A spectral collocation method for the coupled system of nonlinear fractional differential equations

<abstract><p>This paper analyzes the coupled system of nonlinear fractional differential equations involving the caputo fractional derivatives of order $ \alpha\in(1, 2) $ on the interval (0, T). Our method of analysis is based on the reduction of the given system to an equivalent system of integral equations, then the resulting equation is discretized by using a spectral method based on the Legendre polynomials. We have constructed a Legendre spectral collocation method for the coupled system of nonlinear fractional differential equations. The error bounds under the $ L^2- $ and $ L^{\infty}- $norms is also provided, then the theoretical result is validated by a number of numerical tests.</p></abstract>

Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations

In this article, we prove some new common fixed point results under the generalized contraction condition using w-distance and weak altering distance functions. Also, the validity of the results is demonstrated by an example along with numerical experiment for approximating the common fixed point. Later, as applications, the unique common solutions for the system of nonlinear Fredholm integral equations, nonlinear Volterra integral equations and nonlinear fractional differential equations of Caputo type are derived.

On Fuzzy Extended Hexagonal b-Metric Spaces with Applications to Nonlinear Fractional Differential Equations

The focus of this research article is to investigate the notion of fuzzy extended hexagonal b-metric spaces as a technique of broadening the fuzzy rectangular b-metric spaces and extended fuzzy rectangular b-metric spaces as well as to derive the Banach fixed point theorem and several novel fixed point theorems with certain contraction mappings. The analog of hexagonal inequality in fuzzy extended hexagonal b-metric spaces is specified as follows utilizing the function b(c,d): mhc,d,t+s+u+v+w≥mhc,e,tb(c,d)∗mhe,f,sb(c,d)∗mhf,g,ub(c,d)∗mhg,k,vb(c,d)∗mhk,d,wb(c,d) for all t,s,u,v,w>0 and c≠e,e≠f,f≠g,g≠k,k≠d. Further to that, this research attempts to provide a feasible solution for the Caputo type nonlinear fractional differential equations through effective applications of our results obtained.

A Note on the Stability Analysis of Fuzzy Nonlinear Fractional Differential Equations Involving the Caputo Fractional Derivative

In this paper, we present and establish a new result on the stability analysis of solutions for fuzzy nonlinear fractional differential equations by extending Lyapunov’s direct method from the fuzzy ordinary case to the fuzzy fractional case. As an application, several examples are presented to illustrate the proposed stability result.

The Soliton Solutions for Some Nonlinear Fractional Differential Equations with Beta-Derivative

Nonlinear fractional differential equations have gained a significant place in mathematical physics. Finding the solutions to these equations has emerged as a field of study that has attracted a lot of attention lately. In this work, He’s semi-inverse variation method and the ansatz method have been applied to find the soliton solutions for fractional Korteweg–de Vries equation, fractional equal width equation, and fractional modified equal width equation defined by Atangana’s conformable derivative (beta-derivative). These two methods are effective methods employed to get the soliton solutions of these nonlinear equations. All of the calculations in this work have been obtained using the Maple program and the solutions have been replaced in the equations and their accuracy has been confirmed. In addition, graphics of some of the solutions are also included. The found solutions in this study have the potential to be useful in mathematical physics and engineering.