Analytical and Numerical Approaches for Complicated Nonlinear Equations
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ThreeStep Block Method for Solving Nonlinear Boundary Value Problems
Abstract
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.
1. Introduction
Boundary value problems (BVPs) arise in many areas of applied mathematics, for example, application to chemical reactor theory [1] and Bratutype problem [2]. Recently many methods are available to solve BVPs such as Adomian decomposition method, variational iteration method, homotopy perturbation method, and modified homotopy perturbation method. The Adomian decomposition method and variational iteration method were developed by Singh and Kumar [3] and Lu [4], respectively. The homotopy perturbation method that has been introduced by Saadatmandi et al. [5] to solve secondorder BVPs has less iteration compared to Adomian decomposition method. Asadi et al. [6] has extended the modified homotopy perturbation method to solve the nonlinear system of secondorder BVPs. The quintic Bspline collocation method has been modified by Lang and Xu [7] to solve secondorder BVPs. Srivastava et al. [8] developed a numerical algorithm based on the nonpolynomial quintic spline functions for the solution of secondorder BVPs with engineering applications, while Ibraheem and Khalaf [9] have proposed a shooting neural networks algorithm for solving BVPs. Rahman et al. [10] solved numerically secondorder BVPs by the Galerkin method. A smart nonstandard finite difference scheme has been proposed by Erdogan and Ozis [11] for solving secondorder nonlinear BVPs. Besides that, Liu et al. [12] solve BVPs with Neumann type by using polynomial spline approach. The aim of this research is to propose a threestep block method to solve the BVPs directly using multiple shooting techniques with variable step size strategy.
The present paper is organized as follows. In Section 2, we present the derivation of the threestep block method. In Section 3, we show the analysis of the method including the order, consistency, and stability. In Section 4 we show the implementation of the multiple shooting techniques. The numerical results and the discussion are presented in Section 5. Finally, a conclusion is given in Section 6.
2. Derivation of ThreeStep Block Method
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. [13] which uses the method to solve secondorder ordinary differential equations.
The twopoint secondorder boundary value problem is as follows: Dirichlet boundary condition: Neumann boundary condition: Type 1: Type 2: The system of twopoint secondorder boundary value problem: DirichletDirichlet boundary condition: NeumannDirichlet boundary condition: We have divided the interval into a series of blocks with each block containing three points as shown in Figure 1, where is the step size and is the ratio of the step size. Three approximate solutions, , , and are simultaneously computed using the same back values by integrating (1) once and twice over the intervals , , and , respectively. Consider The method is derived by replacing the function in (8) with Lagrange interpolation polynomial where five interpolating points are involved and will be implemented using the variable step size strategy. The choices of the next step size will be restricted to half, double, or the same as the current step size. When the next step size is doubled, the ratio is 0.5 and while the step size remains constant, is 1. If step fails, the current step size is half the previous step size and the ratio is 2. Then the approximate solutions in the block will be recalculated. The corrector formulae for are as follows:
3. Analysis of the 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):
3.1. Order of the Method
Definition 1 (Fatunla [14] and Lambert [15]). The linear multistep method is said to be of order if
where ,
where is the error constant.
Rewrite (9) as (10) with being 4 in matrix form:
From Definition 1, we obtain
Therefore, the order of threestep Adam’s method is five; with error constant .
3.2. Consistency of the Method
Definition 2 (Lambert [15]). The linear multistep method is said to be consistent if it has order .
Since the order of threestep Adam’s method is , therefore, the method is consistent according to Definition 2.
3.3. Stability of the Method
Definition 3 (Lambert [15]). A linear multistep method is zerostable provided the roots , of the first characteristics polynomial specified as satisfies and for those roots with the multiplicity must not exceed two.
Rewrite (9) in matrix form as follows:
From (15), the first characteristic polynomial, , where
According to Definition 3, threestep Adam’s method is zerostable.
3.4. Convergence of the Method
Definition 4 (Lambert [15]). 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.
4. Implementation of the Method
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 into th subinterval. For the Dirichlet boundary condition, the missing initial condition is . Equations (1) and (2) can be written as with initial conditions Therefore, we obtain the th stop conditions The iteration is repeated until we reach the stop conditions, the value of will be generated by threestep iterative method as follows: Detail for the threestep iterative method can be refered to in Yun [16].
