Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Abstract

The mathematical structure of some natural phenomena of nonlinear physical and engineering systems can be described by a combination of fuzzy differential equations that often behave in a way that cannot be fully understood. In this work, an accurate numeric-analytic algorithm is proposed, based upon the use of the residual power series, to investigate the fuzzy approximate solution for a nonlinear fuzzy Duffing oscillator, along with suitable uncertain guesses under strongly generalized differentiability. The proposed approach optimizes the approximate solution by minimizing a residual function to generate r-level representation with a rapidly convergent series solution. The influence, capacity, and feasibility of the method are verified by testing some applications. Level effects of the parameter r are given graphically and quantitatively, showing good agreement between the fuzzy approximate solutions of upper and lower bounds, that together form an almost symmetric triangular structure, that can be determined by central symmetry at r = 1 in a convex region. At this point, the fuzzy number is a convex fuzzy subset of the real line, with a normalized membership function. If this membership function is symmetric, the triangular fuzzy number is called the symmetric triangular fuzzy number. Symmetrical fuzzy estimates of solutions curves indicate a sense of harmony and compatibility around the parameter r = 1. The results are compared numerically with the crisp solutions and those obtained by other existing methods, which illustrate that the suggested method is a convenient and remarkably powerful tool in solving numerous issues arising in physics and engineering.