A Novel Spacetime Collocation Meshless Method for Solving Two- Dimensional Backward Heat Conduction Problems

In this article,a meshless method using the spacetime collocation for solving the two-dimensional backward heat conduction problem(BHCP)is proposed.The spacetime collocation meshless method(SCMM)is to derive the general solutions as the basis functions for the two-dimensional transient heat equation using the separation of variables.Numerical solutions of the heat conduction problem are expressed as a series using the addition theorem.Because the basis functions are the general solutions of the governing equation,the boundary points may be collocated on the spacetime boundary of the domain.The proposed method is verified by conducting several heat conduction problems.We also carry out numerical applications to compare the SCMM with other meshless methods.The results show that the SCMM is accurate and efficient.Furthermore,it is found that the recovered boundary data on inaccessible boundary can be obtained with high accuracy even though the over specified data are provided only at a 1/6 portion of the spacetime boundary.

A Modified Tikhonov Regularization Method for an Axisymmetric Backward Heat Equation

This paper deals with the inverse time problem for an axisymmetric heat equation. The problem is ill-posed. A modified Tikhonov regularization method is applied to formulate regularized solution which is stably convergent to the exact one. estimate between the approximate solution and exact technical inequality and improving a priori smoothness Meanwhile, a logarithmic-HSlder type error solution is obtained by introducing a rather assumption.

Nonlinear Wavelet Methods for High-dimensional Backward Heat Equation

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation Motivated by Reginska＇s work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent.

Effects of backward path parameters on formability in conventional spinning of aluminum hemispherical parts

During multi-pass conventional spinning, roller paths combined with the forward and the backward pass are usually used to improve the material formability. In order to understand the backward spinning process properly, the backward roller paths of hemispherical parts with aluminum alloy 2024-O are analyzed. Finite element model with parameterized conventional spinning roller paths, which are based on quadratic Bezier curves, is developed to explore the evolution of the stress, strain and thinning during the backward processes. Analysis of the simulation results reveals stress and strain features of backward pass spinning. According to the findings, the application of the backward pass can obviously improve the uniformity of wall thickness. Furthermore, references of the parameters in future backward path design are provided.

The evolution of microstructure and texture of AZ80 Mg alloy cup-shaped pieces processed by rotating backward extrusion

This study proposed an effective plastic deformation technique,rotating backward extrusion(RBE),for producing high performance AZ80 magnesium alloy cup-shaped pieces.The RBE process was carried on the Gleeble-3500 test machine at 653 K,and the conventional backward extrusion(CBE)was also conducted for comparison.A detailed microstructure analysis was performed using the optical microscopy(OM)and electron back-scatter diffraction(EBSD).The results shown that the equivalent strain and deformation uniformity of the cup pieces could be substantially increased by the RBE process compared with the CBE process.Furthermore,the RBE process could significantly improve the grain refining capacity and the proportion of dynamic recrystallization(DRX),of which the maximum reduction of grain size was 88.60%,and the maximum increase of DRX proportion was 55.30%in the cup bottom.The main deformation mechanism of the RBE process was the discontinuous DRX(DDRX),while the continuous DRX(CDRX)was also occurred in the cup transition.Compared with the CBE sample,the texture of the cup bottom was weakened for the RBE sample.The microhardness value of the RBE sample was higher than that of the CBE sample,which can be attributed to the grain refinement strengthening.

MULTI-DIMENSIONAL REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THE COMPARISON THEOREM

In this article, we study the multi-dimensional reflected backward stochastic differential equations. The existence and uniqueness result of the solution for this kind of equation is proved by the fixed point argument where every element of the solution is forced to stay above the given stochastic process, i.e., multi-dimensional obstacle, respectively. We also give a kind of multi-dimensional comparison theorem for the reflected BSDE and then use it as the tool to prove an existence result for the multi-dimensional reflected BSDE where the coefficient is continuous and has linear growth.

Nonhomogeneous(H,Q)-Process:The Backward and Forward Equations

As for the backward and forward equation of nonhomogeneous(H, Q) -processes,we proof them in a new way. On the base of that, this paper gives the direct computational formalfor one dimensional distribution of the nonhomogeneous(H, Q) -process.

Rheological behavior,microstructure and hardness of A356 aluminum alloy in semisolid state using backward extrusion process

The influence of temperature on the flow behavior and rheological characteristics of an A356 alloy in the semi-solid state was investigated using backward extrusion process.Experiments were performed at 5 temperatures and 4 different wall thicknesses.Viscosities were determined using the force-displacement graphs obtained form back extrusion tests.As observed experimentally,at a constant temperature,the increase of shear rate results in the decrease of alloy viscosity exponentially.Raising the temperature increases the liquid fraction hence reduces the semi-solid alloy viscosity.Metallographic and image analyses show that,because of low forming speed,liquid has time to escape from solid phase forward the sample wall.This condition is the main reason for the segregation phenomenon seen in the base and walls.Vickers hardness test on samples reveals that the hardness increases with the decrease of temperature and wall thickness.

