Fractional Exponent Research Articles

Symplectic Framework-Based New Analytic Solutions for Thermal Buckling of Temperature-Dependent Moderately Thick Functionally Graded Rectangular Plates

This paper presents some new analytic thermal buckling solutions of temperature-dependent moderately thick functionally graded (FG) rectangular plates with non-Lévy-type constraints within the symplectic solution framework in the Hamiltonian system. An original problem is reduced to the superposition of two constructed subproblems that are analytically solved via the rigorous symplectic elasticity approach with mathematical techniques such as variable separation in the symplectic space and symplectic eigen expansion. The physical neutral surface is employed to remove the stretching-bending coupling in constitutive equations. The main characteristic of the present symplectic superposition strategy is on the rational derivation without assumption on solution forms, which, however, is hard to achieve by conventional semi-inverse methods. Comprehensive benchmark results are presented, including critical buckling temperatures as well as mode shapes of typical non-Lévy-type FG plates in uniform and nonlinear temperature fields, and are verified by available numerical results. The effects of boundary constraint, aspect ratio, volume fraction exponent, and thickness-to-width ratio on buckling temperatures are quantitatively investigated. The occurrence of thermal buckling is further studied by presenting a failure mode map covering a wide range of geometric parameters.

Dead-core solutions to fast diffusion–reaction equation for catalyst slabs with power-law reaction kinetics and external mass transfer resistance

This paper investigates semi-analytic approaches used to solve a two-point boundary value problem for nonlinear fast diffusion–reaction equation for catalytic slabs with external mass resistance. The kinetics considered is the power-law type having a fractional reaction exponent. The semi-analytic approach for dead-core problems with Fickian diffusion is generalized to models with non-Fickian diffusion. The dimensionless steady-state equation for mass conservation in the catalyst slab for a single n-th order chemical reaction and the non-Fickian diffusion model is derived and studied. We show that the dead zone can appear close to the pellet center under certain combinations of the following parameters: slab size, effective diffusivity, mass transfer coefficient, bulk reactant concentration, reaction order, reaction rate constant, and diffusion exponent. We also study the effects of the process parameters on the concentration profiles and length of dead zones. Analytical findings are verified by numerical simulations.

Adaptive control of an unsteady stress oscillation in a functionally graded multiferroic composite thin plate

An unsteady stress oscillation is known to be caused in a functionally graded material (FGM) thin plate by the action of impact loading. Since the stress then changes alternately and intensively from compression to tension as time advances, it may lead to serious damage to the FGM thin plate. Therefore, the development of an analytical model for control of the dynamic stress adapted to the action of impact loading is required. In this paper, adaptive control of an unsteady stress oscillation is investigated for a functionally graded multiferroic composite (FGMC) thin plate subjected to uniform impact pressure. Material properties of the FGMC thin plate are assumed to vary in the thickness direction according to a power law distribution. This elastodynamic problem is analyzed under the assumption that the volume fraction exponent and uniform impact pressure are unknown, whereas a piezoelectric voltage is measured across the thin plate during the first cycle of the oscillation. The magnitudes of the unknown exponent and pressure as well as an appropriate magnetic potential difference for control of the unsteady stress oscillation are considered to be determined within the period of the second cycle from knowledge on the basis of the measured piezoelectric voltage. Finally, numerical calculations have been carried out for the case where the appropriate magnetic potential difference is applied to the FGMC thin plate from the third cycle. Obtained results demonstrate that the maximum amplitude of the controlled unsteady stress oscillation is successfully reduced by about 46% and the performance of the adaptive control is monitored through the piezoelectric sensing. In consequence, the analytical model for adaptive control and monitoring of the unsteady stress oscillation has been proposed.

Ground states for critical fractional Schrödinger‐Poisson systems with vanishing potentials

This paper deals with a class of fractional Schrödinger‐Poisson system with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where is the fractional critical exponent. The problem is set on the whole space, and compactness issues have to be tackled. By employing the mountain pass theorem, concentration‐compactness principle, and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on , , , and .

