International Journal of Differential Equations

International Journal of Differential Equations / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 346036 | 7 pages | https://doi.org/10.1155/2015/346036

Implementation of TAGE Method Using Seikkala Derivatives Applied to Two-Point Fuzzy Boundary Value Problems

Academic Editor: Bashir Ahmad
Received21 Jan 2015
Revised15 Apr 2015
Accepted19 Apr 2015
Published14 Jun 2015

Abstract

Iterative methods particularly the Two-Parameter Alternating Group Explicit (TAGE) methods are used to solve system of linear equations generated from the discretization of two-point fuzzy boundary value problems (FBVPs). The formulation and implementation of the TAGE method are also presented. Then numerical experiments are carried out onto two example problems to verify the effectiveness of the method. The results show that TAGE method is superior compared to GS method in the aspect of number of iterations, execution time, and Hausdorff distance.

1. Introduction

Fuzzy boundary value problems (FBVPs) and treating fuzzy differential equations were one of the major applications for fuzzy number arithmetic [1]. FBVPs can be approached by two types. For instance, the first approach addresses problems in which the boundary values are fuzzy where the solution is still in fuzzy function. Then the second approach is based on generating the fuzzy solution from the crisp solution [2]. To solve these problems, numerical methods obtain their approximate solution. Consequently, in this paper, let two-point linear FBVPs be defined in general form as follows: where is a fuzzy function and , , and are continuous functions on , whereas, and are fuzzy numbers.

Based on the Seikkala derivative [3], (1) will be solved numerically by applying the second-order central finite difference scheme to discretize the two-point linear FBVPs into linear systems. Then the generated linear systems will be solved iteratively by using Two-Parameter Alternating Group Explicit (TAGE) method [4, 5]. By considering the Group Explicit (GE) method for the numerical solution of parabolic and elliptic problems, Evans [6, 7] discovered Alternating Group Explicit method. Later, Sukon and Evans [5] expanded this approach to initiate the TAGE method thus proving that this method is superior compared to AGE method. From previous studies, findings of the papers related to the TAGE iterative method and its variants [813] have shown that TAGE method has been widely used to solve the nonfuzzy problems. Due to the efficiency of the methods, this paper extends the application of TAGE iterative method in solving fuzzy problems. Since the fuzzy linear systems will be constructed, the iterative method becomes the natural option to get a fuzzy numerical solution of the problem.

The outline of the paper is organized as follows. Section 2 will discuss the finite difference method based on the second-order finite difference scheme in discretizing two-point FBVPs, while Section 3 presents the formulation and implementation of the TAGE methods in solving linear systems generated from the second-order finite difference scheme. Section 4 shows some numerical examples and conclusions are given in Section 5.

2. Finite Difference Approximation Equations

To be clear, let be a fuzzy subset of real numbers. It is characterized by the corresponding membership function evaluated at , writing as a number in . -cut of , in which is denoted as a crisp number, can be written as in , for . The interval of the -cut of fuzzy numbers will be written as , for all , since they were always closed and bounded [14]. Suppose is parametric form of fuzzy function . For arbitrary positive integer subdivide the interval , whereas for and .

Denote the value of and at the representative point by at . Thus, by using the second-order central finite difference scheme, problem (1) can be developed aswhich giveBy using parametric form of fuzzy function, (1) can be written asSuppose that and for . ThenBy applying (2a) and (3a), (6a) will be reduced tofor . Meanwhile, by substituting (2b) and (3b) into (6b), we will haveThen, (7a) and (7b) can be rewritten as follows:respectively, for . Since both of (8a) and (8b) have the same form in terms of the equation, except that, based on the interval of the -cuts, the differences are identified only in the upper and lower bounds, it can be rewritten asfor , where

Now, we can express the second-order central finite difference approximation (9) in a matrix form aswithSince this study will deal with an application of the method, the computational method of it will be diagonally dominant matrix and positive definite matrix [15].

3. Two-Parameter Alternating Group Explicit Iterative Method

Based on previous study conducted by Evans, clearly we can see that they have discussed theoretically how to compute the value of parameter given by Mohanty et al. [913]. In this paper, the optimum value of parameters and will be calculated by implementing several numerical experiments, so those optimum values will be found if the number of iterations is smaller.

Family of AGE can be considered efficient to two-step method to solve linear system. None of the researchers had been trying to apply this method in solving fuzzy problem generated from discretization of fuzzy partial difference equation. This paper will discuss the application of this iterative method which will solve the fuzzy linear system given by (1). Consider a class of methods mentioned in [4, 5] which is based on the splitting of the matrix into the sum of its constituent symmetric and positive definite matrices, as follows:where if is odd. Similarly, we define the following matrices:if is even, with . In this paper, we only consider that case is even.

Then (11) becomesThus, the explicit form of TAGE method can be written aswhere are the acceleration parameters, and a pair of and are invertible. From (17), therefore, the implementation of TAGE method is presented in Algorithm 1.

Algorithm 1 (TAGE method). (i)Initialize and .(ii)For , initialize parameters , , , , , , , and .(iii) First Sweep. For ,compute(iv) Second Sweep. For ,compute(v) Convergence Test. If the convergence criterion, that is, , is satisfied, go to Step (vi). Otherwise go back to Step (ii).(vi)Display approximate solutions.