The missing initial condition for the Neumann boundary condition is , therefore the first initial condition is , . The stop condition and the threestep iterative method implementation depend to the value of .
5. Numerical Result
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. [17]; bvp4c: MATLAB solver proposed by Kierzenka and Shampine [18]; MLAM: multilevel augmentation method proposed by Chen [19]; COLHW: collocation method with Haar wavelets proposed by SirajulIslam et al. [20]; SCM: sinccollocation method proposed by Mohamed [21]; LRBFM: local radial basis function method proposed by Mehdi and Ahmad [22]; 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; : .
Problem 1 (nonlinear Dirichlet boundary value problem). Consider
Dirichlet boundary condition: , ; exact solution: ; source: Chen [19].
Problem 2 (nonlinear Neumann boundary value problem). Consider
Neumann boundary condition: Exact solution: ; source: SirajulIslam et al. [20].
Problem 3 (nonlinear system of boundary value problem). Consider
Two Dirichlet boundary condition: Exact solution: , ; source: Mehdi and Ahmad [22].
Problem 4 (nonlinear system of boundary value problem). The equations governing the free convective boundarylayer flow above a heated impermeable horizontal surface are
Mixed boundary condition:
Source: Merkin and Zhang [23].
Problems 1–3 are solved by 3SAM with tolerances , , , , and . In Problem 1, we solved the boundary value problem subject to the Dirichlet type boundary conditions by 3SAM and compared our result with 2P1BVS, MLAM, and bvp4c. In Problem 2, 3SAM solved the boundary value problem subject to the Neumann type boundary conditions and compared our result to 2P1BVS, COLHW, and bvp4c. In Problem 3, 3SAM solved the system of boundary value problem subject to two boundary conditions and compared our result to SCM and LRBFM. 3SAM, 2P1BVS, and bvp4c are implemented by variable step size strategy and controlled by the tolerances while MLAM and COLHW used constant step size.
Tables 1–3 showed the comparison of the numerical result for solving Problems 1–3. We have observed that the accuracy for 3SAM is better as the total number of steps was increased for all problems tested. Firstly, we will discuss the block method (3SAM and 2P1BVS) implemented in variable step size strategy. The total number of steps and the total function call taken by 3SAM is less than 2P1BVS in all problems tested; this is expected because 3SAM can obtain the solution at three points simultaneously while 2P1BVS obtains two points simultaneously per step. We also noticed that the maximum error for 2P1BVS is comparable to or better than 3SAM but the accuracy for 3SAM is still within the tolerance. This is because the total step size of 3SAM is less than 2P1BVS therefore the step size used by 3SAM is larger than 2P1BVS. For example, in Table 1 the maximum error for 3SAM is with 4 steps and the maximum error for 2P1BVS is with 11 steps.



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 12, we choose the initial mesh point as 15. The total function call taken by 3SAM is less than bvp4c in all problems tested. We also noticed that the maximum error for 3SAM is comparable to or better than bvp4c when the tolerance is larger. For the tolerance getting smaller, the accuracy for bvp4c is comparable to or better than 3SAM but the accuracy for 3SAM is still within the tolerance. For example, in Table 2 the maximum error for 3SAM is with 341 total functions call and the maximum error for bvp4c is with 2745 total functions call. The accuracy of 3SAM is still within the tolerance and compared to the bvp4c and 3SAM is cheaper in terms of total function call and execution time.
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 3 the maximum error for 3SAM is with 7 steps, SCM has obtained the maximum error with 30 steps and LRBFM has obtained the maximum error with 61 steps. We also noticed that 3SAM has superiority in terms of execution time compared to MLAM.
In Problem 4, 3SAM solved the system of boundary value problem subject to the NeumannDirichlet type boundary conditions which is a free convective boundarylayer flow in a porous medium above a heated horizontal impermeable surface or below a cooled horizontal impermeable surface where wall temperature is a power function of distance from the origin. Figures 23 display the approximate solution of the value and for selected values in solving Problem 4.
6. Conclusion
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author gratefully acknowledged the financial support of Fundamental Research Grant Scheme (0201131157FR) and MyPhD scholarship from the Ministry of Education Malaysia.
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Copyright © 2014 Phang Pei See 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.