Research Article  Open Access
Pei See Phang, Zanariah Abdul Majid, Fudziah Ismail, Khairil Iskandar Othman, Mohamed Suleiman, "New Algorithm of TwoPoint Block Method for Solving Boundary Value Problem with Dirichlet and Neumann Boundary Conditions", Mathematical Problems in Engineering, vol. 2013, Article ID 917589, 10 pages, 2013. https://doi.org/10.1155/2013/917589
New Algorithm of TwoPoint Block Method for Solving Boundary Value Problem with Dirichlet and Neumann Boundary Conditions
Abstract
Twopoint block method with variable stepsize strategy is presented to obtain the solutions for boundary value problems directly. Dirichlet type and Neumann type of boundary conditions are studied in this paper. Multiple shooting techniques adapted with the threestep iterative method are employed for generating the guessing value. Six boundary value problems are solved using the proposed method, and the numerical results are compared to the existing methods. The results suggest a significant improvement in the efficiency of the proposed methods in terms of the number of steps, execution time, and accuracy.
1. Introduction
Numerical solutions for boundary value problem have great importance in scientific computation, as they were widely used to model the realworld problems. There are several methods that can be used to solve the twopoint boundary value problems such as the Adomian decomposition method modified by Duan and Rach [1] and Ebaid [2] to solve boundary value problem. Based on the Adomian decomposition method, a new analytical and numerical treatment is introduced to investigate linear and nonlinear twopoint boundary value problems. Lang and Xu [3] studied a new quintic Bspline collocation method for linear and nonlinear secondorder boundary value problems. Islam and Shirin [4] used Bernoulli polynomials to find the numerical solutions of the secondorder linear and nonlinear boundary value problems. Besides that, Liu [5] had solved the boundary value problem with Neumann type using polynomial spline approach. The numerical solutions of secondorder boundary value problems by collocation method with the Haar wavelets were presented by SirajulIslam et al. [6]. A new kind of finite difference scheme presented for special secondorder nonlinear twopoint boundary value problem has been proposed by Erdogan and Ozis [7], while Prentice [8] considered the error control in a finite difference solution of a twopoint boundary value problem.
In this paper, we are concerned for solving twopoint secondorder boundary value problems (BVPs) with two types of boundary condition, that is, Dirichlet and Neumann type. Twopoint secondorder boundary value problem is as follows: Dirichlet boundary condition is as follows: Neumann boundary condition is as follows: The BVPs as (1) will be solved using twopoint block method with variable step size. The block methods are commonly used to solve the ODEs such as in Majid et al. [9]. The authors have used the twopoint block onestep method of RungeKutta type to solve the general secondorder ODEs with variable stepsize strategy.
In Majid et al. [10], the authors have solved the BVPs of Dirichlet type only by twopoint block method with constant step size. Hence, in this paper, we aim to extend the work in Majid et al. [10] for solving BVPs of Dirichlet and Neumann type using variable step size. This block method has the advantage to solve the secondorder differential equation directly and obtain two approximate solutions simultaneously in block. We adapted the multiple shooting techniques to obtain the missing initial value, and the threestep iterative method proposed by Yun [11] was employed to generate the missing guessing value. The numerical computations have been performed using the C language.
2. Formulation of the TwoPoint Block Method
In Figure 1, we have divided the interval into a series of blocks with each block containing two points. Both approximate solutions are simultaneously found using the same back values. The approximate solution of and at the point, and , respectively, with step size will be computed simultaneously using three back values at the points, , and with step size rh. The value, and will be obtained by integrate (1) once and twice over the intervals and , respectively, as follows
First point is
Second point is The method is derived using Lagrange interpolation polynomial and the five interpolating points; that is, , , , , and are involved in the corrector formulae of the twopoint block method. The function in (4) and (5) will be replaced by Lagrange interpolating polynomial. Let , replacing and changing the limit of integration from 2 to 1 for the first point and changing the limit of integration to 2 and 0 for the second point.
Evaluate these integrals using MAPLE, the corrector formulae of the twopoint block method will be obtained as follows.
First point is
Second point is The method is the combination of predictor which is one order less than the corrector. The twopoint block method with variable stepsize strategy will be implemented for solving the boundary value problems via multiple shooting techniques.
3. Analysis of the Method
In this section, stability analysis, stability region, order of the method, and error constant of the twopoint block method are discussed.
3.1. Stability Analysis
Definition 1. The method is zero stable provided the roots of the first characteristic polynomial specified as satisfy .
Substitute into (7), and rewrite in matrix from:
The first characteristic polynomial of the twopoint block method is given as follows:
From Definition 1 and (9) the twopoint one block method is zero stable.
3.2. Stability Region
The stability polynomial of the twopoint block method applied to the test equation is as follows:
The stability polynomials of twopoint block method are as follows.
For : where and .
For : where and .
For : where and .
Figure 2 shows the regions of absolute stability for the twopoint block method when the stepsize ratio . The stability region is the bounded shaded region and the region is larger as the stepsize ratio increases. This is expected since smaller stepsize will give larger step size ratio.
3.3. Order and Error Constant
Definition 2. The block method is
where
is the order of the LMM method, and is the local truncation error.
Substituting into (7) and apply the formulae, we obtaining
From Definition 2, the order of twopoint block method is five with error constant
4. Implementation of the Method
4.1. Multiple Shooting Technique
The initial conditions being imposed at the same point in the independent variable , but the boundary conditions are imposed at different values of the independent variable. The idea of 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 into th subinterval.
4.1.1. DirichletType Boundary Condition
The missing initial condition is . Equation (1) can be written as with conditions Therefore, we obtain the th stopping conditions as follows: The iteration is repeated until we reached the stopping conditions, and the value of will be generated by the threestep iterative method as follows: where is the solution of
4.1.2. NeumannType Boundary Condition
The missing initial condition is . Equation (1) can be written as with conditions given as in (19), but condition 1 will be replaced by The stopping conditions are The threestep iterative method is where the value of can be obtained from solving the as follows:
4.2. Variable StepSize Strategy
The choices of the next step size will be restricted to half, doubled or the same as the current step size. The adjustment is based on the local truncation error (LTE). If the local truncation error is less or equal to the tolerance (TOL), the choice for the next step will be double or remain the same. In the code developed, when the next step size is doubled, the ratio is 0.5, while the step size remains constant, r is 1. If this condition fails, the current step size will be halved from the previous step size; that is, r is 2, and the approximate solutions in the block will be recalculate.
Case 1. LTE TOL (successful step).
Substituting in (6) and (7) will produce the following corrector formulae:
Substituting in (6) and (7) will produce the following corrector formulae:
Case 2. LTE TOL (failure step).
Substituting in (6) and (7) will produce the following corrector formulae:
4.3. Algorithm??2P1BVS
Step 1. Set TOL and .
Step 2. For . ?set , and evaluate the approximate values and with direct AdamsBashforth method. ?Compute functions and .
Step 3. While , do Step 4.
Step 4. For . ?Set .?Evaluate the approximate values and with twopoint block method.?Compute functions and .
Step 5. If fulfill stopping condition, go to Step 8.
Step 6. Generate the new guessing values by threestep iterative method. Set .
Step 7. If , set , and go to Step 2.
Step 8. Complete.
This algorithm was developed in C language.
5. Problem Tested
In this section, we have tested the algorithm 2P1BVS to six numerical examples to illustrate its accuracy and efficiency.
Problem 1. We have
Dirichlet boundary condition is as follows: , .
Exact solution is as follows: .
Problem 2. We have
Dirichlet boundary condition is as follows: , .
Exact solution is as follows: .
Problem 3. We have
Neumann boundary condition is as follows: , .
Exact solution is as follows: .
Problem 4. We have
Neumann boundary condition is as follows: , .
Exact solution is as follows: .
Problem 5. We have
Neumann boundary condition: , .
Exact solution is as follows: .
Problem 6 (Troeschâ€™s Problem). We have
Dirichlet boundary condition is as follows: ,.
Troeschâ€™s problem comes from the investigation of the confinement of a plasma column under radiation pressure. The closed form solution to this problem has been given in terms of Jacobian elliptic function by Lin et al. [12].
6. Numerical Result and Discussion
The following notations are used in the tables: ?TOL: Tolerance ?TS: Total number of steps ?MAXE: Maximum error ?Time (sec): Execution time in second ?TFC: Total function calls?2P1BVS: Twopoint block method with variable step size via multiple shooting technique adapted with threestep iterative method ?MLAM: Multilevel augmentation method proposed by Chen [13] ?COLHW: Collocation method with Haar wavelets proposed by SirajulIslam [6] ?FDM: Finite different method proposed by Erdogan and Ozis [7].
In Problems 1 and 2, we solved the boundary value problem with Dirichlettype boundary conditions by 2P1BVS and compare our result with MLAM. In Table 1, the 2P1BVS has superiority in terms of accuracy and execution time when compared to MLAM. For example, the maximum error for the 2P1BVS and MLAM with 15 steps is 1.94 and 2.42 respectively. This result is expected since 2P1BVS solves the secondorder BVPs directly and obtains two approximate solutions simultaneously. In Table 2, we observed that the 2P1BVS can obtain the maximum error 1.07 with 57 steps, but MLAM can obtain the maximum error 5.00 with 255 steps.


