Performance analysis of a modified Newton method for parameterized dual fuzzy nonlinear equations and its application - Sulaiman, Ibrahim Mohammed; Mamat, Mustafa; Malik, Maulana; Nisar, Kottakkaran Sooppy; Elfasakhany, Ashraf (Results in physics, 2022)
In this paper, we are interested in the numerical solution of nonlinear equations, particularly, when the coefficients are fuzzy numbers rather than crisp numbers. These problems often arise in numerous applications and have recently been the subject of several studies, where many researchers parameterized the fuzzy coefficients and proposed numerical method for obtaining the solutions. Most of the numerical methods applied for solving the parameterized fuzzy equations are Newton-like methods that require the computation of Jacobian matrix at every iteration or after every few iterations. However, the Newton search direction may not be a descent direction if the Jacobian J(xk) is not positive definite, and, when J(xk) is singular, the method may fail to converge. Also, there are limited literatures on numerical methods for solution of dual fuzzy nonlinear equations. This paper proposed a variant of Newton’s method for solving parameterized dual fuzzy nonlinear equations. This method introduces a new parameter μk to the Newton’s method to ensure that the search direction is a descent direction even when J(xk) is not positive definite. The proposed method was further applied to solve an application problem. Preliminary results on the considered benchmark and application problems show that the new algorithm effective and promising. •A modified Newton method was proposed for solving dual fuzzy nonlinear equations.•The proposed method introduces a parameter μk to the Newton method.•μk guarantee a descent search direction in the case the of positive definite Jacobian.•The convergence analysis of the proposed method was discussed under suitable conditions.•Preliminary results presented show that the proposed method is efficient and•In addition, the method can find wider application in real-life problems.