International Journal of Differential Equations
Volume 2015 (2015), Article ID 346036, 7 pages
http://dx.doi.org/10.1155/2015/346036

Research Article

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

A. A. Dahalan¹ and J. Sulaiman²

¹Department of Mathematics, Centre for Defence Foundation Studies, National Defence University of Malaysia, 57000 Kuala Lumpur, Malaysia
²Faculty of Science & Natural Resources, Universiti Malaysia Sabah, 88400 Kota Kinabalu, Sabah, Malaysia

Received 21 January 2015; Revised 15 April 2015; Accepted 19 April 2015

Academic Editor: Bashir Ahmad

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.

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 [8–13] 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 as

which give

By using parametric form of fuzzy function, (1) can be written as

Suppose that and for . Then

By applying (2a) and (3a), (6a) will be reduced to

for . Meanwhile, by substituting (2b) and (3b) into (6b), we will have

Then, (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 as

for , where

Now, we can express the second-order central finite difference approximation (9) in a matrix form as

with

Since 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. [9–13]. 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) becomes

Thus, the explicit form of TAGE method can be written as

where 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.

Table 1: Comparison of three parameters between GS, AGE, and TAGE methods at .

Methods
n

512
1024
2048
4096
8192

Problem
Number of iterations
GS
681711
2431928
8548735
29480437
99066551

AGE
96747
354438
1279808
4549671
15883620

TAGE
77377
279463
876061
2879619
10383345

Execution time
GS
48.94
211.19
989.91
5719.20
32465.10

AGE
8.00
39.00
202.00
1310.00
8125.00

TAGE
7.00
31.00
141.00
822.00
5342.00

Hausdorff distance
GS

AGE

TAGE

Problem
Number of iterations
GS
475487
1692329
5930853
20369573
68062962

AGE
67638
247434
891667
3161503
10997813

TAGE
53492
187245
671456
2122064
7505046

Execution time
GS
35.27
155.77
764.09
4457.31
26063.40

AGE
6.00
27.00
141.00
912.00
5676.00

TAGE
5.00
20.00
107.00
608.00
3887.00

Hausdorff distance
GS

AGE

TAGE

Table 2: Comparison of three parameters between GS, AGE, and TAGE methods at .

Methods
n

512
1024
2048
4096
8192

Problem
Number of iterations
GS
682475
2434982
8560953
29529307
99262033

AGE
96840
354815
1281323
4555751
15908020

TAGE
77449
279746
876948
2882382
10399116

Execution time
GS
49.07
211.36
991.23
5874.81
32551.12

AGE
9.00
39.00
202.00
1301.00
8164.00

TAGE
7.00
31.00
141.00
827.00
5402.00

Hausdorff distance
GS

AGE

TAGE

Problem
Number of iterations
GS
476030
1694502
5939547
20404350
68202066

AGE
67704
247701
892745
3165828
11015151

TAGE
53543
187435
672208
2124610
7514448

Execution time
GS
35.26
155.79
756.06
4465.35
25999.98

AGE
6.00
27.00
142.00
903.00
5652.00

TAGE
5.00
21.00
106.00
605.00
3893.00

Hausdorff distance
GS

AGE

TAGE

Table 3: Comparison of three parameters between GS, AGE, and TAGE methods at .

Methods
n

512
1024
2048
4096
8192

Problem
Number of iterations
GS
683007
2437112
8569470
29563373
99398298

AGE
96905
355076
1282378
4559989
15925021

TAGE
77499
279944
877567
2884304
10408307

Execution time
GS
49.25
210.43
988.93
5784.36
32665.34

AGE
9.00
39.00
203.00
1311.00
8152.00

TAGE
6.00
31.00
141.00
837.00
5397.00

Hausdorff distance
GS

AGE

TAGE

Problem
Number of iterations
GS
476410
1696018
5945607
20428592
68299033

AGE
67751
247888
893496
3168843
11027246

TAGE
53578
187569
672733
2126364
7520993

Execution time
GS
35.40
155.80
757.38
4585.51
26078.03

AGE
6.00
27.00
141.00
912.00
5696.00

TAGE
5.00
21.00
107.00
620.00
3900.00

Hausdorff distance
GS

AGE

TAGE

Table 4: Comparison of three parameters between GS, AGE, and TAGE methods at .

Methods
n

512
1024
2048
4096
8192

Problem
Number of iterations
GS
683321
2438369
8574499
29583490
99478766

AGE
96944
355232
1283001
4562489
15935054

TAGE
77528
280061
877932
2885438
10414635

Execution time
GS
49.22
210.33
1026.58
5771.53
32617.94

AGE
8.00
39.00
203.00
1298.00
8186.00

TAGE
7.00
31.00
141.00
835.00
5395.00

Hausdorff distance
GS

AGE

TAGE

Problem
Number of iterations
GS
476633
1696912
5949186
20442908
68356295

AGE
67778
247998
893940
3170624
11034378

TAGE
53599
187647
673042
2127413
7524856

Execution time
GS
35.42
155.72
757.27
4364.75
26127.43

AGE
6.00
27.00
141.00
914.00
5706.00

TAGE
4.00
20.00
107.00
612.00
3937.00

Hausdorff distance
GS

AGE

TAGE

Table 5: Comparison of three parameters between GS, AGE, and TAGE methods at .

Methods
n

512
1024
2048
4096
8192

Problem
Number of iterations
GS
683426
2438784
8576162
29590144
99505380

AGE
96956
355282
1283208
4563320
15938400

TAGE
77538
280098
878054
2885812
10416768

Execution time
GS
49.45
210.66
809.53
5758.67
32519.13

AGE
9.00
39.00
202.00
1313.00
8221.00

TAGE
7.00
31.00
141.00
817.00
5383.00

Hausdorff distance
GS

AGE

TAGE

Problem
Number of iterations
GS
476706
1697208
5950370
20447642
68375230

AGE
67786
248034
894086
3171216
11036748

TAGE
53606
187674
673146
2127768
7526132

Execution time
GS
35.43
155.72
755.20
4615.31
25815.45

AGE
6.00
27.00
141.00
915.00
5662.00

TAGE
4.00
21.00
107.00
613.00
3941.00

Hausdorff distance
GS

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 for

are

respectively.

Problem 2 (see [17]). Consider

where with the boundary conditions and . The exact solutions for

are

respectively.

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 [19–24] and quarter-sweep [25–27] 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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