New Configurations of the Fuzzy Fractional Differential Boussinesq Model with Application in Ocean Engineering and Their Analysis in Statistical Theory

The fractional-order Boussinesq equations(FBSQe)are investigated in this work to see if they can effectively improve the situation where the shallow water equation cannot directly handle the dispersion wave.The fuzzy forms of analytical FBSQe solutions are first derived using the Adomian decomposition method.It also occurs on the sea floor as opposed to at the functionality.A set of dynamical partial differential equations(PDEs)in this article exemplify an unconfined aquifer flow implication.Thismethodology can accurately simulate climatological intrinsic waves,so the ripples are spread across a large demographic zone.The Aboodh transform merged with the mechanism of Adomian decomposition is implemented to obtain the fuzzified FBSQe in R,R^(n) and(2nth)-order involving generalized Hukuhara differentiability.According to the system parameter,we classify the qualitative features of the Aboodh transform in the fuzzified Caputo and Atangana-Baleanu-Caputo fractional derivative formulations,which are addressed in detail.The illustrations depict a comparison analysis between the both fractional operators under gH-differentiability,as well as the appropriate attributes for the fractional-order and unpredictability factorsσ∈[0,1].A statistical experiment is conducted between the findings of both fractional derivatives to prevent changing the hypothesis after the results are known.Based on the suggested analyses,hydrodynamic technicians,as irrigation or aquifer quality experts,may be capable of obtaining an appropriate storage intensity amount,including an unpredictability threshold.

Existence of Solutions of Three-Dimensional Fractional Differential Systems

In this article, we consider the three-dimensional fractional differential system of the form together with the Neumann boundary conditions,? where are the standard Caputo fractional derivatives, . A new result on the existence of solutions for a class of fractional differential system is obtained by using Mawhin’s coincidence degree theory. Suitable examples are given to illustrate the main results.

Quantum entangled fractional Fourier transform based on the IWOP technique

In our previous papers,the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics,and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform.The core function of the coordinate-momentum exchange operators in the addition law of fractional Fourier transform was analyzed too.In this paper,the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators(IWOP)are used to establish the entanglement fractional Fourier transform theory to the extent of quantum.A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

Perturbation Theory of Fractional Lagrangian System and Fractional Birkhoffian System

Perturbation to symmetry and adiabatic invariants are studied for the fractional Lagrangian system and the fractional Birkhoffian system in the sense of Riemann-Liouville derivatives.Firstly,the fractional Euler-Lagrange equation,the fractional Birkhoff equations as well as the fractional conservation laws for the two systems are listed.Secondly,the definition of adiabatic invariant for fractional mechanical system is given,then perturbation to symmetry and adiabatic invariants are established for the fractional Lagrangian system and the fractional Birkhoffian system under the special and general infinitesimal transformations,respectively.Finally,two examples are devoted to illustrate the results.

Fractional derivative statistical damage model of unsaturated expansive soil based on unified hydraulic effect

Unsaturated expansive soil is widely distributed in China and has complex engineering properties.This paper proposes the unified hydraulic effect shear strength theory of unsaturated expansive soil based on the effective stress principle,swelling force principle,and soil–water characteristics.Considering the viscoelasticity and structural damage of unsaturated expansive soil during loading,a fractional hardening–damage model of unsaturated expansive soil was established.The model parameters were established on the basis of the proposed calculation method of shear strength and the triaxial shear experiment on unsaturated expansive soil.The proposed model was verified by the experimental data and a traditional damage model.The proposed model can satisfactorily describe the entire process of the strain-hardening law of unsaturated expansive soil.Finally,by investigating the damage variables of the proposed model,it was found that:(a)when the values of confining pressure and matric suction are close,the coupling of confining pressure and matric suction contributes more to the shear strength;(b)there is a damage threshold for unsaturated expansive soil,and is mainly reflected by strength criterion of infinitesimal body;(c)the strain hardening law of unsaturated expansive soil is mainly reflected by fractional derivative operator.

AN ANALYTICAL SOLUTION FOR THREE-DIMENSIONAL DIFFRACTION OF PLANE P-WAVES BY A HEMISPHERICAL ALLUVIAL VALLEY WITH SATURATED SOIL DEPOSITS

An analytical solution for the three-dimensional scattering and diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits is developed by employing Fourier-Bessel series expansion technique. Unlike previous studies, in which the saturated soil deposits were simulated with the single-phase elastic theory, in this paper, they are simulated with Biot＇s dynamic theory for saturated porous media, and the half space is assumed as a single-phase elastic medium. The effects of the dimensionless frequency, the incidence angle of P-wave and the porosity of soil deposits on the surface displacement magnifications of the hemispherical alluvial valley are investigated. Numerical results show that the existence of a saturated hemispherical alluvial valley has much influence on the surface displacement magnifications. It is more reasonable to simulate soil deposits with Biot＇s dynamic theory when evaluating the displacement responses of a hemispherical alluvial valley with an incidence of P-waves.

