Volume 20, Issue 1

This article discusses several definitions of the fractional Laplace operator L = — (—Δ) α /2 in R d , also known as the Riesz fractional derivative operator; here α ∈ (0,2) and d ≥ 1. This is a core example of a nonlocal pseudo-differential operator, appearing in various areas of theoretical and applied mathematics. As an operator on Lebesgue spaces ℒ p (with p ∈ [1,∞)), on the space 𝒞 0 of continuous functions vanishing at infinity and on the space 𝒞 bu of bounded uniformly continuous functions, L can be defined, among others, as a singular integral operator, as the generator of an appropriate semigroup of operators, by Bochner’s subordination, or using harmonic extensions. It is relatively easy to see that all these definitions agree on the space of appropriately smooth functions. We collect and extend known results in order to prove that in fact all these definitions are completely equivalent: on each of the above function spaces, the corresponding operators have a common domain and they coincide on that common domain.

Many phenomena in inter-disciplinary fields can be explained naturally by coordinated behavior of agents with fractional-order dynamics. Under the assumption that the interconnection topology of all agents has a spanning tree, the consensuses of linear and nonlinear fractional-order multi-agent systems with input time delay are studied, respectively. Based on the properties of Mittag-Leffler function, matrix theory, stability theory of fractional-order differential equations, some sufficient conditions on consensus are derived by using the technique of inequality, which shows that the consensus can be achieved for any bounded input time delay. Finally, two numerical examples are given to illustrate the effectiveness of the theoretical results.

In this paper, the authors study the asymptotic behavior of solutions of higher order fractional differential equations with Caputo-type Hadamard derivatives of the form C,HDarx(t)=e(t)+f(t,x(t)),a> -->1,$$\begin{equation*}^{C,H}\mathcal{D}_{a}^{r}x(t)=e(t)+f(t,x(t)), \quad a\gt1, \end{equation*}$$ where r = n+α–1, α ∊ (0,1), and n ∊ℤ + . They also apply their technique to investigate the oscillatory and asymptotic behavior of solutions of the related integral equation x(t)=e(t)+∫ atln⁡ tsr− 1k(t,s)f(s,x(s))dss,a> -->1,r is as above.$$\begin{equation*}x(t)=e(t)+\int\limits_{a}^{t}\left( \ln \frac{t}{s}\right) ^{r-1}k(t,s)f(s,x(s))\frac{ds}{s}, \quad a\gt1, \quad r\textrm{ is as above}. \end{equation*}$$

We develop a fast space-time finite difference method for space-time fractional diffusion equations by fully utilizing the mathematical structure of the scheme. A circulant block preconditioner is proposed to further reduce the computational costs. The method has optimal-order memory requirement and approximately linear computational complexity. The method is not lossy, as no compression of the underlying numerical scheme has been employed. Consequently, the method retains the stability, accuracy, and, in particular, the conservation property of the underlying numerical scheme. Numerical experiments are presented to show the efficiency and capacity of long time modelling of the new method.

Let Ω be a 𝒞 2 -bounded domain of ℝ d , d = 2,3, and fix Q = (0, T )× Ω with T ∈ (0,+∞]. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ∂ tα u+Au=fb(u)inQwhere1< α < 2,∂ tα $\partial_t^\alpha u+\mathcal A u=f_b(u)\,\,\text{in}\, Q\, \text{where} \,\,1\lt\alpha\lt2, \,\partial_t^\alpha$ corresponds to the Caputo fractional derivative of order α , 𝒜 is an elliptic operator and the nonlinearity f b ∈ 𝒞 1 (ℝ) satisfies f b (0) = 0 and |fb′(u)|⩽ C|u|b− 1 $|{f_b'(u)}| \leqslant C\,|{u}|^{b-1}$ for some b > 1. We first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation ∂ tα u+Au=f(t,x) $\partial_t^\alpha u+\mathcal A u=f(t,x)$ in Q . Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of b > 1. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data.

In this paper, we consider the space-time fractional wave-diffusion equation − ∞ Dxα ,λ u(x,t)=Dtβ u(x,t),x∈ R,t>0, $$ \begin{equation} {{}_{ - \infty }D_x^{\alpha ,\lambda } }u(x,t)=D_t^\beta u(x,t),\quad x\in\mathbb {R},\quad t>0,\nonumber \end{equation} $$ for the tempered Riemann-Liouville derivative of order 1 < α ≤ 2 in space and the Caputo derivative of order 0 < β ≤ α in time. The first fundamental solution of the Cauchy problem is derived by means of the Fourier transform and given in computable series form as well as under the restriction 0 < β ≤ 1 in form of a fast converging integral. We additionally point out two integral representations for accurate evaluation of the two-sided M -Wright function. The derived equations are verified by comparisons with the Fourier spectral method.

The importance of the concept of stability in fractional order system and control has been recognized for some time now. Recently, it has become evident that many conclusions were drawn, but little consensus was reached. Consequently, there is an urgent need for a much deeper understanding of such a concept. With the definition of fractional order positive definite matrix, a set of equivalent and elegant stability criteria are developed via revisiting a stability criterion we proposed before. All the results are formed in terms of linear matrix inequalities. Afterwards, a series of interesting properties of these criteria are revealed profoundly, including completeness, singularity, conservatism, etc. Eventually, a simulation study is provided to validate the effectiveness of the obtained results.

Exploiting the theory of fractional Fourier transform, the wavelet convolution product and existence theorems associated with the n -dimensional wavelet transform are investigated and their properties studied.

The inhomogeneous time-fractional telegraph equation with Caputo derivatives with constant coefficients is considered. For the considered equation, general representation of regular solution in rectangular domain is obtained and the fundamental solution is presented. Using this representation and the properties of the fundamental solution, the Cauchy problem and the main boundary value problems in half-strip and rectangular domains are studied. For the Cauchy problem theorems of existence and uniqueness of solution are proved, and the explicit form of the solution is constructed. The solutions of the investigated problems are constructed in terms of the appropriate Green functions, which are also constructed in explicit form.

We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.

The article considers the controllability of a fractional order diffusion equation. We show that the resulting fractional order diffusion equation is null-controllable. Our method reduces essentially to the study of a moment problem related to the Mittag-Leffler functions. Paley-Wiener type theorems are applied to construct biorthogonal sequence to a family of complex Mittag-Leffler functions.

This paper tackles important problems in stable discretization of commensurate fractional-order continuous-time LTI SISO systems based on the Grünwald-Letnikov (GL) difference. New, analytical stability/instability conditions are given for the GL-discretized systems governed by fractional-order transfer functions. A stability preservation analysis is also performed for a class of finite GL approximators.

We show that the anomalous diffusion equations with a fractional spatial derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution is conceived as integral transform whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and for different initial conditions.

In this note we present a Lyapunov-type inequality for a fractional boundary value problem with anti-periodic boundary conditions, that we show to be a generalization of a classical one. Moreover, we address the issue of further research directions for such type of inequalities.