Abstract
In this research article, we propose efficient numerical iterative methods for estimating roots of interval-valued trapezoidal fuzzy nonlinear equations. Convergence analysis proves that the order of convergence of numerical schemes is 3. Some real-life applications are considered from optimization as numerical test problems which contain interval-valued trapezoidal fuzzy quantities in parametric form. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in literature.
1. Introduction
One of the ancient problems of science and engineering in general and in mathematics is to approximate roots of a nonlinear equation. The nonlinear equations play a major role in the field of engineering, mathematics, physics, chemistry, economics, medicines finance, and in optimization. Many times the particular realization of such type of nonlinear problems involves imprecise and nonprobabilistic uncertainties in the parameter, where the approximations are known due to expert knowledge or due to some experimental data. Due to these reasons, several real-world applications contain vagueness and uncertainties. Therefore, in most of real-world problems, the parameter involved in the system or variables of the nonlinear functions are presented by a fuzzy number or interval-valued trapezoidal fuzzy number. The concept of fuzzy numbers and arithmetic operation with fuzzy numbers were first introduced and investigated in [1–8]. Hence, it is necessary to approximate the roots of fuzzy nonlinear equation.
The standard analytical technique like the Buckley and Qu method [9–12] is not suitable for solving the equations like where , , , , , , ,and are fuzzy numbers and is a fuzzy variable.
We therefore look towards numerical iterative schemes which approximate the roots of fuzzy nonlinear equations. To approximate roots of fuzzy nonlinear equations, Abbsbandy and Asady [13] used Newton’s method, Allahviranloo and Asari [14] used the Newton-Raphson method, Mosleh [15] used the Adomian decomposition method, and Ibrahim et al. give the Levenberg-Marquest method (see also [16–23]).
This research article is aimed at proposing efficient higher order iterative method as compared to well-known classical method, such as the Newton-Raphson method. Numerical test results, CPU time, and log of residual show the dominance efficiency of our newly constructed method over the classical Newton’s method.
This paper is organized in five sections. In Section 2, we recall some fundamental results of interval-valued trapezoidal fuzzy numbers. In Section 3, we propose numerical iterative scheme for approximating roots of interval-valued trapezoidal fuzzy nonlinear equations and its convergence analysis. In Section 4, we illustrate some real-world applications from optimization as numerical test examples to show the performance and efficiency of the constructed method and conclusions in the last section. Section 5 is a conclusion section.
2. Preliminaries
Definition 1. A fuzzy number is a fuzzy set like which satisfies [24–27]. (1) is upper semicontinuous(2) outside some interval (3)There are real numbers such that and (i) is monotonic increasing on (ii) is monotonic decreasing on (iii), for We denote by the set of all fuzzy numbers. An equivalent parametric form is also given in [19] as follows.
Definition 2 [28]. A fuzzy number in parametric form is a pair of function which satisfies the following requirements: (1) is a bounded monotonic increasing left continuous function(2) is a bounded monotonic decreasing left continuous function(3) A popular fuzzy number is the generalized interval-valued trapezoidal fuzzy number , denoted by a fuzzy number with membership function as follows: Assume be the family of all -trapezoidal fuzzy number, i.e.,
Definition 3 [29]. Let and level interval-valued trapezoidal fuzzy number , denoted by is an interval-valued fuzzy number on set with where , , , , and . This interval-valued trapezoidal fuzzy number is shown in Figure 1. Moreover, , which means the grade of membership , and the latest and greatest grade of membership at are and , respectively. We therefore denote the family of all interval-valued trapezoidal fuzzy number by , i.e.,
Definition 4 [29]. A is said to be nonnegative iff and denoted by .
Definition 5 [30]. Two interval-valued trapezoidal fuzzy numbers.
and are said to be equal iff , i.e., and for all .
Definition 6. [30]. Extend addition, scalar multiplication, and extend multiplication in interval-valued trapezoidal fuzzy number are defined as if and and ; then,
Definition 7 [28]. Let . The signed distance between and is .
Definition 8 [31]. Let ; then, alpha-cut set of denotes and is defined by where
3. Construction of Iterative Scheme (MM)
In order to approximate the roots of interval-valued trapezoidal fuzzy nonlinear equation , we propose the following two-step iterative scheme as follows:
Suppose that is the solution of above system and is approximate solutions of the system, t denote the alpha-cut parameter; then,
By using Taylor’s series of about , then we have the following:
If are close to , then are small enough. Assume all partial derivatives of exist and bounded; then, we have the following:
Since are unknown quantities, they are obtained by solving the following equations: where
, , and the next approximation for is found by using recursive scheme as follows:
For initial guess, one can use the fuzzy number
Remark 9. Sequence converges to iff , , , , and .
Lemma 10. Let , and if the sequence of converges to according to Newton’s method, then where
Proof. It is obvious, because for all in convergent case, Hence, it is proved.☐
Finally, it is shown that under certain condition, the MM method for fuzzy equation is cubic convergent. In compact form for , the MM method can be written as follows: where
Theorem 11. Let be -times Frĕchet differential function on a convex set containing the root of ; then, the MM method has cubic convergence and satisfies the following error equation. where .
