Keywords: fractal geometry, fractional derivative, fractional Fourier transform, fractional power of a matrix, self similarity, complex partial differential equation, broadband ultrasound, frequency-dependent attenuation, time domain. 1. Backgrounds The rational behind this model is schematically illustrated below: Fractal geometry (irregular soft tissues) → Fractional Fourier transform (freque...

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...

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...

Many complex systems containing fractal structure can be described by fractional order derivative. By expressing the complex system as fractional order system (system containing fractional order derivative), system’s complex behavior can be simply described with small number of parameters. However, in order to control or identify such system containing unknown variable parameters, online parame...

The unsteady hydro-magnetic free convection flow with heat transfer of a linearly viscous, incompressible, electrically conducting fluid near a moving vertical plate with the constant heat is investigated. The flow domain is the porous half-space and a magnetic field of a variable direction is applied. The Caputo time-fractional derivative is employed in order to introduce a thermal flux consti...

Fractional derivative has a history as long as that of classical calculus, but it is much less popular than it should be. What is the physical meaning of fractional derivative? This is still an open problem. In modeling various memory phenomena, we observe that a memory process usually consists of two stages. One is short with permanent retention, and the other is governed by a simple model of ...

The exact solutions in the wave form are derived for the time fractional KdV and the time fractional Burgers’ equations in conformable fractional derivative sense. The fractional variable change using the fundamental properties of the conformable derivative reduces both equations to some nonlinear ODEs. The predicted solution is assumed to be a finite series form of a function satisfying a part...

We describe a general operational method that can be used in the analysis of fractional initial and boundary value problems with additional analytic conditions. As an example, we derive analytic solutions of some fractional generalisation of differential equations of mathematical physics. Fractionality is obtained by substituting the ordinary integer-order derivative with the Caputo fractional ...