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 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. [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) 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.
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 [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.