NUMERICAL MODELING OF KELVIN-HELMHOLTZ INSTABILITY AND RAYLEIGH-TAYLOR INSTABILITY AT LARGE DEFORMATION STAGE——(Ⅰ) NUMERICAL METHOD AND NUMERICAL RESULTS OF RAYLEIGHTAYLOR INSTABILITY

This paper improves the discrete vortex method for modeling Kelvin-Helmholtz instability and Rayleigh-Tay- lor instability by proper choice of velocity weighted average coefficients, redistribution of markers and successive adding of computational points with the increase of interfacial deformation and gives the numerical results of Rayleigh-Taylor instability. The numerical results show that the first two techniques greatly enhance the ability of the discrete vortex method for modeling large interracial deformations and the last technique greatly reduces the computational amounts of the numerical modeling at large deformation stage. The numerical modeling of Rayleigh- Taylor instability not only reproduces some phenomena such as the roll up at the end part of the spike observed in experiments but also finds some new phenomena such as the splashes at the roll up parts which needs to be tested by experiment.

Numerical analysis of the effects of rock bolts on stress redistribution around a roadway

Besides opening geometry, in situ stress and material properties, opening support also has significant effects on stress redistribution around a roadway. To investigate these effects of rock bolts on the stress redistribution around a roadway, a series of numerical studies were carried out using the finite difference method. Since the stress changes around a roadway caused by rock bolting is small relative to the in situ stress, they cannot obviously be observed in stress contour plots. To overcome this difficulty, a new result processing methodology was developed using the contouring program Surfer. With this methodology, the effects of rock bolts on stress redistribution can obviously be analyzed. Numerical results show that in the three patterns of rock bolts installed in the roof, in the roof and the two lateral sides, and in all the four sides of the rectangular roadway, the maximum stress magnitude of the increase is 0.931 MPa, 2.46 MPa,and 6.5 MPa, respectively; the bolt number of 5 can form an integrated ground arch; the appropriate length and pre-tensioned force of the rock bolt is 2.0 m and 60 k N, respectively. What is more, the ground arch action under the function of rock bolting is able to be effectively examined. The rock bolts dramatically increase the minor principal stress around a roadway which results in significant increase in material strength. Consequently, the major principal stress that the material can carry will greatly increase.With adequate supports, an integrated ground arch which is critical for the stability of roadway will be formed around the roadway.

A Neural Study of the Fractional Heroin Epidemic Model

This works intends to provide numerical solutions based on the nonlinear fractional order derivatives of the classical White and Comiskey model(NFD-WCM).The fractional order derivatives have provided authentic and accurate solutions for the NDF-WCM.The solutions of the fractional NFD-WCM are provided using the stochastic computing supervised algorithm named Levenberg-Marquard Backpropagation(LMB)based on neural networks(NNs).This regression approach combines gradient descent and Gauss-Newton iterative methods,which means finding a solution through the sequences of different calculations.WCM is used to demonstrate the heroin epidemics.Heroin has been on-growth world wide,mainly in Asia,Europe,and the USA.It is the fourth foremost cause of death due to taking an overdose in the USA.The nonlinear mathematical system NFD-WCM discusses the overall circumstance of different drug users,such as suspected groups,drug users without treatment,and drug users with treatment.The numerical results of NFD-WCM via LMB-NNs have been substantiated through training,testing,and validation measures.The stability and accuracy are then checked through the statistical tool,such asmean square error(MSE),error histogram,and fitness curves.The suggested methodology’s strength is demonstrated by the high convergence between the reference solutions and the solutions generated by adding the efficacy of a constructed solver LMB-NNs,with accuracy levels ranging from 10?9 to 10?10.

The Perturbed Riemann Problem for a Geometrical Optics System

The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved.By studying the interactions among of the delta-shock,vacuum,and contact discontinuity,fourteen kinds of structures of Riemann solutions are obtained.The compound wave solutions consisting of delta-shocks,vacuums,and contact discontinuities are found.The single and double closed vacuum cavitations develop in solutions.Furthermore,it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data.Finally,the numerical results completely coinciding with theoretical analysis are presented.

Mathematical Analysis of Novel Coronavirus (2019-nCov) Delay Pandemic Model

In this manuscript,the mathematical analysis of corona virus model with time delay effect is studied.Mathematical modelling of infectious diseases has substantial role in the different disciplines such as biological,engineering,physical,social,behavioural problems and many more.Most of infectious diseases are dreadful such as HIV/AIDS,Hepatitis and 2019-nCov.Unfortunately,due to the non-availability of vaccine for 2019-nCov around the world,the delay factors like,social distancing,quarantine,travel restrictions,holidays extension,hospitalization and isolation are used as key tools to control the pandemic of 2019-nCov.We have analysed the reproduction number𝐑𝐑𝐧𝐧𝐧𝐧𝐧𝐧𝐧𝐧of delayed model.Two key strategies from the reproduction number of 2019-nCov model,may be followed,according to the nature of the disease as if it is diminished or present in the community.The more delaying tactics eventually,led to the control of pandemic.Local and global stability of 2019-nCov model is presented for the strategies.We have also investigated the effect of delay factor on reproduction number𝐑R_(nCov).Finally,some very useful numerical results are presented to support the theoretical analysis of the model.

