Volume 15, Issue 3

In this work we extend a result from the literature on fractional continuous-time linear systems to the case of implicit fractional continuous-time state space models, based on the Caputo fractional derivative. The solution of the problem is derived using the Laplace transform.

This paper studies a new class of boundary value problems of nonlinear fractional differential inclusions of order q ∈ (1, 2] with fractional separated boundary conditions. New existence results are obtained for this class of problems by using some standard fixed point theorems. A possible generalization for the inclusion problem with fractional separated integral boundary conditions is also discussed. Some illustrative examples are presented.

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

A fractional Hamiltonian formalism is introduced for the recent combined fractional calculus of variations. The Hamilton-Jacobi partial differential equation is generalized to be applicable for systems containing combined Caputo fractional derivatives. The obtained results provide tools to carry out the quantization of nonconservative problems through combined fractional canonical equations of Hamilton type.

In this paper we give an answer for the following question: when $$AC_{a + }^\alpha \subset AC_{a + }^\beta ,$$ where AC a+α = AC a+α([a, b], ℝn), AC a+β = AC a+β([a, b], ℝn) are the sets of functions possessing the fractional Riemann-Liouville derivatives of order α, β, respectively.

We study analytically a generalized fractional Langevin equation. General formulas for calculation of variances and the mean square displacement are derived. Cases with a three parameter Mittag-Leffler frictional memory kernel are considered. Exact results in terms of the Mittag-Leffler type functions for the relaxation functions, average velocity and average particle displacement are obtained. The mean square displacement and variances are investigated analytically. Asymptotic behaviors of the particle in the short and long time limit are found. The model considered in this paper may be used for modeling anomalous diffusive processes in complex media including phenomena similar to single file diffusion or possible generalizations thereof. We show the importance of the initial conditions on the anomalous diffusive behavior of the particle.

This paper studies a new class of anti-periodic boundary value problems of fractional differential equations with nonlinear term depending on lower order fractional derivative. Some existence and uniqueness results are obtained by applying some standard fixed point principles. Several examples are given to illustrate the results.

This paper establishes a product rule for fractional derivatives of a realvalued function defined on a finite dimensional Euclidean vector space. The proof uses Fourier transforms.

This paper deals with the Ornstein-Uhlenbeck (O-U) process driven by the fractional Brownian motion (fBm). Based on the fractional Itô formula, we present the corresponding fBm-driven Fokker-Planck equation for the nonlinear stochastic differential equations driven by an fBm. We then apply it to establish the evolution of the probability density function (PDF) of the fBm-driven O-U process. We further obtain the closed form of such PDF by combining the Fourier transform and the method of characteristics. Interestingly, the obtained PDF has an infinite variance which is significantly different from the classical O-U process. We reveal that the fBm-driven O-U process can describe the heavy-tailedness or anomalous diffusion. Moreover, the speed of the sub-diffusion is inversely proportional to the viscosity coefficient, while is proportional to the Hurst parameter. Finally, we carry out numerical simulations to verify the above findings.

We introduce an integral transform related to a Fourier sine-Fourier - Fourier cosine generalized convolution and prove a Watson type theorem for the transform. As applications we obtain solutions of some integral equations in closed form.

The authors study a type of nonlinear fractional boundary value problem with non-homogeneous integral boundary conditions. The existence and uniqueness of positive solutions are discussed. An example is given as the application of the results.