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

Abstract

In this article, an analytical reliable treatment based on the concept of residual error functions is employed to address the series solution of the differential logistic system in the fractional sense. The proposed technique is a combination of the generalized Taylor series and minimizing the residual error function. The solution methodology depends on the generation of a fractional expansion in an effective convergence formula, as well as on the optimization of truncated errors, Resqjt, through the use of repeated Caputo derivatives without any restrictive assumptions of system nature. To achieve this, some logistic patterns are tested to demonstrate the reliability and applicability of the suggested approach. Numerical comparison depicts that the proposed technique has high accuracy and less computational effect and is more efficient.

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

“…Meanwhile, the same procedure can be performed for other cases. To do that, let p and D 1 1 p are (1)-differentiable, that is, D 1 n p(t) and D 2 n,m p(t) exists, therefore, according to the RPS approach [25][26][27][28][29], the solutions of the converted crisp system (5) at t 0 = 0 can be given by the following forms:…”

“…It has been successfully used to establish reliable approximate solutions of many physical and engineering problems, including crisp initial value problems, differential algebraic equations system, singular initial value problems of nonlinear systems, and a fractional stiff system [20][21][22][23]. This approach aims to construct series solutions expansion, by minimizing the residual functions in computing the desirable unknown coefficients of these solutions, which typically produces the solutions in rapidly convergent series forms with no need linearization or any limitation on the nature of the problem and its classification [24][25][26][27][28][29][30][31][32][33].…”

Residual Power Series Technique for Simulating Fractional Bagley–Torvik Problems Emerging in Applied Physics

“…According the FRPS method [24][25][26][27], let us assume that the solutions of IVPs (8) and (9) can be written by…”

“…Thus, using the procedures of the RPS algorithm [25][26][27][28], the 4th RPS approximate solution of FBTEs (13) and (14) can be given by ω 4 (t) = 2t 4α Γ(4α+1) . Consequently, the RPS solution at α = 1/2 will be ω(t) = t 2 , which is fully compatible with the exact solution investigated earlier in [32].…”

“…Later, RPSM has been used in generating a fractional power series (FPS) solutions for strongly nonlinear FDEs in the form of a rapidly convergent with a minimum size of calculations without any restrictive hypotheses. Thus, this adaptive can be used as an alternative technique in solving several nonlinear problems arising in engineering and physics [24][25][26][27][28].…”