Fractional Laplacian Of Order Research Articles

Lattice Dynamical Systems Associated with a Fractional Laplacian

We derive optimal well-posedness results and explicit representations of solutions in terms of special functions for the linearized version of the equationfor some constant where denotes the Caputo fractional derivative in time of order β and denotes the discrete fractional Laplacian of order We also prove a comparison principle. A special case of this equation is the discrete Fisher- equation with and without delay. We show that if for every and the function is concave on f(x, s) is nonnegative for every and satisfiesfor some then the system has a nonnegative unique solution u satisfying for every and Our results include cubic nonlinearities and incorporate new results for the discrete Newell-Whitehead-Segel equation. We use Lévy stable processes as well as Mittag-Leffler, Wright and modified Bessel functions to describe the solutions of the linear lattice model, providing a useful framework for further study. For the nonlinear model, we use a generalization of the upper–lower solution method for reaction–diffusion equations in order to prove existence and uniqueness of solutions.

Comprehensive comparison based on meta-heuristic algorithms for approximation of the fractional-order Laplacian sα as a weighted sum of first-order high-pass filters

To implement an approximation of the fractional order Laplacian operator sα as a weighted sum of high pass filter sections, it is essential to extract the cutoff frequencies and filter gains of each section in order to achieve the lowest error possible. Therefore, in this work, five meta-heuristic optimization algorithms are tested in this problem based on a weighted sum objective function. The employed algorithms include the: ant-lion optimizer, cuckoo search algorithm, flower pollination algorithm, whale optimizer, and multi-verse algorithm. A comparison is carried out between the results of these algorithms based on the relative percentage error in magnitude and phase of the obtained approximation in order to endorse the most recommended algorithm. The main outcome is that the Cuckoo search and ant-lion optimizers are capable of identifying the required filter parameters with the least phase and magnitude relative error and with a higher convergence rate.

Existence of nontrivial solutions for fractional Schrödinger equations with critical or supercritical growth

In this paper, we study the following fractional Schrödinger equation with critical or supercritical growth urn:x-wiley:mma:media:mma5441:mma5441-math-0001 where 0 < s < 1, N > 2s, λ > 0, , , ( − Δ)s denotes the fractional Laplacian of order s and f is a continuous superlinear but subcritical function. Under some suitable conditions, we prove that the equation has a nontrivial solution for small λ > 0 by variational methods. Our main contribution is related to the fact that we are able to deal with the case .

Interior regularity results for zeroth order operators approaching the fractional Laplacian

In this article we are interested in interior regularity results for the solution $${\mu _ \in } \in C(\bar \Omega )$$ of the Dirichlet problem $$\{ _{\mu = 0in{\Omega ^c},}^{{I_ \in }(\mu ) = {f_ \in }in\Omega }$$ where Ω is a bounded, open set and $${f_ \in } \in C(\bar \Omega )$$ for all є ∈ (0, 1). For some σ ∈ (0, 2) fixed, the operator $$\mathcal{I}_{\in}$$ is explicitly given by $${I_ \in }(\mu ,x) = \int_{{R^N}} {\frac{{[\mu (x + z) - \mu (x)]dz}}{{{ \in ^{N + \sigma }} + |z{|^{N + \sigma }}}}} ,$$ which is an approximation of the well-known fractional Laplacian of order σ, as є tends to zero. The purpose of this article is to understand how the interior regularity of uє evolves as є approaches zero. We establish that uє has a modulus of continuity which depends on the modulus of fє, which becomes the expected Holder profile for fractional problems, as є → 0. This analysis includes the case when fє deteriorates its modulus of continuity as є → 0.

Approximation of the Fractional-Order Laplacian <inline-formula> <tex-math notation="LaTeX">$s^\alpha$ </tex-math> </inline-formula> As a Weighted Sum of First-Order High-Pass Filters

A new approximation method of the fractional-order Laplacian operator ${s}^{\alpha }$ is introduced. The approximation is based on a weighted sum of first-order filter sections and its analytical proof is given. The optimal high-pass filter section parameters that cover six frequency decades are obtained using the flower pollination algorithm while the effect of the number of filter sections on the accuracy of the approximation is investigated. Approximations of fractional-order capacitors of orders ${\alpha =0.5}$ and ${\alpha =0.7}$ synthesized in Foster-II form are given as a validating example. Further, an active emulator of a fractional-order differentiator function based on this technique is also proposed and experimentally validated.