Proof. Let and , then by Taylor series of in the neighborhood of if and exist. Then, and This gives Expanding about we have the following: Hence, the theorem is proved.☐
A well-known existing method in literature for solving triangular fuzzy nonlinear equation is classical Newton Raphson’s method. Interval-valued trapezoidal fuzzy version of well-known Newton method [13] (abbreviated as NN) for finding roots of interval-valued trapezoidal fuzzy nonlinear equation is as follows:
where
4. Numerical Applications
Here, we present examples to illustrate the performance and efficiency of MM and NN methods for approximating roots of interval-valued trapezoidal fuzzy nonlinear equations. Examples 1–3 are considered from Buckley and Qu [9]. All the computations are performed using CAS Maple 18 with 64 digits floating point arithmetic with stopping criteria as follows. Analytical, numerical approximate solutions, computational order of convergence [32], computational time in second, and residual error graph of interval-valued trapezoidal fuzzy nonlinear equation used in Examples 1–3 are shown in Figures 2–8(a) and 8(c), respectively. Algorithm 1 shows the implementation of MM iterative method on CAS Maple18. where represents the absolute error. We take .
(a)
(b)
(c)
In Figure 2, left shows analytical solution of interval-valued trapezoidal fuzzy nonlinear equation used in Example 1, center shows for Example 2, and right shows for Example 3, respectively.
| ||||||||||||||||||
Figure 3 shows computational order of convergence of iterative methods MM and NN for finding roots of interval-valued trapezoidal fuzzy nonlinear equations used in Examples 1–3, respectively.
In Figure 3, MM1-MM4 and NN1-NN4 show computational order of convergence of iterative method MM and NN for approximating roots of interval-valued trapezoidal fuzzy nonlinear equations used in Examples 1–3, respectively.
Figure 4 shows computational time in seconds of iterative methods MM and NN for finding roots of interval-valued trapezoidal fuzzy nonlinear equations used in Examples 1–3, respectively.
In Figure 4, MM1-MM4 and NN1-NN4 show computational time in seconds of iterative method MM and NN for finding roots of interval-valued trapezoidal fuzzy nonlinear equation used in Examples 1–3, respectively.
Example 1 Application in optimization (a profit maximization problem). A corporation company wishes to invest one million dollar at fuzzy interest rate to earn maximum profit, so that after a year, they may withdraw 25000$ approximately and after two years 900000$ left. Find so that will be sufficient to cover and where is an interval-valued trapezoidal fuzzy number whose support lies between After a year, the amount in the account will be At the end of second year, total amount left is or where and . Therefore, we have to solve or where . For fuzzy interest rate substituting values of , , and in above equation, we have the following: Without any loss of generality, assume that is positive; then, the parametric form of this equation is as follows:
Table 1 clearly shows the dominance behavior of MM over NN in terms of absolute error on the same number of iterations for Example 1.
Table 2 shows analytical solutions for Example 1.
Figure 5 shows initial guessed values, analytical, and numerical approximate solution graph of iterative methods MM and NN for interval-valued trapezoidal fuzzy nonlinear equation used in Example 1.
To obtain initial guess, we use above system for and ; therefore,
Consequently, , , , , and . After 4 iterations, we obtain the solution with the maximum error less than . Now suppose is negative, we have the following:
For , we have ; therefore negative root does not exist.
Example 2. Consider the interval-valued trapezoidal fuzzy nonlinear equation Without any loss of generality, assume that is positive; then, the parametric form of this equation is as follows: or
Table 3 clearly shows the dominance behavior of MM over NN in terms of absolute error on the same number of iterations for Example 2.
Table 4 shows analytical solutions for Example 2.
Figure 6 shows initial guessed values, analytical, and numerical approximate solution graph of iterative methods MM and NN for interval-valued trapezoidal fuzzy nonlinear equation used in Example 2.
To obtain initial guess, we use above system for and ; therefore,
Consequently, , , , , and . After 4 iterations, we obtain the solution with the maximum error less than . Now suppose is negative, we have
For , we have , therefore, negative root does not exist.
Example 3. Consider the interval-valued trapezoidal fuzzy nonlinear equation Without any loss of generality, assume that is positive; then, the parametric form of this equation is as follows: or
Table 5 clearly shows the dominance behavior of MM over NN in terms of absolute error on the same number of iterations for Example 3.
Table 6 shows analytical solutions for Example 3.
Figure 7 shows initial guessed values, analytical, and numerical approximate solution graph of iterative methods MM and NN for interval-valued trapezoidal fuzzy nonlinear equation used in Example 3.
To obtain initial guess, we use above system for and ; therefore,
Consequently, , , , , and . After 4 iterations, we obtain the solution with the maximum error less than . Now suppose is negative, we have
For , we have hence ; therefore, negative root does not exist.
Figures 8(a)–8(c) show residual falls for iterative methods MM and NN for interval-valued trapezoidal fuzzy nonlinear equation used in Examples 1–3, respectively.
5. Conclusion
In this research paper, we constructed highly efficient two-step numerical iterative method to approximate roots of interval-valued trapezoidal fuzzy nonlinear equations. A set of real-life applications from optimization are considered as a numerical test examples showing the practical performance and dominance efficiency of MM over NN method on the same number of iterations. From Tables 1–6 and Figures 1–8, we observe that numerical results of MM methods are better in terms of absolute error and CPU time as compared to NN method. Considering the same ways as in this article, we can establish higher order and efficient numerical iterative methods for solving system of fuzzy nonlinear equations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Authors’ Contributions
All authors contributed equally in the preparation of this manuscript.
Acknowledgments
The first author would like to thank the Deanship of Scientific Research at Majmaah University for supporting this work under Project Number R-2021-170.