IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi Publishing Corporation 10.1155/2015/346036 346036 Research Article Implementation of TAGE Method Using Seikkala Derivatives Applied to Two-Point Fuzzy Boundary Value Problems Dahalan A. A. 1 Sulaiman J. 2 Ahmad Bashir 1 Department of Mathematics, Centre for Defence Foundation Studies National Defence University of Malaysia 57000 Kuala Lumpur Malaysia upnm.edu.my 2 Faculty of Science & Natural Resources Universiti Malaysia Sabah 88400 Kota Kinabalu, Sabah Malaysia ums.edu.my 2015 1462015 2015 21 01 2015 15 04 2015 19 04 2015 1462015 2015 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.

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 . 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 . 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: (1)x~t+ptx~t+qtx~t=ft,ta,b,x~a=σ,x~b=ω,where x~t is a fuzzy function and f(t), p(t), and q(t) are continuous functions on a,b, whereas, σ and ω are fuzzy numbers.

Based on the Seikkala derivative , (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  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  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 x~ be a fuzzy subset of real numbers. It is characterized by the corresponding membership function evaluated at t, writing x~t as a number in [0,1]. α-cut of x~, in which α is denoted as a crisp number, can be written as x~α in xx~tα, for 0<α1. The interval of the α-cut of fuzzy numbers will be written as x~α=x_α,x¯α, for all α, since they were always closed and bounded . Suppose x_,x¯ is parametric form of fuzzy function x. For arbitrary positive integer n subdivide the interval atb, whereas ti=a+ihi=0,1,2,,n for i and h=b-a/n.

Denote the value of x and x_,x¯ at the representative point tii=0,1,2,,n by xi at xi_,xi¯. Thus, by using the second-order central finite difference scheme, problem (1) can be developed as(2a)xi_xi-1_-2xi_+xi+1_h2,(2b)xi¯xi-1¯-2xi¯+xi+1¯h2,(3a)xi_xi+1_-xi-1_2h,(3b)xi¯xi+1¯-xi-1¯2h,which give(4)xi=xi_,xi¯,xi=xi_,xi¯.By using parametric form of fuzzy function, (1) can be written as(5a)xi_=fti-ptixi-qtixi_,(5b)xi¯=fti-ptixi-qtixi¯.Suppose that pti>0 and qti>0 for i=0,1,2,,n. Then(6a)xi_+ptixi_+qtixi_=fti,(6b)xi¯+ptixi¯+qtixi¯=fti.By applying (2a) and (3a), (6a) will be reduced to(7a)xi-1_-2xi_+xi+1_h2+ptixi+1_-xi-1_2h+qtixi_=ftifor i=1,2,,n-1. Meanwhile, by substituting (2b) and (3b) into (6b), we will have(7b)xi-1¯-2xi¯+xi+1¯h2+ptixi+1¯-xi-1¯2h+qtixi¯=fti.Then, (7a) and (7b) can be rewritten as follows:(8a)2-hptixi-1_+2h2qti-4xi_+2+hptixi+1_=2h2fti,(8b)2-hptixi-1¯+2h2qti-4xi¯+2+hptixi+1¯=2h2fti,respectively, for i=1,2,,n-1. Since both of (8a) and (8b) have the same form in terms of the equation, except that, based on the interval of the α-cuts, the differences are identified only in the upper and lower bounds, it can be rewritten as(9)ρixi-1+βixi+φixi+1=Fifor i=1,2,,n-1, where(10)ρi=2-hpti,βi=2h2qti-4,φi=2+hpti,Fi=2h2fti.

Now, we can express the second-order central finite difference approximation (9) in a matrix form as(11)Ax=bwith(12)A=β1φ1ρ2β2φ2ρ3β3φ3ρn-3βn-3φn-3ρn-2βn-2φn-2ρn-1βn-1n-1×n-1,x=x1x2xn-2xn-1T,b=f1-ρ1x0f2fn-2fn-1-φn-1xnT.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 .