Improved Dual Algorithm for Constrained Optimization Problems

One class of effective methods for the optimization problem with inequality constraints are to transform the problem to a unconstrained optimization problem by constructing a smooth potential function. In this paper, we modifies a dual algorithm for constrained optimization problems and establishes a corresponding improved dual algorithm; It is proved that the improved dual algorithm has the local Q-superlinear convergence; Finally, we performed numerical experimentation using the improved dual algorithm for many constrained optimization problems, the numerical results are reported to show that it is valid in practical computation.

An exact algorithm for optimal redundancy in a series system with multiple component choices

In this paper, an exact algorithm was proposed for optimal redundancy in a series system with multiple component choices. A reformulation of the nonseparable reliability function was approximated by a separable integer programming problem. The resulting separable nonlinear integer programming problem is used to compute upper bounds by Lagrangian relaxation and dual search. A special partition scheme was derived to reduce the duality gap in a branch-and-bound process, thus ensure the convergence of the algorithm. Computational results show that the algorithm is efficient for solving this class of reliability optimization problems.

AN ANALYTICAL SOLUTION TO LARGE DEFLECTION EQUATIONS OF SIMPLY－SUPPORTED RECTANGULAR HYPERBOLOIDAL SHALLOW SHELLS OF ORTHOTROPIC COMPOSITES

Based on the product rule of the Fourier series and some relevant results inreferences[1, 2]. a method on solving the large deflection equations of plates and shells by means of the Fourier series is proposed in the present paper,Applying this method .we derive a type solution to the Navier’s solution of the nonlinear differential equations of the rectangular hyperboloidal shallow shells of the orthotropic compositessimply supported .This solution is suitable for plates and shells with large deflection orsmall deflection whether it is isotropic or orthotropic.Their data processing results are correlative with those found in the classical examples and from the experiments.

Fractional thermoelasticity applications for porous asphaltic materials

A new mathematical model of poro-thermoelasticity has been constructed in the context of a new consideration of heat conduction with fractional order.One-dimensional application for a poroelastic half-space saturated with fluid is considered.The surface of the halfspace is assumed to be traction-free,permeable,and subjected to heating.The Laplace transform technique is used to solve the problem.The inversion of the Laplace transform will be obtained numerically and the numerical values of the temperature,stresses,strains,and displacements will be illustrated graphically for the solid and the liquid.

Natural convection of an alumina-water nanofluid inside an inclined wavy-walled cavity with a non-uniform heating using Tiwari and Das' nanofluid model

The present study is devoted to numerical analysis of natural convective heat transfer and fluid flow of alumina-water nanofluid in an inclined wavy-walled cavity under the effect of non-uniform heating. A single-phase nanofluid model with experimental correlations for the nanofluid viscosity and thermal conductivity has been included in the mathematical model. The considered governing equations formulated in dimensionless stream function, vorticity, and temperature have been solved by the finite difference method. The cavity inclination angle and irregular walls(wavy and undulation numbers)are very good control parameters for the heat transfer and fluid flow. Nowadays, optimal parameters are necessary for the heat transfer enhancement in different practical applications. The effects of the involved parameters on the streamlines and isotherms as well as on the average Nusselt number and nanofluid flow rate have been analyzed. It has been found that the heat transfer rate and fluid flow rate are non-monotonic functions of the cavity inclination angle and undulation number.

THERMAL POST-BUCKLING OF FUNCTIONALLY GRADED MATERIAL TIMOSHENKO BEAMS

Analysis of thermal post-buckling of FGM （Functionally Graded Material） Timoshenko beams subjected to transversely non-uniform temperature rise is presented. By accurately considering the axial extension and transverse shear deformation in the sense of theory of Timoshenko beam, geometrical nonlinear governing equations including seven basic unknown functions for functionally graded beams subjected to mechanical and thermal loads were formulated. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using a shooting method, the obtained nonlinear boundary value problem was numerically solved and thermal buckling and post-buckling response of transversely nonuniformly heated FGM Timoshenko beams with fixed-fixed edges were obtained. Characteristic curves of the buckling deformation of the beam varying with thermal load and the power law index are plotted. The effects of material gradient property on the buckling deformation and critical temperature of beam were discussed in details. The results show that there exists the tension-bend coupling deformation in the uniformly heated beam because of the transversely non-uniform characteristic of materials.

FURTHER STUDY ON A DUAL ALGORITHM

The dual algorithm for minimax problems is further studied in this paper.The resulting theoretical analysis shows that the condition number of the corresponding Hessian of the smooth modified Lagrange function with changing parameter in the dual algorithm is proportional to the reciprocal of the parameter,which is very important for the efficiency of the dual algorithm.At last,the numerical experiments are reported to validate the analysis results.