In Problems 3 and 4, 2P1BVS solved the boundary value problem with Neumanntype boundary conditions when was compared to COLHW. The maximum errors for 2P1BVS in Tables 3 and 4 are comparable to COLHW for the larger tolerance. For example, in Table 3, the maximum error for 2P1BVS with 12 steps and COLHW with 16 steps is 3.64 and 2.91, respectively. As the tolerance is getting smaller, the maximum error and total number of steps for 2P1BVS are setting better than COLHW. In Table 4, the maximum error for 2P1BVS with 108 steps and COLHW with 512 steps is 3.47 and 6.34, respectively.


Table 5 displays the numerical results for Problem 5 with larger interval at and for solving Problem 5. The 2P1BVS managed to obtain the accuracy within the given tolerances. We also observed that the 2P1BVS manages to converge rapidly for the larger interval, for example, at TOL = , the execution time is 0.000551 and 0.001083 seconds when and , respectively.

We consider the cases when and to solve the Troesch problem at TOL = . The numerical results in Tables 6 and 7 show that the 2P1BVS manage to solve the problem as accurate as or better than FDM. Figures 3, 4, 5, and 6 display the comparison of the maximum errors versus the total steps for the numerical results in Tables 1â€“7.


7. Conclusion
In this paper, we have shown that the proposed twopoint block method using variable step size (2P1BVS) is suitable for solving directly twopoint secondorder boundary value problems in Dirichlet and Neumanntypeboundary conditions.
Acknowledgments
The author gratefully acknowledged the financial support of Fundamental Research Grant Scheme (FRGS) and MyPhD. scholarship from the Ministry of Higher Education Malaysia.
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Copyright
Copyright © 2013 Pei See Phang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.