Computational Optimization of Residual Power Series Algorithm for Certain Classes of Fuzzy Fractional Differential Equations

Abstract

This paper aims to present a novel optimization technique, the residual power series (RPS), for handling certain classes of fuzzy fractional differential equations of order 1 < ≤ 2 under strongly generalized differentiability. The proposed technique relies on generalized Taylor formula under Caputo sense aiming at extracting a supportive analytical solution in convergent series form. The RPS algorithm is significant and straightforward tool for creating a fractional power series solution without linearization, limitation on the problem's nature, sort of classification, or perturbation. Some illustrative examples are provided to demonstrate the feasibility of the RPS scheme. The results obtained show that the scheme is simple and reliable and there is good agreement with exact solution.

An Analytical Numerical Method for Solving Fuzzy Fractional Volterra Integro-Differential Equations

“…Therefore, at β = 1, the approximated solutions of OFVIDEs (20) and (21) can be written as: The coefficients of the FRPS approximate solutions for the system (25) and (26): To obtain the value of the coefficients c n and d n , n = 1, 2, . .…”

“…The FRPS technique developed in [19] is considered an easy and applicable tool to create a power series solution for strongly linear and nonlinear equations without being linearized, discretized, or exposed to perturbation [20][21][22][23][24][25][26][27]. This technique is featured by the following characteristics: firstly, the method provides the solutions in Taylor expansions; therefore, the exact solutions will be available when the solutions are polynomials.…”

Numerical Modeling of Coupled Partial Differential Equations Using Residual Error Functions

“…This problem is a special case of the American option pricing problem under regime switching (1). We assume that, there are only two regimes (i = 1, 2) and introduce a time-reverse transformation t = T − τ .…”

“…The RPSM was first developed for solving first-order fuzzy differential equations. Later, it has been successfully applied to find numerical solutions for other equations, including ordinary and partial differential equations, nonlinear systems of singular initial value problems, pantograph delay differential equation, fractional differential equations, fuzzy fractional differential models [1,2,3,4,5,12,13,16,17,18,20,21,22].…”

Advanced Analytical Treatment of Fractional Logistic Equations Based on Residual Error Functions

“…In this research work, a reliable numerical treatment, the fractional residual power series method (FRPSM), is suggested for solving a sort of fractional logistic system. e FRPSM is an easy and reliable tool to obtain the values of unknown coefficients of desired fractional series solution for different types of linear and nonlinear FDEs without discretization, perturbation, and linearization by solving sequence of algebraic system [16][17][18][19][20][21][22][23]. e FRPS technique is primarily applied using the residual error concept and the repetition of Caputo derivatives to obtain the appropriate series solution by choosing a fit initial data, whereas the gained series solution and all fractional derivatives are valid for all mesh points of the domain of interest.…”