THEORY AND APPLICATION OF FRACTIONAL STEP CHARACTERISTIC FINITE DIFFERENCE METHOD IN NUMERICAL SIMULATION OF SECOND ORDER ENHANCED OIL PRODUCTION

A kind of second-order implicit fractional step characteristic finite difference method is presented in this paper for the numerically simulation coupled system of enhanced （chemical） oil production in porous media. Some techniques, such as the calculus of variations, energy analysis method, commutativity of the products of difference operators, decomposition of high-order difference operators and the theory of a priori estimates are introduced and an optimal order error estimates in l^2 norm is derived. This method has been applied successfully to the numerical simulation of enhanced oil production in actual oilfields, and the simulation results ate quite interesting and satisfactory.

Effects of Fiber Volume Fraction on Damping Properties of Three-Dimensional and Five-Directional Braided Composites

The effects of fiber volume fraction on damping properties of carbon fiber three-dimensional and five-directional( 3D-5Dir)braided carbon fiber / epoxyres in composite cantilever beams were studied by experimental modal analysis method. Meanwhile,carbon fiber plain woven laminated / epoxy resin composites with different fiber volume fraction were concerned for comparison. The experimental result of braided specimens shows that the first three orders of natural frequency increase and the first three orders of the damping ratios of specimens decrease, when the fiber volume fraction increases. Furthermore,larger fiber volume fraction will be valuable for the better anti-exiting property of braided composites,and get an opposite effect on dissipation of vibration energy. The fiber volume fraction is an important factor for vibration performance design of braided composites. The comparison between the braided specimens and laminated specimens reveals that 3D braided composites have a wider range of damping properties than laminated composites with the same fiber volume fractions.

Three-Dimensional Static Analysis of Nanoplates and Graphene Sheets by Using Eringen’s Nonlocal Elasticity Theory and the Perturbation Method

A three-dimensional(3D)asymptotic theory is reformulated for the static analysis of simply-supported,isotropic and orthotropic single-layered nanoplates and graphene sheets(GSs),in which Eringen’s nonlocal elasticity theory is used to capture the small length scale effect on the static behaviors of these.The perturbation method is used to expand the 3D nonlocal elasticity problems as a series of two-dimensional(2D)nonlocal plate problems,the governing equations of which for various order problems retain the same differential operators as those of the nonlocal classical plate theory(CST),although with different nonhomogeneous terms.Expanding the primary field variables of each order as the double Fourier series functions in the in-plane directions,we can obtain the Navier solutions of the leading-order problem,and the higher-order modifications can then be determined in a hierarchic and consistent manner.Some benchmark solutions for the static analysis of isotropic and orthotropic nanoplates and GSs subjected to sinusoidally and uniformly distributed loads are given to demonstrate the performance of the 3D nonlocal asymptotic theory.

An Improved Three-Dimensional Model for Emotion Based on Fuzzy Theory

Emotion Model is the basis of facial expression recognition system. The constructed emotional model should not only match facial expressions with emotions, but also reflect the location relationship between different emotions. In this way, it is easy to understand the current emotion of an individual through the analysis of the acquired facial expression information. This paper constructs an improved three-dimensional model for emotion based on fuzzy theory, which corresponds to the facial features to emotions based on the basic emotions proposed by Ekman. What’s more, the three-dimensional model for motion is able to divide every emotion into three different groups which can show the positional relationship visually and quantitatively and at the same time determine the degree of emotion based on fuzzy theory.

The Theory and Application of Upwind Finite Difference Fractional Steps Procedure for Seawater Intrusion

Numerical simulation and theoretical analysis of seawater intrusion is the mathematical basis for modern environmental science. Its mathematical model is the nonlinear coupled system of partial differential equations with initial-boundary problems. For a generic case of a three-dimensional bounded region, two kinds of finite difference fractional steps pro- cedures are put forward. Optimal order estimates in norm are derived for the error in the approximation solution. The present method has been successfully used in predicting the consequences of seawater intrusion and protection projects.

On plane Λ-fractional linear elasticity theory

Non-local plane elasticity problems are discussed in the context of Λ-fractional linear elasticity theory. Adapting the Λ-fractional derivative along with the Λ-fractional space, where geometry and mechanics are valid in the conventional way, non-local plane elasticity problems are solved with the help of biharmonic functions. Then, the results are transferred into the initial plane.Applications are presented to homogeneous and the fractional beam bending problem.