Variable thickness thin-walled rotating blades made of functionally graded porous materials

Within this study, the statics and dynamics performance of turbomachinery variable thickness rotating blades, composed of thin-walled, functionally graded metal-ceramic materials including cross-sectional distributed porosity is optimized. The pre-twisted, tapered, Thin-Wall Rotating Beams (TWRB) with constant angular velocity are considered within this context. For the optimization purpose, a genetic algorithm (GA) is implemented in a nonlinear constraint optimization problem. The goal is to exploit desirable thickness variations with improved dynamic, static, and buckling behavior of the variable thickness functionally graded material (FGM) blades. Hamilton’s principle was applied using the Euler–Lagrange dynamic system of governing equation, while the extended Galerkin’s method (EGM) was applied to solve the equations. Two variables in thickness function for each wall of the beam box (a total of eight variables) are considered in the optimization procedure. The effects of some parameters, such as the FGM volume fraction exponent, angular velocity, the pre-twist angle, porosity coefficient, and taper ratios on the mechanical behavior of the variable thickness FG beams are studied. The optimization result shows that applying the variable thickness concept can significantly increase the 1st and 2nd natural frequencies (by 60.4%, 60.22%) and reduce the weight (by 31.05%) of the system compared to the reference constant thickness beam while the increase in chordwise and flapwise deformations (by 8.16%, 9.93%) under combined loading is in an acceptable range, respectively.

Some classes of complete permutation polynomials in the form of (xpm − x + δ ) s + axpm + bx over F p 2m

Abstract Several classes of complete permutation polynomials (CPPs) in the form of (xpm − x+ δ ) s + axpm + bx over F p 2m are described in this paper. For fractional exponent s, the permutation properties of the given polynomials are proved by the AGW criterion and for determining the bijectivity of the corresponding piecewise functions. For exponent ssatisfying s= i(pm − 1) + pj , the main proofs here depend on the AGW criterion and the quantity of solutions for certain linearised equations. Several results generalise some earlier results on permutation polynomials (PPs).

Existence and multiple solutions for the critical fractional p‐Kirchhoff type problems involving sign‐changing weight functions

The aim of this paper is to study the existence and multiplicity of nonnegative solutions for the following critical Kirchhoff equation involving the fractional p‐Laplace operator . More precisely, we consider where is an open bounded domain with Lipschitz boundary with m > 1, a > 0, b > 0, dimension is the fractional critical Sobolev exponent, and the parameters λ > 0, 0 < s < 1 < q < p < ∞. Applying the Nehari manifold, fibering maps and Krasnoselskii genus theory, we investigate the existence and multiplicity of nonnegative solutions.

Regularity and upper semicontinuity of pullback attractors for non-autonomous Rao–Nakra beam

In this paper we study the long-time dynamics of a non-autonomous Rao–Nakra sandwich beam. The governing equations of Rao–Nakra sandwich beam consist of two wave equations for the longitudinal displacements of the top and bottom layers, and one Euler–Bernoulli beam equation for the transversal displacement. Under quite general assumptions on nonlinear damping and sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions. We also establish a Lipschitz stability result. We prove the existence of pullback attractors in natural space energy. Finally, we prove the regularity of the family of pullback attractors and its upper semicontinuous with respect to the fractional exponent γ ∈ (0, 1/2).

Vibration and stability characteristics of functionally graded sandwich plates with/without porosity subjected to localized edge loadings