4. Numerical Experiments

Two examples of FBVPs are considered to verify the effectiveness of GS, AGE, and TAGE methods. For comparison purposes, three parameters were observed that are number of iterations, execution time (in seconds), and Hausdorff distance (as mentioned in Definition 2). Based on these two problems, numerical results for GS, AGE, and TAGE methods have been recorded in Tables 1 to 5.


Methodsn
5121024204840968192

Problem Number of iterationsGS681711243192885487352948043799066551
AGE967473544381279808454967115883620
TAGE77377279463876061287961910383345
Execution timeGS48.94211.19989.915719.2032465.10
AGE8.0039.00202.001310.008125.00
TAGE7.0031.00141.00822.005342.00
Hausdorff distanceGS
AGE
TAGE

Problem Number of iterationsGS475487169232959308532036957368062962
AGE67638247434891667316150310997813
TAGE5349218724567145621220647505046
Execution timeGS35.27155.77764.094457.3126063.40
AGE6.0027.00141.00912.005676.00
TAGE5.0020.00107.00608.003887.00
Hausdorff distanceGS
AGE
TAGE


Methodsn
5121024204840968192

Problem Number of iterationsGS682475243498285609532952930799262033
AGE968403548151281323455575115908020
TAGE77449279746876948288238210399116
Execution timeGS49.07211.36991.235874.8132551.12
AGE9.0039.00202.001301.008164.00
TAGE7.0031.00141.00827.005402.00
Hausdorff distanceGS
AGE
TAGE

Problem Number of iterationsGS476030169450259395472040435068202066
AGE67704247701892745316582811015151
TAGE5354318743567220821246107514448
Execution timeGS35.26155.79756.064465.3525999.98
AGE6.0027.00142.00903.005652.00
TAGE5.0021.00106.00605.003893.00
Hausdorff distanceGS
AGE
TAGE


Methodsn
5121024204840968192

Problem Number of iterationsGS683007243711285694702956337399398298
AGE969053550761282378455998915925021
TAGE77499279944877567288430410408307
Execution timeGS49.25210.43988.935784.3632665.34
AGE9.0039.00203.001311.008152.00
TAGE6.0031.00141.00837.005397.00
Hausdorff distanceGS
AGE
TAGE

Problem Number of iterationsGS476410169601859456072042859268299033
AGE67751247888893496316884311027246
TAGE5357818756967273321263647520993
Execution timeGS35.40155.80757.384585.5126078.03
AGE6.0027.00141.00912.005696.00
TAGE5.0021.00107.00620.003900.00
Hausdorff distanceGS
AGE
TAGE


Methodsn
5121024204840968192

Problem Number of iterationsGS683321243836985744992958349099478766
AGE969443552321283001456248915935054
TAGE77528280061877932288543810414635
Execution timeGS49.22210.331026.585771.5332617.94
AGE8.0039.00203.001298.008186.00
TAGE7.0031.00141.00835.005395.00
Hausdorff distanceGS
AGE
TAGE

Problem Number of iterationsGS476633169691259491862044290868356295
AGE67778247998893940317062411034378
TAGE5359918764767304221274137524856
Execution timeGS35.42155.72757.274364.7526127.43
AGE6.0027.00141.00914.005706.00
TAGE4.0020.00107.00612.003937.00
Hausdorff distanceGS
AGE
TAGE


Methodsn
5121024204840968192

Problem Number of iterationsGS683426243878485761622959014499505380
AGE969563552821283208456332015938400
TAGE77538280098878054288581210416768
Execution timeGS49.45210.66809.535758.6732519.13
AGE9.0039.00202.001313.008221.00
TAGE7.0031.00141.00817.005383.00
Hausdorff distanceGS
AGE
TAGE

Problem Number of iterationsGS476706169720859503702044764268375230
AGE67786248034894086317121611036748
TAGE5360618767467314621277687526132
Execution timeGS35.43155.72755.204615.3125815.45
AGE6.0027.00141.00915.005662.00
TAGE4.0021.00107.00613.003941.00
Hausdorff distanceGS
AGE
TAGE

Definition 2 (see [16]). Given two minimum bounding rectangles and , a lower bound of the Hausdorff distance from the elements confined by to the elements confined by is defined as

Problem 1. Consider where with the boundary conditions and . The exact solutions forarerespectively.

Problem 2 (see [17]). Consider where with the boundary conditions and . The exact solutions forarerespectively.

5. Conclusions

In this paper, TAGE method was used to solve linear systems which arise from the discretization of two-point FBVPs using the second-order central finite difference scheme. The results show that TAGE method is more superior in terms of the number of iterations, execution time, and Hausdorff distance compared to the AGE and GS methods. Since TAGE is well suited for parallel computation, it can be considered as a main advantage because this method has groups of independent task which can be implemented simultaneously. It is hoped that the capability of the proposed method will be helpful for the further investigation in solving any multidimensional fuzzy partial differential equations [18]. Basically the results of this paper can be classified as one of full-sweep iteration. Apart from the concept of the full-sweep iteration, further investigation of half-sweep [1924] and quarter-sweep [2527] iterations can also be considered in order to speed up the convergence rate of the standard proposed iterative methods. Other than that, further study will be extended to solve nonlinear problem by combining Newton-Raphson method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This paper was funded by National Defence University of Malaysia.

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Copyright © 2015 A. A. Dahalan and J. Sulaiman. 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.

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