Determination of the Backward Predictability Limit and Its Relationship with the Forward Predictability Limit

In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space. The forward predictability mainly focuses on the forward evolution of initial errors superposed on the initial state over time, while the backward predictability is mainly concerned with when the given state can be predicted before this state happens. From the results, there is a negative correlation between the local forward and backward predictability limits. That is, the forward predictability limits are higher when the backward predictability limits are lower, and vice versa. We also find that the sum of forward and backward predictability limits of each state tends to fluctuate around the average value of sums of the forward and backward predictability limits of sufficient states.Furthermore, the average value is constant when the states are sufficient. For different chaotic systems, the average value is dependent on the chaotic systems and more complex chaotic systems get a lower average value. For a single chaotic system,the average value depends on the magnitude of initial perturbations. The average values decrease as the magnitudes of initial perturbations increase.

Non-equilibrium turbulent phenomena in the ?ow over a backward-facing ramp

Non-equilibrium turbulence phenomena have raised great interests in recent years. Significant efforts have been devoted to non-equilibrium turbulence properties in canonical flows, e.g., grid turbulence, turbulent wakes, and homogeneous isotropic turbulence(HIT). The non-equilibrium turbulence in non-canonical flows, however, has rarely been studied due to the complexity of the flows. In the present contribution, a directnumerical simulation(DNS) database of a turbulent flow is analyzed over a backwardfacing ramp, the flow near the boundary is demonstrated, and the non-equilibrium turbulent properties of the flow in the wake of the ramp are presented by using the characteristic parameters such as the dissipation coefficient C and the skewness of longitudinal velocity gradient Sk, but with opposite underlying turbulent energy transfer properties. The equation of Lagrangian velocity gradient correlation is examined, and the results show that non-equilibrium turbulence is the result of phase de-coherence phenomena, which is not taken into account in the modeling of non-equilibrium turbulence. These findings are expected to inspire deeper investigation of different non-equilibrium turbulence phenomena in different flow conditions and the improvement of turbulence modeling.

Blocking backward reaction on hydrogen evolution cocatalyst in a photosystem Ⅱ hybrid Z-scheme water splitting system

Photocatalytic Z-scheme water splitting is considered as a promising approach to produce solar hydrogen.However,the forward hydrogen production reaction is often impeded by backward reactions.In the present study,in a photosystem Ⅱ-integrated hybrid Z-scheme water splitting system,the backward hydrogen oxidation reaction was significantly suppressed by loading a PtCrOx cocatalyst on a ZrO2/TaON photocatalyst.Due to the weak chemisorption and activation of molecular hydrogen on PtCrOx,where Pt is stabilized in the oxidized forms,Pt^Ⅱ and Pt^Ⅳ,hydrogen oxidation is inhibited.However,it is remarkably well-catalyzed by the metallic Pt cocatalyst,thereby rapidly consuming the produced hydrogen.This work describes an approach to inhibit the backward reaction in the photosystem Ⅱ-integrated hybrid Z-scheme water splitting system using Fe(CN)6^3-/Fe(CN)6^4-redox couple as an electron shuttle.

Theoretical models for designing a 220-GHz folded waveguide backward wave oscillator

In this paper, the basic equations of beam-wave interaction for designing the 220 GHz folded waveguide （FW） backward wave oscillator （BWO） are described. On the whole, these equations are mainly classified into small signal model （SSM）, large signal model （LSM）, and simplified small signal model （SSSM）. Using these linear and nonlinear one-dimensional （1D） models, the oscillation characteristics of the FW BWO of a given configuration of slow wave struc- ture （SWS） can be calculated by numerical iteration algorithm, which is more time efficient than three-dimensional （3D） particle-in-cell （PIC） simulation. The SSSM expressed by analytical formulas is innovatively derived for determining the initial values of the FW SWS conveniently. The dispersion characteristics of the FW are obtained by equivalent circuit analysis. The space charge effect, the end reflection effect, the lossy wall effect, and the relativistic effect are all considered in our models to offer more accurate results. The design process of the FW BWO tube with output power of watt scale in a frequency range between 215 GHz and 225 GHz based on these 1D models is demonstrated. The 3D PIC method is adopted to verify the theoretical design results, which shows that they are in good agreement with each other.

Discrimination geochemical interaction effects on mineralization at the polymetallic Glojeh deposit,NW Iran by interative backward quadratic modeling

The physico-chemical exchanges between hydrothermal fluids and the host rock are usually controlled by elemental interaction effects.A criterion-based backward elimination approach applies the iterative regression analysis and analysis of variance to investigate the geochemical features of the polymetallic Glojeh(Au-Ag-CuPb-Zn)deposit in NW Iran.A statistical definition of the elemental interaction effects(X(i-j)^2,Xi×Xj)could elucidate the relationship between variables and the performance of a full quadratic polynomial model(QPM).The model optimization procedure was carried out by the removal of insignificant predictors(P value 95%CL)based on R^2(pred.)criterion.In order to straighten the convergent trend with R^2 and R^2(adj.),R^2(pred.)gradually increased from 0%to 77.8%by 15-steps optimization.The miniature-scale geochemical changes indicate double ordinal Au(Ag,Pb)and Au(Ag,Zn)interactions within the vein and host rock,in QPM.Results show that the Au(Pb-Zn)commonly presents ordinal effect at the vein and disordinal interaction at the host rock.This ordinal-disordinal interaction revealed that elements Pb and Zn have similar geochemical features during mineralization.In addition,Akima's polynomial contour map confirms the results from Pb-Zn interaction effects by dependency tracing between Au-Pb-Zn at different populations.However,it is noteworthy that Pb and Zn occur together in the second phase of Pb-Zn-Cu±(Ag±Cd)sulfide mineralization at Glojeh,which implies intergrowth and interaction of Pb-Zn on Au concentration.Pb and Zn demonstrate relatively high mobility and are generally concentrated in the near surface zones.Nb is an immobile element during alteration and high content Hg zone is mainly restricted to narrow stripes above ore vein and veinlets.

