Volume 24, Issue 5

Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or anomalous behavior. This has driven a desire for a vector calculus that includes nonlocal and fractional gradient, divergence and Laplacian type operators, as well as tools such as Green’s identities, to model subsurface transport, turbulence, and conservation laws. In the literature, several independent definitions and theories of nonlocal and fractional vector calculus have been put forward. Some have been studied rigorously and in depth, while others have been introduced ad-hoc for specific applications. The goal of this work is to provide foundations for a unified vector calculus by (1) consolidating fractional vector calculus as a special case of nonlocal vector calculus, (2) relating unweighted and weighted Laplacian operators by introducing an equivalence kernel, and (3) proving a form of Green’s identity to unify the corresponding variational frameworks for the resulting nonlocal volume-constrained problems. The proposed framework goes beyond the analysis of nonlocal equations by supporting new model discovery, establishing theory and interpretation for a broad class of operators, and providing useful analogues of standard tools from the classical vector calculus.

A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the accuracy of the fractional derivative, a novel numerical method for the Caputo fractional derivative is proposed. The present adaptive memory method significantly reduces the requirement for computational memory for storing function values at past time points and also significantly improves the accuracy by calculating convolution weights to function values at past time points which can be non-uniformly distributed in time. The superior accuracy of the present method to the accuracy of the previously reported methods is identified by deriving numerical errors analytically. The sub-diffusion process of a time-fractional diffusion equation is simulated to demonstrate the accuracy as well as the computational efficiency of the present method.

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space C p is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation ∑ i=1mdixα iDα iu(x)+(xβ − ν 2)u(x)=0,α i> -->0,β > -->0, $$\begin{array}{} \displaystyle \sum_{i=1}^{m}d_i\, x^{\alpha_i}D^{\alpha_i} u(x) + (x^\beta - \nu^2)u(x)=0, \, \, \, \alpha_i \gt 0,\, \beta \gt 0, \end{array}$$ which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients d i . Our theory leads to a threshold admissible value for ν 2 , which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.

In this paper we establish the existence of solutions for the nonlinear abstract Cauchy problem of order α ∈ (1, 2), where the fractional derivative is considered in the sense of Caputo. The autonomous and nonautonomous cases are studied. We assume the existence of an α -resolvent family for the homogeneous linear problem. By using this α -resolvent family and appropriate conditions on the forcing function, we study the existence of classical solutions of the nonhomogeneus semilinear problem. The non-autonomous problem is discussed as a perturbation of the autonomous case. We establish a variation of the constants formula for the nonautonomous and nonhomogeneous equation.

Generalized summability results are obtained regarding formal solutions of certain families of linear moment integro-differential equations with time variable coefficients. The main result leans on the knowledg e of the behavior of the moment derivatives of the elements involved in the problem. A refinement of the main result is also provided giving rise to more accurate results which remain valid in wide families of problems of high interest in practice, such as fractional integro-differential equations.

We prove the wellposedness of a distributed-order space-fractional diffusion equation with variably distribution and its support, which could adequately model the challenging phenomena such as the anomalous diffusion in multiscale heterogeneous porous media, and smoothing properties of its solutions. We develop and analyze a collocation scheme for the proposed model based on the proved smoothing properties of the solutions. Furthermore, we approximately expand the stiffness matrix by a sum of Toeplitz matrices multiplied by diagonal matrices, which can be employed to develop the fast solver for the approximated system. We prove that it suffices to apply O (log N ) terms of expansion to retain the accuracy of the numerical discretization of degree N , which reduces the storage of the stiffness matrix from O ( N 2 ) to O ( N log N ), and the computational cost of matrix-vector multiplication from O ( N 2 ) to O ( N log 2 N ). Numerical results are presented to verify the effectiveness and the efficiency of the fast method.