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 r given by Mohanty et al. . In this paper, the optimum value of parameters r1 and r2 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 A into the sum of its constituent symmetric and positive definite matrices, as follows:(13)A=G1+G2,where (14) G 1 = g 1 φ 1 ρ 2 g 2 g 3 φ 3 ρ 4 g 4 g n - 2 φ n - 2 ρ n - 1 g n - 1 , G 2 = g 1 g 2 φ 2 ρ 3 g 3 g n - 3 φ n - 3 ρ n - 2 g n - 2 g n - 1 if n is odd. Similarly, we define the following matrices: (15) G 1 = g 1 φ 1 ρ 2 g 2 g n - 3 φ n - 3 ρ n - 2 g n - 2 g n - 1 , G 2 = g 1 g 2 φ 2 ρ 3 g 3 g n - 2 φ n - 2 ρ n - 1 g n - 1 if n is even, with gi=βi/2i=1,2,,n-1. In this paper, we only consider that case n is even.

Then (11) becomes(16)G1+G2x=b.Thus, the explicit form of TAGE method can be written as(17)xk+1/2=G1+r1I-1b-G2-r1Ixk,xk+1=G2+r2I-1b-G1-r2Ixk+1/2,where r1,r2>0 are the acceleration parameters, and a pair of G1+r1I and G2+r2I are invertible. From (17), therefore, the implementation of TAGE method is presented in Algorithm 1.

Algorithm 1 (TAGE method).

Initialize U~(0)0 and ε10-10.

For i=1p,2p,,n-p, initialize parameters ρi, βi, φi, fi, r1, r2, G1, and G2.

First Sweep. For i=1p,2p,,n-p,

compute(18)xk+1/2=G1+r1I-1b-G2-r1Ixk.

Second Sweep. For i=1p,2p,,n-p,

compute(19)xk+1=G2+r2I-1b-G1-r2Ixk+1/2.

Convergence Test. If the convergence criterion, that is, U~k+1-U~kε, is satisfied, go to Step (vi). Otherwise go back to Step (ii).

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.

Comparison of three parameters between GS, AGE, and TAGE methods at α=0.00.

Methods n
512 1024 2048 4096 8192
Problem 1 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 2.6560 e - 06 1.0624 e - 05 4.2497 e - 05 1.6999 e - 04 6.7995 e - 04
AGE 3.2355 e - 07 1.3084 e - 06 5.2674 e - 06 2.1148 e - 05 8.4787 e - 05
TAGE 2.4955 e - 07 9.8491 e - 07 3.0435 e - 06 1.0607 e - 05 4.5689 e - 05

Problem 2 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 2.4952 e - 06 7.7115 e - 06 3.0279 e - 05 1.2097 e - 04 4.8386 e - 04
AGE 8.3545 e - 07 1.0823 e - 06 3.7861 e - 06 1.5058 e - 05 6.0327 e - 05
TAGE 7.7615 e - 07 8.1054 e - 07 2.6440 e - 06 1.0853 e - 05 3.2986 e - 05

Comparison of three parameters between GS, AGE, and TAGE methods at α=0.25.

Methods n
512 1024 2048 4096 8192
Problem 1 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 2.6560 e - 06 1.0624 e - 05 4.2497 e - 05 1.6999 e - 04 6.7995 e - 04
AGE 3.2355 e - 07 1.3083 e - 06 5.2675 e - 06 2.1148 e - 05 8.4786 e - 05
TAGE 2.4955 e - 07 9.8490 e - 07 3.0417 e - 06 1.0452 e - 05 4.5526 e - 05

Problem 2 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 2.4650 e - 06 7.7039 e - 06 3.0277 e - 05 1.2097 e - 04 4.8386 e - 04
AGE 8.0517 e - 07 1.0748 e - 06 3.7841 e - 06 1.5058 e - 05 6.0327 e - 05
TAGE 7.4595 e - 07 8.0323 e - 07 2.6424 e - 06 1.0749 e - 05 3.2626 e - 05

Comparison of three parameters between GS, AGE, and TAGE methods at α=0.50.