This article investigates the influence of porosity and localized edge loads on the vibration and buckling characteristics of functionally graded material (FGM) plates using the finite element (FE) method. The analysis is carried out by choosing a single-layer FGM and two different types of FGM sandwich plates in such a way that there is no material discontinuity along the thickness direction. The porosity imperfections are accounted for in this study as criteria of stiffness reduction and are incorporated using modified power law distribution. The vibration and buckling responses are studied by considering four types of localized edge loads on plates with different porosity distributions. The application of different types of localized edge loads on the plate leads to the development of nonuniform in-plane stresses. Hence, they are computed first by using a dynamic approach before obtaining the buckling loads. The accuracy of the FE formulation is first validated for FGM plates by comparing the natural frequencies and the critical buckling loads obtained in the present investigation with the solutions already available in the literature. After validating the accuracy, detailed parametric studies have been performed on plates with varying volume fraction exponent, porosity distribution, porosity index, side-to-thickness ratio, load width ratio, aspect ratio, and support condition, and the results are presented with appropriate conclusions. A probabilistic sensitivity analysis is carried out to identify the significant parameter affecting the buckling load and natural frequency of porous FGM plates subjected to localized edge loads, which considerably aids in the design of the FGM plates.

Convolution series and the generalized convolution Taylor formula

In this paper, we deal with the convolution series that are a far reaching generalization of the conventional power series and the power series with the fractional exponents including the Mittag-Leffler type functions. Special attention is given to the most interesting case of the convolution series generated by the Sonine kernels. In this paper, we first formulate and prove the second fundamental theorem for the general fractional integrals and the $n$-fold general sequential fractional derivatives of both the Riemann-Liouville and the Caputo types. These results are then employed for derivation of two different forms of a generalized convolution Taylor formula for representation of a function as a convolution polynomial with a remainder in form of a composition of the $n$-fold general fractional integral and the $n$-fold general sequential fractional derivative of the Riemann-Liouville and the Caputo types, respectively. We also discuss the generalized Taylor series in form of convolution series and deduce the formulas for its coefficients in terms of the $n$-fold general sequential fractional derivatives.

Arbitrarily accelerating space-time wave packets.

All known realizations of optical wave packets that accelerate along their propagation axis, such as Airy wave packets in dispersive media or wave-front-modulated X-waves, exhibit a constant acceleration; that is, the group velocity varies linearly with propagation. Here we synthesize space-time wave packets that travel in free space with arbitrary axial acceleration profiles, including group velocities that change with integer or fractional exponents of the distance. Furthermore, we realize a composite acceleration profile: the wave packet accelerates from an initial to a terminal group velocity, before decelerating back to the initial value. These never-before-seen optical-acceleration phenomena are produced using the same experimental arrangement that precisely sculpts the wave packet's spatio-temporal spectral structure.

Seascape origin of Richards growth.

First proposed as an empirical rule over half a century ago, the Richards growth equation has been frequently invoked in population modeling and pandemic forecasting. Central to this model is the advent of a fractional exponent γ, typically fitted to the data. While various motivations for this nonanalytical form have been proposed, it is still considered foremost an empirical fitting procedure. Here, we find that Richards-like growth laws emerge naturally from generic analytical growth rules in a distributed population, upon inclusion of (i) migration (spatial diffusion) among different locales, and (ii) stochasticity in the growth rate, also known as "seascape noise." The latter leads to a wide (power law) distribution in local population number that, while smoothened through the former, can still result in a fractional growth law for the overall population. This justification of the Richards growth law thus provides a testable connection to the distribution of constituents of the population.

High-Pressure ESR Spectroscopy: On the Rotational Motion of Spin Probes in Pressurized Ionic Liquids.

We report high-pressure (up to 50 MPa) ESR-spectroscopic investigations on the rotational correlation times of the nitroxide radicals 2,2,6,6-tetramethylpiperidine 1-oxyl (TEMPO), 4-hydroxy-2,2,6,6-tetramethylpiperidine 1-oxyl (TEMPOL), and 4-amino-2,2,6,6-tetramethylpiperidine 1-oxyl (ATEMPO) in the ionic liquids 1-ethyl-3-methylimidazolium tetrafluoroborate (emimBF4), 1-butyl-3-methylimidazolium hexafluorophosphate (bmimPF6), 1-butyl-3-methylimidazolium tetrafluoroborate (bmimBF4), 1-methyl-3-octylimidazolium tetrafluoroborate (omimBF4), and 1-methyl-3-octylimidazolium hexafluorophosphate (omimPF6). The activation volumes (38.5–56.6 Å3) determined from pressure dependent rotational diffusion coefficients agree well with the pressure dependent viscosities of the ionic liquids. Experimentally, the fractional exponent of the generalized Stokes–Einstein–Debye relation is found to be close to one.

