On the generalized fractional Laplacian

The objective of this paper is, for the first time, to extend the fractional Laplacian (−△) s u(x) over the space Ck (Rn ) (which contains S(Rn ) as a proper subspace) for all s > 0 and s ≠ 1, 2, …, based on the normalization in distribution theory, Pizzetti’s formula and surface integrals in Rn . We further present two theorems showing that our extended fractional Laplacian is continuous at the end points 1, 2, … . Two illustrative examples are provided to demonstrate computational techniques for obtaining the fractional Laplacian using special functions, Cauchy’s residue theorem and integral identities. An application to defining the Riesz derivative in the classical sense at odd numbers is also considered at the end.

Weighted fractional Hardy operators and their commutators on generalized Morrey spaces over quasi-metric measure spaces

We study commutators of weighted fractional Hardy-type operators within the frameworks of local generalized Morrey spaces over quasi-metric measure spaces for a certain class of “radial” weights. Quasi-metric measure spaces may include, in particular, sets of fractional dimentsions. We prove theorems on the boundedness of commutators with CMO coefficients of these operators. Given a domain Morrey space 𝓛 p,φ (X) for the fractional Hardy operators or their commutators, we pay a special attention to the study of the range of the exponent q of the target space 𝓛 q,ψ (X). In particular, in the case of classical Morrey spaces, we provide the upper bound of this range which is greater than the known Adams exponent.

A strong maximum principle for the fractional laplace equation with mixed boundary condition

In this work we prove a strong maximum principle for fractional elliptic problems with mixed Dirichlet–Neumann boundary data which extends the one proved by J. Dávila (cf. [11]) to the fractional setting. In particular, we present a comparison result for two solutions of the fractional Laplace equation involving the spectral fractional Laplacian endowed with homogeneous mixed boundary condition. This result represents a non–local counterpart to a Hopf’s Lemma for fractional elliptic problems with mixed boundary data.

Some properties of the fractal convolution of functions

We consider the fractal convolution of two maps f and g defined on a real interval as a way of generating a new function by means of a suitable iterated function system linked to a partition of the interval. Based on this binary operation, we consider the left and right partial convolutions, and study their properties. Though the operation is not commutative, the one-sided convolutions have similar (but not equal) characteristics. The operators defined by the lateral convolutions are both nonlinear, bi-Lipschitz and homeomorphic. Along with their self-compositions, they are Fréchet differentiable. They are also quasi-isometries under certain conditions of the scale factors of the iterated function system. We also prove some topological properties of the convolution of two sets of functions. In the last part of the paper, we study stability conditions of the dynamical systems associated with the one-sided convolution operators.

Convergence rate estimates for the kernelized predictor corrector method for fractional order initial value problems

This manuscript presents a kernelized predictor corrector (KPC) method for fractional order initial value problems, which replaces linear interpolation with interpolation by a radial basis function (RBF) in a predictor-corrector scheme. Specifically, the class of Wendland RBFs is employed as the basis function for interpolation, and a convergence rate estimate is proved based on the smoothness of the particular kernel selected. Use of the Wendland RBFs over Mittag-Leffler kernel functions employed in a previous iteration of the kernelized method removes the problems encountered near the origin in [11]. This manuscript performs several numerical experiments, each with an exact known solution, and compares the results to another frequently used fractional Adams-Bashforth-Moulton method. Ultimately, it is demonstrated that the KPC method is more accurate but requires more computation time than the algorithm in [4].

Recent developments on the realization of fractance device

A detailed analysis of the recent developments on the realization of fractance device is presented. A fractance device which is used to exhibit fractional order impedance properties finds applications in many branches of science and engineering. Realization of fractance device is a challenging job for the people working in this area. A term fractional order element, constant phase element, fractor, fractance, fractional order differintegrator, fractional order differentiator can be used interchangeably. In general, a fractance device can be realized in two ways. One is using rational approximations and the other is using capacitor physical realization principle. In this paper, an attempt is made to summarize the recent developments on the realization of fractance device. The various mathematical approximations are studied and a comparative analysis is also performed using MATLAB. Fourth order approximation is selected for the realization. The passive and active networks synthesized are simulated using TINA software. Various physical realizations of fractance device, their advantages and disadvantages are mentioned. Experimental results coincide with simulated results.

An inverse problem approach to determine possible memory length of fractional differential equations

It is a fundamental problem to determine a starting point in fractional differential equations which reveals the memory length in real life modeling. This paper describes it by an inverse problem. Fixed point theorems such as Krasnoselskii’s and Schauder type’s and nonlinear alternative for single–valued mappings are presented. Through existence analysis of the inverse problem, the range of the initial value points and the memory length of fractional differential equations are obtained. Finally, three examples are demonstrated to support the theoretical results and numerical solutions are provided.

Continuous dependence of fuzzy mild solutions on parameters for IVP of fractional fuzzy evolution equations

In this article, we are concerned with the VIP of fractional fuzzy evolution equations in the space of triangular fuzzy numbers. The continuous dependence of two kinds of fuzzy mild solutions on initial values and orders for the studied problem is obtained. The results obtained in this paper improve and extend some related conclusions on this topic.

Inverse problems for diffusion equation with fractional Dzherbashian-Nersesian operator

Fractional Dzherbashian-Nersesian operator is considered and three famous fractional order derivatives named after Riemann-Liouville, Caputo and Hilfer are shown to be special cases of the earlier one. The expression for Laplace transform of fractional Dzherbashian-Nersesian operator is constructed. Inverse problems of recovering space dependent and time dependent source terms of a time fractional diffusion equation with involution and involving fractional Dzherbashian-Nersesian operator are considered. The results on existence and uniqueness for the solutions of inverse problems are established. The results obtained here generalize several known results.