Recovering the historical distribution for nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space

In this paper, we investigate the problem of recovering the historical distribution for a nonlinear space-fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a fractional Laplacian of order α ∈ ( 1 ∕ 2 , 1 ] , which is usually used to model the anomalous diffusion. The problem is severely ill-posed. To regularize the problem, we propose a modified version of the Tikhonov regularization method. A stability estimate of Hölder type is established. Finally, several numerical examples based on the finite difference approximation and the discrete Fourier transform are presented to illustrate the theoretical results.

A metaheuristic optimization approach to discretize the fractional order Laplacian operator without employing a discretization operator

A single-step procedure to obtain first and higher order discrete-time models in terms of infinite impulse response templates of the fractional order Laplacian operator sα, where 0<α<1, is proposed in this paper. The Moth Flame Optimization (MFO) algorithm based rational approximations are generated using a discretization operator-free method. Solution accuracy and the convergence performance of MFO are extensively compared with several other advanced evolutionary algorithms. Simulations justify the improved modelling accuracy of the proposed models over the recently published designs. The effects due to the finite word length leading to truncated filter coefficients are also considered, and the design stability robustness is demonstrated. The efficacy of the proposed model as a fractional order proportional-derivative controller is also validated.

Statistics of zero crossings in rough interfaces with fractional elasticity

We study numerically the distribution of zero crossings in one-dimensional elastic interfaces described by an overdamped Langevin dynamics with periodic boundary conditions. We model the elastic forces with a Riesz-Feller fractional Laplacian of order z=1+2ζ, such that the interfaces spontaneously relax, with a dynamical exponent z, to a self-affine geometry with roughness exponent ζ. By continuously increasing from ζ=-1/2 (macroscopically flat interface described by independent Ornstein-Uhlenbeck processes [Phys. Rev. 36, 823 (1930)PHRVAO0031-899X10.1103/PhysRev.36.823]) to ζ=3/2 (super-rough Mullins-Herring interface), three different regimes are identified: (I) -1/2<ζ<0, (II) 0<ζ<1, and (III) 1<ζ<3/2. Starting from a flat initial condition, the mean number of zeros of the discretized interface (I) decays exponentially in time and reaches an extensive value in the system size, or decays as a power-law towards (II) a subextensive or (III) an intensive value. In the steady state, the distribution of intervals between zeros changes from an exponential decay in (I) to a power-law decay P(ℓ)∼ℓ^{-γ} in (II) and (III). While in (II) γ=1-θ with θ=1-ζ the steady-state persistence exponent, in (III) we obtain γ=3-2ζ, different from the exponent γ=1 expected from the prediction θ=0 for infinite super-rough interfaces with ζ>1. The effect on P(ℓ) of short-scale smoothening is also analyzed numerically and analytically. A tight relation between the mean interval, the mean width of the interface, and the density of zeros is also reported. The results drawn from our analysis of rough interfaces subject to particular boundary conditions or constraints, along with discretization effects, are relevant for the practical analysis of zeros in interface imaging experiments or in numerical analysis.

Ground states for fractional Schrödinger equations with critical growth

In this paper, we study the following critical fractional Schrödinger equation: (−Δ)su+V(x)u=|u|2s*−2u+λf(x,u), x∈RN, where λ &amp;gt; 0, 0 &amp;lt; s &amp;lt; 1, N &amp;gt; 2s, 2s*=2NN−2s, (−Δ)s denotes the fractional Laplacian of order s, and f is a continuous superlinear but subcritical function. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by the Nehari method.

Three solutions for a fractional Schrödinger equation with vanishing potentials

In this paper, we study the following fractional Schrödinger equation ( − Δ ) s u + V ( x ) u = K ( x ) f ( u ) + λ W ( x ) | u | p − 2 u , x ∈ R N , where λ > 0 is a parameter, ( − Δ ) s denotes the fractional Laplacian of order s ∈ ( 0 , 1 ) , N > 2 s , W ∈ L 2 2 − p ( R N , R + ) , 1 < p < 2 , V , K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Under some mild assumptions, we prove that the above equation has three solutions.

Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth

In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrodinger–Poisson system: \begin{eqnarray*} \left\{ \begin{array}{ll} \epsilon^{2s}(-\triangle)^{s} {u}+ V(x)u+\phi u =f(u)+|u|^{2^*_{s}-2}u &\mbox{in}\,\,\R^3, \\[2.5mm] \epsilon^{2t}(-\triangle)^{t}{\phi}=u^2 &\mbox{in}\,\, \R^3, \end{array} \right. \end{eqnarray*} where ϵ > 0 is a small parameter, (− △ ) α denotes the fractional Laplacian of order α = s,t ∈ (0,1), where 2 α ∗ 6/3−2α is the fractional critical exponent in Dimension 3; V ∈ C 1 (ℝ 3 ,ℝ + ) and f is subcritical. We first prove that for ϵ > 0 sufficiently small, the system has a positive ground state solution. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small ϵ . Moreover, each positive solution u ϵ converges to the least energy solution of the associated limit problem and concentrates around a global minimum point of V .

Violation of an f -sum rule with generalized kinetic energy

Motivated by the normal state of the cuprates in which the f-sum rule increases faster than a linear function of the particle density, we derive a conductivity sum rule for a system in which the kinetic energy operator in the Hamiltonian is a general function of the momentum squared. Such a kinetic energy arises in scale invariant theories and can be derived within the context of holography. Our derivation of the f-sum rule is based on the gauge couplings of a non-local Lagrangian in which the kinetic operator is a fractional Laplacian of order $\alpha$. We find that the f-sum rule in this case deviates from the standard linear dependence on the particle density. We find two regimes. At high temperatures and low densities, the sum rule is proportional to $nT^{\frac{\alpha-1}{\alpha}}$ where $T$ is the temperature. At low temperatures and high densities, the sum rule is proportional to $n^{1+\frac{2(\alpha-1)}{d}}$ with $d$ being the number of spatial dimensions. The result in the low temperature and high density limit, when $\alpha < 1$, can be used to qualitatively explain the behavior of the effective number of charge carriers in the cuprates at various doping concentrations.

Exact discretization of fractional Laplacian

An exact discretization of fractional-order Laplacian for N-dimensional space is suggested. Particular solutions of fractional-order partial difference equations with the proposed discrete Laplace operators are suggested.

On a class of fractional Schrödinger equations in with sign-changing potential

We are interested in finding solutions to a class of problems involving the fractional Laplacian operator. Specifically, we study the equation where , denotes the fractional Laplacian of order s, , V(x) is a continuous and unbounded potential which may change sign, and the nonlinearity is a continuous function which may be unbounded in x since its growth is controlled by V(x) and has subcritical growth in in the sense of the Sobolev embedding. Assuming suitable conditions under V(x) and and applying a approach variational, we prove the existence of the multiplicity of solution for this equation.

Existence and multiplicity of positive solutions for fractional Schrödinger equations with critical growth

In this paper, we study the existence and multiplicity of positive solutions for the fractional Schrödinger equations ε2α(−Δ)αu+V(x)u=∣u∣2α∗−2u+σg(x,u),x∈RN, where ε and σ are positive parameters, 0<α<1, (−Δ)α denotes the fractional Laplacian of order α, N>2α and 2α∗=2NN−2α is the fractional critical exponent, V∈C(RN,R+), g∈C(RN×R,R). By the variational methods we study the existence and multiplicity of positive solutions. Our main results improve corresponding results in Shang and Zhang (2014).

Existence and multiplicity solutions of fractional Schrödinger equation with competing potential functions

In this paper, we study the existence and multiplicity of solutions for the following fractional Schrödinger equationwhere h is a positive parameter, , ; denotes the fractional Laplacian of order s, , V, K and Q are competing functions. We first prove that the equation has a nontrivial nonnegative ground state solution and then investigate the relation between the number of solutions and the topology of the global minima set of a suitable ground energy function for sufficiently small. Then, we obtain a sufficient condition for the nonexistence of ground state solution. Finally, we establish an embedding theorem involving fractional weighted Sobolev spaces and use it to get a ground state solution under the case V, K and Q may decay to zero as .