Thermally Induced Bistable Functionally Graded Nanocomposite Plate

This paper explores the feasibility of using functionally graded carbon nanotube-reinforced composite (FG-CNTRC) in designing bistable plates. Single-walled carbon nanotubes (SWCNTs) in this nanocomposite are assumed to have a functionally graded distribution across the thickness direction following a power law. By varying the distribution type, volume fraction, and volume fraction exponent of SWCNTs, laminates with multiple thermal properties can be generated. A higher-order Rayleigh–Ritz model is presented to investigate the bistability and buckling behaviours of the proposed bistable plates. Their out-of-plane displacements and snap-through forces are predicted using this model. In the buckling analysis, transverse concentrated forces are applied to the panel corners to induce snap-through. To trace the load-displacement path during the loading process, the arc-length method is used to solve derived nonlinear equilibrium equations. The analytical results are compared using nonlinear finite element (FE) analysis for validation. A comprehensive parametric study is conducted to analyse the effects of curing temperature, distribution type, volume fractions, and volume fraction exponents of SWCNTs on the bistable behaviours of FG-CNTRC plates. It is found that, by varying these factors, bistable FG-CNTRC plates with multiple stable shapes and a wide range of snap-through forces can be generated. FG- CNTRC thus offers a rich design space for producing bistable plates.

Measurement-Induced Dark State Phase Transitions in Long-Ranged Fermion Systems.

We identify an unconventional algebraic scaling phase in the quantum dynamics of long-range hopping, free fermions, which are exposed to continuous local measurements. The algebraic phase occurs for hopping decay exponents 1<p≲3/2, and features an algebraic entanglement entropy growth, and a slow algebraic decay of the density-density correlation function, both with a fractional exponent. It is separated from a critical phase with logarithmic entanglement growth at small, and an area law phase with constant entanglement entropy at large monitoring rates. A perturbative renormalization group analysis predicts that the transitions to the long-range phase correspond to an unconventional, modified sine-Gordon theory. Exact numerical simulations of the monitored wave functions are in excellent agreement with an analytical replica field theory approach, which confirms the view of the measurement-induced phase transition as a quantum phase transition in the dark state of an effective, non-Hermitian Hamiltonian.

Effects of Cone Angles on Nonlinear Vibration Responses of Functionally Graded Shells

The nonlinear vibration responses of functionally graded shells with different cone angles under external load are studied in this paper. Firstly, the Voigt model was employed to describe the physical properties along the thickness direction of FGMs conical shells. Then, the motion equations were derived based on the first shear deformation theory, von Kármán geometric nonlinearity and Hamilton’s principle. Next, the Galerkin method was applied to discretize the motion equations and the governing equations was simplified into a single degree of freedom nonlinear vibration differential equation based on the Volmir’s assumption. Finally, the nonlinear motion equation was solved by the harmonic balance method and Runge-Kutta method, and the amplitude frequency response characteristic curves of FGMs conical shell were obtained. The effects of different material distribution function and the ceramic volume fraction exponents on the amplitude frequency response curve of conical shell were discussed. The bifurcation diagrams of conical shells with different cone angles and time process diagrams and phase diagrams with different excitation amplitudes were described. The motion characteristic was characterized by Poincaré map. The results show that FGMs conical shell presents the nonlinear characteristics of "hardening" spring. The chaotic motion of FGMs conical shell is restrained and FGMs conical shell is not prone to produce motion instability with the increase of cone angle. The motion of FGMs conical shell presents a process from periodic motion to multi-periodic motion and then to chaos with the increase of excitation amplitude.