The effect of fractional thermoelasticity on a two-dimensional problem of a mode I crack in a rotating fiber-reinforced thermoelastic medium

This article is concerned with the effect of rotation on the general model of the equations of the generalized thermoe- lasticity for a homogeneous isotropic elastic half-space solid, whose surface is subjected to a Mode-I crack problem. The fractional order theory of thermoelasticity is used to obtain the analytical solutions for displacement components, force stresses, and temperature. The boundary of the crack is subjected to a prescribed stress distribution and temperature. The normal mode analysis technique is used to solve the resulting non-dimensional coupled governing equations of the problem. The variations of the considered variables with the horizontal distance are illustrated graphically. Some particular cases are also discussed in the context of the problem. Effects of the fractional parameter, reinforcement, and rotation on the varia- tions of different field quantities inside the elastic medium are analyzed graphically. Comparisons are made between the results in the presence and those in the absence of fiber-reinforcing, rotating and fractional parameters.

A New Approach for the Exact Solutions of Nonlinear Equations of Fractional Order via Modified Simple Equation Method

In this article, the modified simple equation method has been extended to celebrate the exact solutions of nonlinear partial time-space differential equations of fractional order. Firstly, the fractional complex transformation has been implemented to convert nonlinear partial fractional differential equations into nonlinear ordinary differential equations. Afterwards, modified simple equation method has been implemented, to find the exact solutions of these equations, in the sense of modified Riemann-Liouville derivative. For applications, the exact solutions of time-space fractional derivative Burgers’ equation and time-space fractional derivative foam drainage equation have been discussed. Moreover, it can also be concluded that the proposed method is easy, direct and concise as compared to other existing methods.

Positive Solutions for Fractional Differential Equations with Multi-Point Boundary Value Problems

In this paper, a fractional multi-point boundary value problem is considered. By using the fixed point index theory and Krein-Rutman theorem, some results on existence are obtained.

Chaotic synchronization for a class of fractional-order chaotic systems

In this paper, a very simple synchronization method is presented for a class of fractional-order chaotic systems only via feedback control. The synchronization technique, based on the stability theory of fractional-order systems, is simple and theoretically rigorous.

Evaluation of Left Ventricular Function by Three-Dimensional Speckle-Tracking Echocardiography in Patients with Chronic Kidney Failure

Objective:To establish a quantitative evaluation of the left ventricle's systolic function in patients with chronic kidney failure(CKF)by three-dimensional speckle-tracking echocardiography.Methods:Two-dimensional and three-dimensional transthoracic echocardiography was performed on 30 patients with CKF.The ejection fraction,mass and global peak longitudinal strain,global circumferential strain,global area strain,and global radial strain of the left ventricle were calculated.Results:The ejection fraction,mass and global peak longitudinal strain(GLS),global circumferential strain(GCS),global area strain(GAS),and global radial strain(GRS)in the CKF group were significantly lower than those in the control group.Simultaneously,the GLS,GCS,GAS and GRS were well correlated with the ejection fraction.For patients with normal ejection fraction in the CKF group,the GLS,GCS,GAS and GRS were lower than those in the control group,while the left ventricular mass was significantly higher in CKF patients than in the control group.For patients with hypertension in the CKF group,ejection fraction,GLS,GCS,GAS and GRS calculated using three-dimensional echocardiography were significantly lower than those in patients with normal blood pressure;however,the myocardial mass was higher.Conclusions:The parameters(GLS,GCS,GAS and GRS)calculated using three-dimensional speckle-tracking software were lower in the CKF group.Simultaneously,the left ventricular mass was higher in CFK patients than in the control group,thus showing that the myocardial contraction function was impaired and that myocardial remodeling had occurred.

Remarks on the Solution of Laplace’s Differential Equation and Fractional Differential Equation of That Type

We discuss the solution of Laplace’s differential equation by using operational calculus in the framework of distribution theory. We here study the solution of that differential Equation with an inhomogeneous term, and also a fractional differential equation of the type of Laplace’s differential equation.

Seismic responses of a hemispherical alluvial valley to SV waves: a three-dimensional analytical approximation

Abstract An analytical solution to the three-dimen-sional scattering and diffraction of plane SV-waves by a saturated hemispherical alluvial valley in elastic half-space is obtained by using Fourier-Bessel series expan-sion technique. The hemispherical alluvial valley with saturated soil deposits is simulated with Biot＇s dynamic theory for saturated porous media. The following conclusions based on numerical results can be drawn： （1） there are a significant differences in the seismic response simulation between the previous single-phase models and the present two-phase model; （2） the nor-malized displacements on the free surface of the alluvial valley depend mainly on the incident wave angles, the dimensionless frequency of the incident SV waves and the porosity of sediments; （3） with the increase of the incident angle, the displacement distributions become more complicated; and the displacements on the free surface of the alluvial valley increase as the porosity of sediments increases.

Function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems

This paper investigates the function projective synchronization between fractional-order chaotic systems and integer-order chaotic systems using the stability theory of fractional-order systems. The function projective synchronization between three-dimensional （3D） integer-order Lorenz chaotic system and 3D fractional-order Chen chaotic system are presented to demonstrate the effectiveness of the proposed scheme.