Iterative methods particularly the TwoParameter Alternating Group Explicit (TAGE) methods are used to solve system of linear equations generated from the discretization of twopoint 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.
Fuzzy boundary value problems (FBVPs) and treating fuzzy differential equations were one of the major applications for fuzzy number arithmetic [
Based on the Seikkala derivative [
The outline of the paper is organized as follows. Section
To be clear, let
Denote the value of
Now, we can express the secondorder central finite difference approximation (
Based on previous study conducted by Evans, clearly we can see that they have discussed theoretically how to compute the value of parameter
Family of AGE can be considered efficient to twostep 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 (
Then (
Initialize
For
compute
compute
Display approximate solutions.
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
Comparison of three parameters between GS, AGE, and TAGE methods at
Methods 



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 





Comparison of three parameters between GS, AGE, and TAGE methods at
Methods 



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 





Comparison of three parameters between GS, AGE, and TAGE methods at
Methods 



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 





Comparison of three parameters between GS, AGE, and TAGE methods at
Methods 



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 





Comparison of three parameters between GS, AGE, and TAGE methods at
Methods 



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 





Given two minimum bounding rectangles
In this paper, TAGE method was used to solve linear systems which arise from the discretization of twopoint FBVPs using the secondorder 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 [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper was funded by National Defence University of Malaysia.