Abstract
The nonlinear equation is a fundamentally important area of study in mathematics, and the numerical solutions of the nonlinear equations are also an important part of it. Fuzzy sets introduced by Zedeh are an extension of classical sets, which have several applications in engineering, medicine, economics, finance, artificial intelligence, decision-making, and so on. The most special types of fuzzy sets are fuzzy numbers. The important fuzzy numbers are trapezoidal fuzzy and triangular fuzzy numbers, which have several applications. In this research article, we propose an efficient numerical iterative method for estimating roots of fuzzy nonlinear equations, which are based on the special type of fuzzy number called triangular fuzzy number. Convergence analysis proves that the order of convergence of the numerical method is three. Some real-life applications are considered as numerical test problems from engineering, which contain fuzzy quantities in the parametric form. Engineering models include fractional conversion of nitrogen-hydrogen feed into ammonia and Van der Waal’s equation for calculating the volume and pressure of a gas and motion of the object under constant force of gravity. Numerical illustrations are given to show the dominance efficiency of the newly constructed iterative schemes as compared to existing methods in the literature.
Highlights
One of the ancient problems of science and engineering in general and in mathematics in particular is to approximate roots of nonlinear equations. e nonlinear equations play a major role in the field of engineering, mathematics, physics, chemistry, economics, medicines, and finance
Several real world applications contain vagueness and uncertainties. erefore, in most of real world problems, the parameters involved in the system or variables of the nonlinear equations are presented by a fuzzy number. e concepts of fuzzy numbers and arithmetic operation with fuzzy numbers were first introduced and investigated in [1,2,3,4,5,6,7,8,9,10]
It is necessary to approximate the root of fuzzy nonlinear equation: F(x) c
Summary
One of the ancient problems of science and engineering in general and in mathematics in particular is to approximate roots of nonlinear equations. e nonlinear equations play a major role in the field of engineering, mathematics, physics, chemistry, economics, medicines, and finance. 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. Iterative methods presented by them have low convergence order to approximate the roots of fuzzy nonlinear equations. Engineers and mathematicians look towards those numerical methods which are more efficient and with high convergence order and low computational cost, or time, to approximate the roots of highly nonlinear fuzzy equations.
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