Methods n
512 1024 2048 4096 8192
Problem 1 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 2.6560 e - 06 1.0624 e - 05 4.2497 e - 05 1.6999 e - 04 6.7995 e - 04
AGE 3.2355 e - 07 1.3084 e - 06 5.2675 e - 06 2.1148 e - 05 8.4785 e - 05
TAGE 2.4955 e - 07 9.8489 e - 07 3.0398 e - 06 1.0293 e - 05 4.5445 e - 05

Problem 2 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 2.4346 e - 06 7.6963 e - 06 3.0275 e - 05 1.2097 e - 04 4.8386 e - 04
AGE 7.7486 e - 07 1.0672 e - 06 3.7823 e - 06 1.5057 e - 05 6.0327 e - 05
TAGE 7.1585 e - 07 7.9587 e - 07 2.6408 e - 06 1.0642 e - 05 3.2256 e - 05

Comparison of three parameters between GS, AGE, and TAGE methods at α=0.75.

Methods n
512 1024 2048 4096 8192
Problem 1 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 2.6560 e - 06 1.0624 e - 05 4.2497 e - 05 1.6999 e - 04 6.7995 e - 04
AGE 3.2355 e - 07 1.3083 e - 06 5.2675 e - 06 2.1148 e - 05 8.4786 e - 05
TAGE 2.4958 e - 07 9.8485 e - 07 3.0382 e - 06 1.0131 e - 05 4.5365 e - 05

Problem 2 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 2.4044 e - 06 7.6888 e - 06 3.0273 e - 05 1.2097 e - 04 4.8386 e - 04
AGE 7.4463 e - 07 1.0596 e - 06 3.7803 e - 06 1.5057 e - 05 6.0327 e - 05
TAGE 6.8564 e - 07 7.8857 e - 07 2.6393 e - 06 1.0531 e - 05 3.1874 e - 05

Comparison of three parameters between GS, AGE, and TAGE methods at α=1.00.

Methods n
512 1024 2048 4096 8192
Problem 1 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 2.6559 e - 06 1.0624 e - 05 4.2497 e - 05 1.6999 e - 04 6.7995 e - 04
AGE 3.2354 e - 07 1.3084 e - 06 5.2674 e - 06 2.1148 e - 05 8.4783 e - 05
TAGE 2.4955 e - 07 9.8492 e - 07 3.0366 e - 06 9.9651 e - 06 4.5321 e - 05

Problem 2 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 2.3742 e - 06 7.6812 e - 06 3.0271 e - 05 1.2097 e - 04 4.8386 e - 04
AGE 7.1441 e - 07 1.0521 e - 06 3.7785 e - 06 1.5056 e - 05 6.0326 e - 05
TAGE 6.5552 e - 07 7.8117 e - 07 2.6376 e - 06 1.0417 e - 05 3.1481 e - 05
Definition 2 (see [<xref ref-type="bibr" rid="B19">16</xref>]).

Given two minimum bounding rectangles P and Q, a lower bound of the Hausdorff distance from the elements confined by P to the elements confined by Q is defined as(20)HausDistLBP,Q=MaxMinDistfα,Q:fαFaces  Of  P.

Problem 1. Consider (21)xt=k~-6t,t0,1,where k~α=k_α,k¯α=0.75+0.25α,1.25-0.25α with the boundary conditions x~0=0 and x~1=1. The exact solutions for(22a)x_t;α=k_α-6t,(22b)x¯t;α=k¯α-6tare(23a)x_t;α=k_α-t3+2t,(23b)x¯t;α=k¯α-t3+2t,respectively.

Problem 2 (see ). Consider (24)xt-4xt=k~4cosh1,t0,1,where k~α=k_α,k¯α=0.75+0.25α,1.25-0.25α with the boundary conditions x~0=0 and x~1=0. The exact solutions for(25a)x_t;α-4x_t;α=k_α4cosh1,(25b)x¯t;α-4x¯t;α=k¯α4cosh1are(26a)x_t;α=k_αcosh2t-1-cosh1,(26b)x¯t;α=k¯αcosh2t-1-cosh1,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 . 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  and quarter-sweep  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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