In this paper, some theorems of the classical power series are generalized for the fractional power series. Some of these theorems are constructed by using Caputo fractional derivatives. Under some constraints, we proved that the Caputo fractional derivative can be expressed in terms of the ordinary derivative. A new construction of the generalized Taylor’s power series is obtained. Some applic...

The fractal derivative is adopted to describe the non-linear fractional wave equation in a space. A variational principle successfully established by semi-inverse method. two-scale method and He?s exp-function are used solve equation, good result obtained.

In this paper, a numerical approach for the fractional order proportional-integral-derivative controller (FO-PID) design for the unstable first order time delay system is proposed. The controller design is based on the system time delay. In order to obtain the relation between the controller parameters and the time delay, for several amounts of the plant time delay and the fractional derivative...

this paper presents conditions for the existence and multiplicity of positive solutions for a boundary value problem of a nonlinear fractional differential equation. we show that it has at least one or two positive solutions. the main tool is krasnosel'skii fixed point theorem on cone and fixed point index theory.

Abstract: In this paper, we studied the numerical solution of a time-fractional Korteweg–de Vries (KdV) equation with new generalized fractional derivative proposed recently. The fractional derivative employed in this paper was defined in Caputo sense and contained a scale function and a weight function. A finite difference/collocation scheme based on Jacobi–Gauss–Lobatto (JGL) nodes was applie...

Abstract: A fractional derivative system identification approach for modeling battery dynamics is presented in this paper, where fractional derivatives are applied to approximate non-linear dynamic behavior of a battery system. The least squares-based state-variable filter (LSSVF) method commonly used in the identification of continuous-time models is extended to allow the estimation of fractio...

In this article, we consider the multi-point boundary-value problem for nonlinear fractional differential equations with p-Laplacian operator: D 0+ φp(D α 0+ u(t)) = f(t, u(t), Dα−2 0+ u(t), Dα−1 0+ u(t), D 0+ u(t)), t ∈ (0, 1), u(0) = u′(0) = D 0+ u(0) = 0, Dα−1 0+ u(1) = m X i=1 σiD α−1 0+ u(ηi), where 2 < α ≤ 3, 0 < β ≤ 1, 3 < α+β ≤ 4, Pm i=1 σi = 1, D α 0+ is the standard Riemann-Liouville ...

In this paper, a simple and effective stabilization method for integrating time delay systems using fractional order PD controllers μ s k k s C d p + = ) ( is proposed. The presented method is based on finding the stability regions according to the fractional orders of the derivative element in the range of (0, 2). These regions are computed by using three stability boundaries: Real Root Bounda...

We study a linear-quadratic optimal control problem involving a parabolic equation with fractional diffusion and Caputo fractional time derivative of orders s ∈ (0, 1) and γ ∈ (0, 1], respectively. The spatial fractional diffusion is realized as the Dirichlet-to-Neumann map for a nonuniformly elliptic operator. Thus, we consider an equivalent formulation with a quasi-stationary elliptic problem...

We introduce stable multi-domain spectral penalty methods suitable for solving fractional partial differential equations with fractional derivatives of any order. The fractional derivative is approximated in each sub domain using orthogonal polynomials and stability of the scheme is achieved through weakly imposed boundary and interface conditions by using a penalty term. We discuss accuracy an...