We propose a threestep block method of Adam’s type to solve nonlinear secondorder twopoint boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of secondorder boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the threestep iterative method. The boundary value problem will be solved without reducing to firstorder equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.
Boundary value problems (BVPs) arise in many areas of applied mathematics, for example, application to chemical reactor theory [
The present paper is organized as follows. In Section
The general twopoint secondorder BVP and the system of secondorder BVP subject to two kinds of boundary condition which are Dirichlet type and Neumann type will be solved directly by threestep block method. The proposed method is the extended of block method proposed by Majid et al. [
The twopoint secondorder boundary value problem is as follows:
Type 1:
Type 2:
Threestep Adam’s method.
In this section we will discuss the order, consistency, stability, and convergence of the threestep Adam’s method. The threestep Adam’s method belongs to the class of linear multistep method (LMM):
The linear multistep method is said to be of order
Rewrite (
The linear multistep method is said to be consistent if it has order
Since the order of threestep Adam’s method is
A linear multistep method is zerostable provided the roots
Rewrite (
The linear multistep method is convergent if and only if it is consistent and zerostable.
Since the consistency and zerostable of the method have been established, then the threestep Adam’s method is convergent.
The threestep block method of Adam’s type (3SAM) will be implemented for solving the boundary value problems via multiple shooting techniques. The idea for shooting technique is to form the initial condition from the boundary condition with the guessing value. Multiple shooting techniques are indeed a combination of several shooting techniques by dividing the given interval
The missing initial condition for the Neumann boundary condition is
In this section, four problems are tested to study the accuracy and the efficiency of the developed codes. The results obtained by the proposed method are compared to the existing method. The following notations are used in the tables:
3SAM: Threestep Adam's method variable step size via multiple shooting techniques adapted with threestep iterative method;
2P1BVS: twopoint block method with variable step size proposed by Phang et al. [
bvp4c: MATLAB solver proposed by Kierzenka and Shampine [
MLAM: multilevel augmentation method proposed by Chen [
COLHW: collocation method with Haar wavelets proposed by SirajulIslam et al. [
SCM: sinccollocation method proposed by Mohamed [
LRBFM: local radial basis function method proposed by Mehdi and Ahmad [
TOL: tolerance;
TS: total number of steps;
MAXE: maximum error;
TFC: total function call;
TIME: execution time in seconds;
RELTOL: the tolerance use to measure the error relative use by bvp4c;
ABSTOL: the absolute error tolerances use by bvp4c;
MP: total mesh point use by bvp4c;
—: no data in the references;
Consider
Dirichlet boundary condition:
exact solution:
source: Chen [
Consider
Neumann boundary condition:
Exact solution:
source: SirajulIslam et al. [
Consider
Two Dirichlet boundary condition:
Exact solution:
source: Mehdi and Ahmad [
The equations governing the free convective boundarylayer flow above a heated impermeable horizontal surface are
Mixed boundary condition:
Source: Merkin and Zhang [
Problems
Tables
Comparison of the numerical result for solving Problem
3SAM  TOL  1 
1 
1 
1 
1 
TS  4  6  11  22  45  
MAXE  3.40 
5.26 
8.17 
2.36 
6.06  
TFC  127  184  319  616  1237  
Time  0.000139  0.000151  0.000216  0.000307  0.000470  


2P1BVS  TOL  1 
1 
1 
1 
1 
TS  11  16  26  32  57  
MAXE  7.66 
2.47 
3.55 
1.16 
1.07  
TFC  265  377  593  737  1321  
Time  0.000141  0.000162  0.000227  0.000275  0.000444  


bpv4c  ABSTOL  1 
1 
1 
1 
1 
RELTOL  1.00  1 
1 
1 
1  
MP  15  15  19  69  233  
MAXE  6.70 
3.35 
3.14 
1.27 
1.25  
TFC  157  187  374  1568  5043  
Time  0.0313  0.0469  0.0313  0.0938  0.1563  


MLAM 

1/15  1/31  1/63  1/127  1/255 
TS  15  31  63  127  255  
MAXE  2.46 
3.94 
4.94 
6.17 
5.00  
Time  0.109  0.235  0.328  0.687  1.953 
Comparison of the numerical result for solving Problem
3SAM  TOL  1 
1 
1 
1 
1 
TS  4  6  10  18  37  
MAXE  8.55 
8.37 
9.68 
1.05 
1.09  
TFC  41  61  98  173  341  
Time  0.000082  0.000101  0.000131  0.000229  0.000388  


2P1BVS  TOL  1 
1 
1 
1 
1 
TS  12  16  23  36  59  
MAXE  1.72 
1.33 
2.55 
6.20 
1.72  
TFC  65  89  129  205  341  
Time  0.000090  0.000109  0.000141  0.000218  0.000315  


bpv4c  ABSTOL  1 
1 
1 
1 
1 
RELTOL  1.00  1 
1 
1 
1  
MP  15  15  29  43  160  
MAXE  3.60 
4.16 
3.17 
6.14 
2.50  
TFC  228  254  491  689  2745  
Time  0.0156  0.0313  0.0313  0.0469  0.1406  


COLHW 

1/16  1/32  1/64  1/128  1/256 
TS  16  32  64  128  256  
MAXE  5.95 
1.55 
3.98 
1.01 
2.53 
Comparison of the numerical result for solving Problem
3SAM  TOL  1 
1 
1 
1 
1 
TS  4  5  7  13  24  
MAXE  3.79 
7.14 
3.57 
8.38 
2.01  
TFC  114  153  231  366  672  
Time  0.000203  0.000231  0.000286  0.000396  0.000412  


SCM  TS  10  20  30  40  — 
MAXE  4.06 
1.70 
8.14 
6.40 
—  


LRBFM  TS  21  41  61  —  — 
MAXE  2.10 
2.37 
2.01 
—  — 
Next we discuss the comparison between 3SAM and bvp4c. The bvp4c is a MATLAB solver which uses the collocation formula and a mesh of points to divide the interval of integration into subintervals. If the solution does not satisfy the tolerance, the solver adapts the mesh and repeats the process. In Problems
Finally we discuss the comparison between 3SAM with the method implemented using constant step size. The total number of steps taken by 3SAM is less than MLAM, COLHW, SCM, and LRBFM. This is expected because the 3SAM is using the variable step size strategy while MLAM, COLHW, SCM, and LRBFM are using the constant step size. We noticed that as the tolerance was getting smaller, 3SAM has obtained better accuracy compared to MLAM, COLHW, SCM, and LRBFM. For example, in Table
In Problem
Approximate solution of the value
Approximate solution of the value
In this paper, we have shown that the proposed threestep block method of Adam’s type using variable step size with multiple shooting technique is suitable for solving nonlinear secondorder twopoint boundary value problems of Dirichlet type, Neumann type, and mixed type of boundary conditions. The numerical results showed that the proposed block method has superiority in terms of accuracy, total function call, total steps, and execution time.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The author gratefully acknowledged the financial support of Fundamental Research Grant Scheme (0201131157FR) and MyPhD scholarship from the Ministry of Education Malaysia.