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.

Solutions of Fractional Verhulst Model by Modified Analytical and Numerical Approaches

“…This section is devoted to introducing some necessary definitions and mathematical preliminaries of fractional calculus, especially those related to Caputo operator, and basics of the generalised fractional power series. For more details, the reader can refer to [ 8 , 11 , 35 , 42 ] for more details about fractional derivatives. Throughout this chapter, we deal with the following spaces: , ; .…”

New horizon in fuzzy distributions: statistical distributions in continuous domains generated by Choquet integral

“… 2019 ; Agahi 2021 ; Alshammari et al. 2020 ; Beliakov et al. 2019 ; Faigle and Grabisch 2011 ; Grabisch 2016 ; Labreuche and Grabisch 2018 ; Sugeno 2013 , 2015 ; Torra 2017 ; Torra and Narukawa 2016 ; Torra et al.…”

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

“…Exploring the analytic solution of FDEs and FIDEs is difficult in most cases, even though abundant efforts have been introduced recently to develop emerging numerical and approximate-analytical techniques for finding out the solutions to linear and nonlinear fractional problems. Among these methods, the kernel Hilbert space method [29], the Haar wavelet method [30], the Adomian decomposition method [31], the homotopy analysis method [32], the finite difference method [33], the Taylor series expansion method [34], the collocation method [35], the Aboodh transform decomposition method [36], and the residual fractional power series (FPS) method [37,38] have been reproduced. The FPS method is one of the semianalytical techniques which befits both linear and nonlinear FDEs [39][40][41].…”