Vibrational Resonance and Electrical Activity Behavior of a Fractional-Order FitzHugh–Nagumo Neuron System

Making use of the numerical simulation method, the phenomenon of vibrational resonance and electrical activity behavior of a fractional-order FitzHugh–Nagumo neuron system excited by two-frequency periodic signals are investigated. Based on the definition and properties of the Caputo fractional derivative, the fractional L1 algorithm is applied to numerically simulate the phenomenon of vibrational resonance in the neuron system. Compared with the integer-order neuron model, the fractional-order neuron model can relax the requirement for the amplitude of the high-frequency signal and induce the phenomenon of vibrational resonance by selecting the appropriate fractional exponent. By introducing the time-delay feedback, it can be found that the vibrational resonance will occur with periods in the fractional-order neuron system, i.e., the amplitude of the low-frequency response periodically changes with the time-delay feedback. The weak low-frequency signal in the system can be significantly enhanced by selecting the appropriate time-delay parameter and the fractional exponent. In addition, the original integer-order model is extended to the fractional-order model, and the neuron system will exhibit rich dynamical behaviors, which provide a broader understanding of the neuron system.

Numerical study of Zika model as a mosquito-borne virus with non-singular fractional derivative

Zika virus infection, which is closely related to dengue, is becoming a global threat to human society. The transmission of the Zika virus from one human to another is spread by bites of Aedes mosquitoes. Recent studies also reveal the fact that the Zika virus can be transmitted by sexual interaction. In this paper, we use the fractional derivative with Mittag–Leffler non-singular kernel to study Zika virus transmission dynamics. This fractional is also known as the Atangana–Baleanu Caputo (ABC) derivative which is employed for the resulting system of ordinary differential equations. We investigate the proposed Zika virus model by using the Legendre spectral method. The model parameters are estimated and validated numerically by investigating the effect of fractional order exponent on various cases such as Susceptible human, infected human, asymptomatic carrier, exposed human, and recovered human. Numerical results obtained with the proposed method have been compared with exact solutions, showing in all parameters a very satisfactory agreement.

Fractional nonlinear surface impurity in a 2D lattice

We study the formation of localized modes around a generalized nonlinear impurity which is located at the boundary of a semi-infinite square lattice, and where we replace the standard discrete Laplacian by a fractional one, characterized by a fractional exponent 0<α<1 where α=1 marks the standard, non-fractional case. We specialize to two impurity cases: impurity at an “edge” and impurity at a “corner” and use the formalism of lattice Green functions to obtain in closed form the bound state energy and its mode amplitude. It is found that, for any fractional exponent and for impurity strengths above a certain threshold, there is always a single bound state for the linear impurity, while for the nonlinear (cubic) case, up to two bound states are possible. At small fractional exponents, the energy of the impurity mode becomes directly proportional to the impurity strength.

Design and finite element analysis of femoral stem prosthesis using functional graded materials

Conventionally biometals were used for design and development of bioimplants. However, the Young’s Modulus (YM) of these bioimplants is higher than that of a natural bone. Asymmetric load transfer from a bone to the bioimplant results in aseptic loosening and stress shielding. Here-in, the use of functionally graded materials (FGM) has been introduced to design the femoral stem prosthesis as a model bioimplant using computational biomechanics. The material properties variations in these FGMs in longitudinal and radial directions are explored to minimize the aseptic loosening and stress-shielding that plays a vital role in defining the performance and longevity of the prosthesis. Three groups of FGM (Ti-HA, SS316L-HA and CoCr alloy-HA) have been explored to design the stem prosthesis and the finite element analysis (FEA) was carried out using computational biomechanics. The stress distribution profile in the designed stem prosthesis demonstrated an increase in the stress values with an increase in the volume fraction exponent. The results corroborated with the stress distribution obtained from the simulation results of a cortico-cancellous bone. The stress distribution in the Ti-HA prosthesis is observed to be more uniform than CoCr-HA and SS316L-HA prosthesis. In addition, the reduced number of stress shielding points were observed for the Ti-HA prosthesis when compared with the CoCr-HA and SS 316 L-HA stem prostheses. Hence, the results suggested that the Ti-HA prosthesis could be considered as a mechanically stable prosthesis and the same could offer safe design for further development of a femoral bioimplant.