Taming Hyperchaos with Exact Spectral Derivative Discretization Finite Difference Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

Four discrete models, using the exact spectral derivative discretization finite difference (ESDDFD) method, are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index , the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is, in part, an artifact of the CEFD method, as it is absent from the ESDDFD models.

Taming Hyperchaos with ESDDFD Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

Four discrete models using the exact spectral derivative discretization finite difference (ESDDFD) method are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index , the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is in part an artifact of the CEFD method as it is absent from the ESDDFD models.

Taming Hyperchaos with ESDDFD Discretization of a Conformable Fractional Derivative Financial System with Market Confidence and Ethics Risk

Four discrete models using the exact spectral derivative discretization finite difference (ESDDFD) method are proposed for a chaotic five-dimensional, conformable fractional derivative financial system incorporating ethics and market confidence. Since the system considered was recently studied using the conformable Euler finite difference (CEFD) method and found to be hyperchaotic, and the CEFD method was recently shown to be valid only at fractional index , the source of the hyperchaos is in question. Through numerical experiments, illustration is presented that the hyperchaos previously detected is in part an artifact of the CEFD method as it is absent from the ESDDFD models.

A New Conformable Fractional Derivative and Applications

This paper is motivated by some papers treating the fractional derivatives. We introduce a new definition of fractional derivative which obeys classical properties including linearity, product rule, quotient rule, power rule, chain rule, Rolle’s theorem, and the mean value theorem. The definition D α f t = lim h ⟶ 0 f t + h e α − 1 t − f t / h , for all t > 0 , and α ∈ 0,1 . If α = 0 , this definition coincides to the classical definition of the first order of the function f .

Generalized Hukuhara Conformable Fractional Derivative and its Application to Fuzzy Fractional Partial DifferentialEquations

The main focus of this paper is to develop an efficient analytical method to obtain the traveling wave fuzzy solution for the fuzzy generalized Hukuhara conformable fractional equations by considering the type of generalized Hukuhara conformable fractional differentiability of the solution. To achieve this, the fuzzy conformable fractional derivative based on the generalized Hukuhara differentiability is defined, and several properties are brought on the topic, such as switching points and the fuzzy chain rule. After that, a new analytical method is applied to find the exact solutions for two famous mathematical equations: the fuzzy fractional Wave equation and the fuzzy fractional Diffusion equation. The present work is the first report in which the fuzzy traveling wave method is used to design an analytical method to solve these fuzzy problems. The final examples are asserted that our new method is applicable and efficient.

On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative

In the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), the authors have used the Krasnoselskii fixed point theorem for showing the existence of mild solutions of an abstract class of conformable fractional differential equations of the form: $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t))$ d α d t α [ d α x ( t ) d t α ] = A x ( t ) + f ( t , x ( t ) ) , $t\in [0,\tau ]$ t ∈ [ 0 , τ ] subject to the nonlocal conditions $x(0)=x_{0}+g(x)$ x ( 0 ) = x 0 + g ( x ) and $\frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x)$ d α x ( 0 ) d t α = x 1 + h ( x ) , where $\frac{d^{\alpha }(\cdot)}{dt^{\alpha }}$ d α ( ⋅ ) d t α is the conformable fractional derivative of order $\alpha \in\, ]0,1]$ α ∈ ] 0 , 1 ] and A is the infinitesimal generator of a cosine family $(\{C(t),S(t)\})_{t\in \mathbb{R}}$ ( { C ( t ) , S ( t ) } ) t ∈ R on a Banach space X. The elements $x_{0}$ x 0 and $x_{1}$ x 1 are two fixed vectors in X, and f, g, h are given functions. The present paper is a continuation of the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) in order to use the Darbo–Sadovskii fixed point theorem for proving the same existence result given in (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) [Theorem 3.1] without assuming the compactness of the family $(S(t))_{t>0}$ ( S ( t ) ) t > 0 and any Lipschitz conditions on the functions g and h.

Explicit Exact Solutions of Space-time Fractional Drinfel’d-Sokolov-Wilson Equations

The space-time fractional Drinfel’d-Sokolov-Wilson equations (DSWEs) has attracted many researchers’ attention in recent years. In this study, combining the (G’/G,1/G)-expansion method and a fractional wave transformation, we derive abundant explicit exact solutions of the DSWEs with the conformable fractional derivative. All of the resulting solutions include triangle, hyperbolic and rational function type. It shows this technique is effective and reliable to find exact solutions of other similar nonlinear fractional partial differential equations (NFPDEs).