 , 2 , 3 , 4 , 5 .

The analytical solution is

(52) u ( t ) = e − t .

Two errors E cp and M cp for distinct number of CPs and GPs are shown in Table 6. Figure 4 represent the comparison of approximate and exact solution for 32 number of CPs. The E cp is decreased up to order × 10 − 14 , whereas the results of Wazwaz [36] are decreased up to × 10 − 09 . The convergence rate is also calculated, which is approximately equal to two (Table 7). The comparison of approximate and exact solution is given in Figure 5 for 32 number of CPs.

Table 6

E cp , R cp , E gp , M cp , and M gp for Problem 4

J N = 2 J + 1 E cp R cp E gp M cp M gp
1 4 2.422818 × 10−11 — 1.121325 × 10−14 1.211702 × 10−11 5.607176 × 10−15
2 8 1.480859 × 10−11 0.7103 7.438494 × 10−15 5.347973 × 10−12 2.654397 × 10−15
3 16 5.478839 × 10−12 1.4345 2.775558 × 10−15 1.555463 × 10−12 7.913990 × 10−16
4 32 1.645017 × 10−12 1.7358 8.881784 × 10−16 4.040732 × 10−13 2.338726 × 10−16
5 64 4.493628 × 10−13 1.8721 3.330669 × 10−16 1.019971 × 10−13 1.279469 × 10−16
6 128 1.172951 × 10−13 1.9377 3.330669 × 10−16 2.555701 × 10−14 1.118863 × 10−16
7 256 2.997602 × 10−14 1.9683 4.440892 × 10−16 6.391976 × 10−15 1.275700 × 10−16
Figure 4 Comparison of numerical and analytical solutions for 32 CPs of Problem 4.
Figure 4

Comparison of numerical and analytical solutions for 32 CPs of Problem 4.

Table 7

E cp , R cp , E gp , M cp , and M gp for Problem 5

J N = 2 J + 1 E cp R cp E gp M cp M gp
1 4 5.640377 × 10−11 — 2.531308 × 10−14 2.820869 × 10−11 1.266044 × 10−14
2 8 3.268852 × 10−11 0.7870 1.598721 × 10−14 1.180128 × 10−11 5.751769 × 10−15
3 16 1.194245 × 10−11 1.4527 6.217249 × 10−15 3.385745 × 10−12 1.807312 × 10−15
4 32 3.577583 × 10−12 1.7390 1.332268 × 10−15 8.766552 × 10−13 4.775249 × 10−16
5 64 9.769963 × 10−13 1.8726 8.881784 × 10−16 2.211255 × 10−13 3.964289 × 10−16
6 128 2.549072 × 10−13 1.9384 2.332268 × 10−16 5.538092 × 10−14 4.807407 × 10−16
7 256 6.572520 × 10−14 1.9555 1.142386 × 10−16 1.372139 × 10−14 4.466837 × 10−16
Figure 5 Comparison of numerical and analytical solutions for 32 CPs of Problem 5.
Figure 5

Comparison of numerical and analytical solutions for 32 CPs of Problem 5.

Problem 5. Consider the nonlinear twelfth-order BVP [37]

(53) u ( 12 ) ( t ) = 1 2 e − t u 2 ( t ) , t ∈ [ 0 , 1 ] , u ( 0 ) = u ( 2 ) ( 0 ) = u ( 4 ) ( 0 ) = u ( 6 ) ( 0 ) = u ( 8 ) ( 0 ) = u ( 10 ) ( 0 ) = 2 , u ( 1 ) = u ( 2 ) ( 1 ) = u ( 4 ) ( 1 ) = u ( 6 ) ( 1 ) = u ( 8 ) ( 1 ) = u ( 10 ) ( 1 ) = 2 .

The analytical solution is

(54) u ( t ) = 2 e t .

Broyden’s technique is used for solution of this problem. Two errors E cp and M cp for distinct number of CPs and GPs are shown in Table 4. From this table, it is evident that by increasing the number of CPs the errors E cp and M cp are decreased. The convergence rate is also calculated, which is approximately equal to two.

5 Conclusion

In this article, the HWC technique is used to find the solution of linear and nonlinear twelfth-order BVPs. The maximum absolute errors for distant number of discrete CPs and GPs are shown for each example in tables. The convergence rate is also calculated, which is approximately equal to two. The maximum absolute errors of present method is compared with Siddiqi and Akram [34], Siddiqi and Twizell [35], and Wazwaz [36]. The results show that the HWC technique is better than the other techniques available in the literature. MATLAB software is used for all computational work.

Acknowledgments

Taif University Researchers Supporting Project number (TURSP-2020/20), Taif University, Taif, Saudi Arabia.