The linear operator c + (−Δ) α /2 , where c > 0 and (−Δ) α /2 is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg–de Vries equation. We establish a relation of the Green function of this linear operator with the Mittag–Leffler function, which was previously used in the context of the Riemann–Liouville and Caputo fractional derivatives. By using this relation, we prove that the Green function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every c > 0 and every α ∈ (0, 2]. On the other hand, we argue from numerical approximations that in the case of α ∈ (2, 4], the Green function is positive and single-lobe for small c and non-positive and non-single lobe for large c .

This paper presents a novel First Order Plus Fractional Diffusive Delay (FOPFDD) model, capable of modeling delay dominant systems with high accuracy. The novelty of the FOPFDD is the Fractional Diffusive Delay (FDD) term, an exponential delay of non-integer order α , i.e. e −( Ls ) α in Laplace domain. The special cases of α = 0.5 and α = 1 have already been investigated thoroughly. In this work α is generalized to any real number in the interval ]0, 1[. For α = 0.5, this term appears in the solution of distributed diffusion systems, which will serve as a source of inspiration for this work. Both frequency and time domain are investigated. However, regarding the latter, no closed-form expression of the inverse Laplace transform of the FDD can be found for all α , so numerical tools are used to obtain an impulse response of the FDD. To establish the algorithm, several properties of the FDD term have been proven: firstly, existence of the term, secondly, invariance of the time integral of the impulse response, and thirdly, dependency of the impulse response’s energy on α . To conclude, the FOPFDD model is fitted to several delay-dominant, diffusive-like resistors-capacitors (RC) circuits to show the increased modeling accuracy compared to other state-of-the-art models found in literature. The FOPFDD model outperforms the other approximation models in accurately tracking frequency response functions as well as in mimicking the peculiar delay/diffusive-like time responses, coming from the interconnection of a large number of discrete subsystems. The fractional character of the FOPFDD makes it an ideal candidate for an approximate model to these large and complex systems with only a few parameters.

In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α -Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation.

This paper investigates the structure of solutions to the BVP of a class of fractional ordinary differential equations involving both fractional derivatives (R-L or Caputo) and fractional Laplacian with variable coefficients. This family of equations generalize the usual fractional diffusion equation and fractional Laplace equation. We provide a deep insight to the structure of the solutions shared by this family of equations. The specific decomposition of the solution is obtained, which consists of the “good” part and the “bad” part that precisely control the regularity and singularity, respectively. Other associated properties of the solution will be characterized as well.

This paper deals with system identification for continuous-time multiple-input single-output (MISO) fractional differentiation models. An output error optimization algorithm is proposed for estimating all parameters, namely the coefficients and the differentiation orders. Given the high number of parameters to be estimated, the output error method can converge to a local minimum. Therefore, an initialization procedure is proposed to help the convergence to the optimum by using three variants of the algorithm. Moreover, a new definition of structured-commensurability (or S-commensurability) has been introduced to cope with the differentiation order estimation. First, a global S-commensurate order is estimated for all subsystems. Then, local S-commensurate orders are estimated (one for each subsystem). Finally the S-commensurability constraint being released, all differentiation orders are further adjusted. Estimating a global S-commensurate order greatly reduces the number of parameters and helps initializing the second variant, where local S-commensurate orders are estimated which, in turn, are used as a good initial hit for the last variant. It is known that such an initialization procedure progressively increases the number of parameters and provides good efficiency of the optimization algorithm. Monte Carlo simulation analysis are provided to evaluate the performances of this algorithm.

The identification problem for system with distributed-order derivative was considered. The order-weight distribution was approximated by piecewise linear functions. Then the discretized order-weight distribution was solved in frequency domain by using the least square technique based on the Moore-Penrose inverse matrix. Finally, five representative numerical examples were used to illustrate the validity of the method. The identification results are satisfactory, especially for the continuous order-weight distributions. In addition, the overlapped Bode magnitude frequency responses from the identified and exact transfer functions imply the effectiveness of the method.

We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. We also show by a contour integration that the Green function is completely monotone on the positive part of the periodic domain.