Meridional Circulation with Latitude Bands of Long-Lived Cyclones in Jupiter’s Convective Atmosphere

Mayr et al. [1] proposed that the vertical velocities in the global scale meridional circulation can produce distinct latitude bands where Jovian vortices like the white and brown are observed, and we present here a brief review of the mechanism. The observed life times of the ovals are much longer than the estimated spin-down times, which indicates that the vortices must be sustained through the release of internal energy. Like Jupiter’s Great Red Spot (GRS), the white/brown ovals are treated like terrestrial hurricanes or cyclones, which are generated by convection. The planetary energy Jupiter emits is transferred by convection, and under this condition the upward motions in the meridional circulation, around the equator for example, release energy from below and decrease the convective instability to suppress the formation of cyclones. But the downward motions in the circulation, near 20° latitude for example, carry energy down so that the convective instability is amplified to produce a dynamical environment that is favorable for the development of cyclones like the GRS and white/brown ovals. This picture is supported by an analysis of results from a numerical model of Jupiter’s alternating jets (Chan and Mayr [2]). Generated by alternating vertical winds in the meridional circulation, the vertical temperature variations reveal distinct latitude bands with enhanced convective instability, most prominent at high latitudes where long-lived circumpolar cyclones are observed from the Juno spacecraft.

A fractional-order model of HIV infection： Numerical solution and comparisons with data of patients

In this paper, a fractional-order model which describes the human immunodeficiency type-1 virus （HIV-1） infection is presented. Numerical solutions are obtained using a generalized Euler method （GEM） to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. We show that the model established in this paper possesses non-negative solutions. Comparisons between the results of the fractional-order model, the results of the integer model and the measured real data obtained from 10 patients during primary HIV-1 infection are presented. These compar- isons show that the results of the fractional-order model give predictions to the plasma virus load of the patients better than those of the integer model.

Modeling and scientific computing for the transmission dynamics of Avian influenza with half-saturated incidence

This paper presents the mathematical analysis of the dynamical system for avian influenza.The proposed model considers a nonlinear dynamical model of birds and human.The half-saturated incidence rate is used for the transmission of avian influenza infection.Rigorous mathematical results are presented for the proposed models.The local and global dynamics of each model are presented and proven that when R0<1,then the disease-free equilibrium of each model is stable both locally and globally,and when R0>1,then the endemic equilibrium is stable both locally and globally.The numerical results obtained for the proposed model shows that influenza could be eliminated from the community if the threshold is not greater than unity.

Global stability of SEIVR epidemic model with generalized incidence and preventive vaccination

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number R0. If the basic reproduction number R0〈 1, the disease- free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number R0 〉 1, the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.

Natural convective heat transfer and nanofluid flow in a cavity with top wavywall and corner heater

A numerical analysis of natural convection of nanofluid in a wavy-walled enclosure with an isothermal comer heater has been carried out. The cavity is heated from the left bottom comer and cooled from the top wavy wall while the rest walls are adiaba- tic. Mathematical model has been formulated using the single-phase nanofluid approach. Main efforts have been focused on the effects of the dimensionless time, Rayleigh number, undulation number, nanoparticle volume fraction and length of comer heaters on the fluid flow and heat transfer inside the cavity. Numerical results have been presented in the form of streamlines, isotherms, velocity and temperature profiles, local and average Nusselt numbers. It has been found that nanoparticle volume fraction essentially affects both fluid flow and heat transfer while undulation number changes significantly only the heat transfer rate.

A FEASIBLE SEMISMOOTH GAUSS-NEWTON METHOD FOR SOLVING A CLASS OF SLCPS

In this paper, we consider a class of the stochastic linear complementarity problems （SLCPs） with finitely many elements. A feasible semismooth damped Gauss-Newton al- gorithm for the SLCP is proposed. The global and locally quadratic convergence of the proposed algorithm are obtained under suitable conditions. Some numerical results are reported in this paper, which confirm the good theoretical properties of the proposed al- gorithm.Mathematics subject classification： 90C33, 65K10.

Diagonalized Chebyshev Rational Spectral Methods for Second-Order Elliptic Problems on Unbounded Domains

Diagonalized Chebyshev rational spectral methods for solving second-order elliptic problems on the half/whole line are proposed.Some Sobolev bi-orthogonal rational basis functions are constructed which lead to the diagonalization of discrete systems.Accordingly,both the exact solutions and the approximate solutions can be represented as infinite and truncated Fourier-like Chebyshev rational series.Numerical results demonstrate the effectiveness of the suggested approaches.

EFFICIENT AND ACCURATE CHEBYSHEV DUAL-PETROV-G ALERKIN METHODS FOR ODD-ORDER DIFFERENTIAL EQUATIONS

Efficient and accurate Chebyshev dual-Petrov-Galerkin methods for solving first-order equation,third-order equation,third-order KdV equation and fifth-order Kawahara equa-tion are proposed.Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems.Accordingly,both the exact solutions and the approximate solutions are expanded as an infinite and truncated Fourier-like series,respec-tively.Numerical experiments illustrate the effectiveness of the suggested approaches.