  1. Competing interests: The authors declared that no competing interest exists regarding this manuscript.

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Received: 2020-09-04
Revised: 2020-10-08
Accepted: 2020-10-29
Published Online: 2020-12-24

© 2020 Rohul Amin et al., published by De Gruyter

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

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https://doi.org/10.1515/phys-2020-0205
Keywords for this article
boundary value problem; Gauss elimination method; collocation method; Haar wavelet
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Articles in the same Issue

  1. Regular Articles
  2. Model of electric charge distribution in the trap of a close-contact TENG system
  3. Dynamics of Online Collective Attention as Hawkes Self-exciting Process
  4. Enhanced Entanglement in Hybrid Cavity Mediated by a Two-way Coupled Quantum Dot
  5. The nonlinear integro-differential Ito dynamical equation via three modified mathematical methods and its analytical solutions
  6. Diagnostic model of low visibility events based on C4.5 algorithm
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  10. Impact of double-diffusive convection and motile gyrotactic microorganisms on magnetohydrodynamics bioconvection tangent hyperbolic nanofluid
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  17. Gaussons: optical solitons with log-law nonlinearity by Laplace–Adomian decomposition method
  18. Adaptive magnetic suspension anti-rolling device based on frequency modulation
  19. Dynamic response characteristics of 93W alloy with a spherical structure
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  62. A clustering algorithm based on nonuniform partition for WSNs
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  73. Energy equilibrium analysis in the effervescent atomization
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  79. Kinetics and thermodynamics of SO2 adsorption on metal-loaded multiwalled carbon nanotubes
  80. Effect of the inner-surface baffles on the tangential acoustic mode in the cylindrical combustor
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  82. Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications
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  84. Exact optical solitons of the perturbed nonlinear Schrödinger–Hirota equation with Kerr law nonlinearity in nonlinear fiber optics
  85. Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water
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  88. Stable solutions to the nonlinear RLC transmission line equation and the Sinh–Poisson equation arising in mathematical physics
  89. Calculation of focal values for first-order non-autonomous equation with algebraic and trigonometric coefficients
  90. Influence of interfacial electrokinetic on MHD radiative nanofluid flow in a permeable microchannel with Brownian motion and thermophoresis effects
  91. Standard routine techniques of modeling of tick-borne encephalitis
  92. Fractional residual power series method for the analytical and approximate studies of fractional physical phenomena
  93. Exact solutions of space–time fractional KdV–MKdV equation and Konopelchenko–Dubrovsky equation
  94. Approximate analytical fractional view of convection–diffusion equations
  95. Heat and mass transport investigation in radiative and chemically reacting fluid over a differentially heated surface and internal heating
  96. On solitary wave solutions of a peptide group system with higher order saturable nonlinearity
  97. Extension of optimal homotopy asymptotic method with use of Daftardar–Jeffery polynomials to Hirota–Satsuma coupled system of Korteweg–de Vries equations
  98. Unsteady nano-bioconvective channel flow with effect of nth order chemical reaction
  99. On the flow of MHD generalized maxwell fluid via porous rectangular duct
  100. Study on the applications of two analytical methods for the construction of traveling wave solutions of the modified equal width equation
  101. Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method
  102. A powerful numerical technique for treating twelfth-order boundary value problems
  103. Fundamental solutions for the long–short-wave interaction system
  104. Role of fractal-fractional operators in modeling of rubella epidemic with optimized orders
  105. Exact solutions of the Laplace fractional boundary value problems via natural decomposition method
  106. Special Issue on 19th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering
  107. Joint use of eddy current imaging and fuzzy similarities to assess the integrity of steel plates
  108. Uncertainty quantification in the design of wireless power transfer systems
  109. Influence of unequal stator tooth width on the performance of outer-rotor permanent magnet machines
  110. New elements within finite element modeling of magnetostriction phenomenon in BLDC motor
  111. Evaluation of localized heat transfer coefficient for induction heating apparatus by thermal fluid analysis based on the HSMAC method
  112. Experimental set up for magnetomechanical measurements with a closed flux path sample
  113. Influence of the earth connections of the PWM drive on the voltage constraints endured by the motor insulation
  114. High temperature machine: Characterization of materials for the electrical insulation
  115. Architecture choices for high-temperature synchronous machines
  116. Analytical study of air-gap surface force – application to electrical machines
  117. High-power density induction machines with increased windings temperature
  118. Influence of modern magnetic and insulation materials on dimensions and losses of large induction machines
  119. New emotional model environment for navigation in a virtual reality
  120. Performance comparison of axial-flux switched reluctance machines with non-oriented and grain-oriented electrical steel rotors
  121. Erratum
  122. Erratum to “Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications”
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