The Application of UBET in Backward Simulation of a Blade

A UBET backward simulation system, which has been used to simulate the forging process of a blade, is developed. It has been proved to be helpful to shape the preform of a blade accurately and properly. This paper emphasizes on the deductions and solutions of the velocity field, strain-rate field, plastic power and friction power between elements and the dies during simulation.

Quantitative Comparison of Predictabilities of Warm and Cold Events Using the Backward Nonlinear Local Lyapunov Exponent Method

The backward nonlinear local Lyapunov exponent method(BNLLE)is applied to quantify the predictability of warm and cold events in the Lorenz model.Results show that the maximum prediction lead times of warm and cold events present obvious layered structures in phase space.The maximum prediction lead times of each warm(cold)event on individual circles concentric with the distribution of warm(cold)regime events are roughly the same,whereas the maximum prediction lead time of events on other circles are different.Statistical results show that warm events are more predictable than cold events.

Backward Calculation Based on the Advection and Diffusion of Oil Spills on the Sea Surface

In the light of the problem of oil pollution brought about by ships, in this paper we present the concept of backward tracing oil spills. In the course of backward calculation of the two-dimensional convection & diffusion equation, on the one hand, the advection term itself has the strong unilateral property, which means information in the upper reaches is transmitted downstream via the advection term; on the other hand, because of the opposite direction of calculation, it is essential for information to be conveyed upstream by means of the advection term. In addition, unlike that in the forward calculation, the diffusion term in the backward calculation is prone to accumulate errors, and thus renders the whole scheme unstable. Therefore, we adopt the central difference to deal with both the convectional term and the diffusion term. By examining two practical examples (1) under the unlimited boundary condition, and (2) under the limited boundary condition, it is proven that this method could achieve fundamentally satisfactory results not only in the open ocean but also in the closed or semi-closed bay.

The Effects of a Backward Bifurcation on a Continuous Time Markov Chain Model for the Transmission Dynamics of Single Strain Dengue Virus

Global incidence of dengue, a vector-borne tropical disease, has seen a dramatic increase with several major outbreaks in the past few decades. We formulate and analyze a stochastic epidemic model for the transmission dynamics of a single strain of dengue virus. The stochastic model is constructed using a continuous time Markov chain (CTMC) and is based on an existing deterministic model that suggests the existence of a backward bifurcation for some values of the model parameters. The dynamics of the stochastic model are explored through numerical simulations in this region of bistability. The mean of each random variable is numerically estimated and these are compared to the dynamics of the deterministic model. It is observed that the stochastic model also predicts the co-existence of a locally asymptotically stable disease-free equilibrium along with a locally stable endemic equilibrium. This co-existence of equilibria is important from a public health perspective because it implies that dengue can persist in populations even if the value of the basic reproduction number is less than unity.

A Comparison Theorem for Solution of the Fully Coupled Backward Stochastic Differential Equations

The comparison theorems of solutions for BSDEs in fully coupled forward-backward stochastic differential equations (FBSDEs) are studied in this paper, here in the fully coupled FBSDEs the forward SDEs are the same structure.

A New Second Order Numerical Scheme for Solving Forward Backward Stochastic Differential Equations with Jumps

In this paper, we propose a new second order numerical scheme for solving backward stochastic differential equations with jumps with the generator linearly depending on . And we theoretically prove that the convergence rates of them are of second order for solving and of first order for solving and in norm.

Compact finite difference schemes for the backward fractional Feynman–Kac equation with fractional substantial derivative

The fractional Feynman-Kac equations describe the distributions of functionals of non-Brownian motion, or anomalous diffusion, including two types called the forward and backward fractional Feynman-Kac equations, where the nonlocal time-space coupled fractional substantial derivative is involved. This paper focuses on the more widely used backward version. Based on the newly proposed approximation operators for fractional substantial derivative, we establish compact finite difference schemes for the backward fractional Feynman-Kac equation. The proposed difference schemes have the q-th(q = 1, 2, 3, 4) order accuracy in temporal direction and fourth order accuracy in spatial direction, respectively. The numerical stability and convergence in the maximum norm are proved for the first order time discretization scheme by the discrete energy method, where an inner product in complex space is introduced. Finally, extensive numerical experiments are carried out to verify the availability and superiority of the algorithms. Also, simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed to further confirm the effectiveness of the proposed methods.