Multiple solutions of nonlinear Schrödinger equation with the fractional Laplacian

In this paper we are devoted to a time-independent fractional Schrödinger equation (−Δ)αu+V(x)u=f(x,u)in RN, where (−Δ)α stands for the fractional Laplacian of order α∈(0,1), f is either asymptotically linear or superquadratic growth. Under appropriate assumptions on V and f, we prove the existence of infinitely many nontrivial high or small energy solutions, which extend and complement previously known results in the literature.

Soliton solutions for fractional Schrödinger equations

In this paper, we study the existence of a soliton type solution for the fractional Schrödinger equation ( − Δ ) s u + V ( x ) u + [ ( − Δ ) s u 2 ] u = λ | u | p − 1 u , u > 0 , x ∈ R N , where 0 < s < 1 , ( − Δ ) s denotes the fractional Laplacian of order s , N > 2 s , 2 s ∗ = 2 N N − 2 s , 1 < p < 2 s ∗ − 1 . We prove that the equation has a solution by using a constrained minimization argument.

Introduction to the special issue on IEEE-MWSCAS 2013

It is our pleasure to introduce this special issue on the 56th International IEEE Midwest Symposium on Circuits and Systems (MWSCAS) that was held in Columbus, OH, August 2013. Out of the 436 paper submissions from 36 countries, 372 papers were accepted for presentation in ten special lecture sessions, 38 regular lecture sessions, and 27 poster sessions. The theme of the conference was on ‘‘Advancing the Frontiers of Green Technology’’. In addition to the typical analog, digital and mixed signal circuits sessions, power management and energy harvesting sessions were particularly tied to the conference theme, The paper entitled ‘‘Design of High-Performance Integrated Dimmable LED Driver for High-Brightness SolidState Lighting Applications’’ by Zhidong Liu and Hoi Lee, was selected from that session to present a high-voltage dimmable LED driver in 0.35 lm technology for highbrightness solid-state lighting applications. The driver presented in the paper achieves significant improvements in power efficiency (peak efficiency of 98.8 %) and current accuracy. The improvement in efficiency was realized by eliminating the power diode and the sensing resistor in the power stage, thereby significantly reducing conduction losses in these components. A simultaneous peak and valley current sensing scheme is introduced in the paper to improve current sensing accuracy. The proposed scheme minimizes LED current errors under a wide range of input voltages and LEDs. Analog computer aided design (CAD) and circuit synthesis is a topic of considerable importance to the circuits and systems community (CAS). The paper by Reyad ElKhazali on the Biquadratic Approximation of FractionalOrder Laplacian Operators, introduces an improved algorithm to approximate the fractional Laplace operator. The presented biquadratic transfer function shows a flat phase characteristics with the frequency band of interest, with a bandwidth of approximation that can be extended by cascading several biquadratic modules. In addition to the several numerical examples given, the novel algorithm is also used for synthesizing fractional-order inductors and capacitors. For Analog to Digital Converters, two papers were selected for this special issue. In the paper by Simran Singh, Mikko Valkama, Michael Epp, Lauri Anttila, Wolfgang Schlecker, and Elmar Ingber, entitled, ‘‘Digital Correction of Frequency Response Mismatches in 2-Channel TimeInterleaved ADCs Using Adaptive I/Q Signal Processing’’, the authors propose a novel adaptive compensation architecture to address the frequency response mismatch of 2-channel time-interleaved ADCs. Time interleaved ADC (TI-ADC) are gaining a lot of momentum given the technological limitations on having a single core ADC running at high sampling rate without sacrificing resolution. However, the analog component variations and temperature drift within the parallel ADCs in TI-ADC introduce its own frequency dependent amplitude and phase mismatch, limiting the effective resolution and spurious free dynamic range (SFDR) of the ADC. In the paper, an adaptive method is proposed to detect the frequency response mismatch inside the frequency band of interest by utilizing digital I/Q mismatch estimation and correction algorithms. M. Hella (&) Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, 110 8th St., Troy, NY 12180, USA e-mail: hellam@ecse.rpi.edu

Infinitely many weak solutions for a fractional Schrödinger equation

In this paper we are concerned with the fractional Schrödinger equation ( − Δ ) α u+V(x)u=f(x,u), x∈ R N , where 0<α<1, N>2α, ( − Δ ) α stands for the fractional Laplacian of order α, V is a positive continuous potential, and f is a continuous subcritical nonlinearity. We obtain the existence of infinitely many weak solutions for the above problem by the fountain theorem in critical point theory.