Solving Potential Problems Using the Best Approximation Method

In anomaly-free quantum field theories the integrand in the bosonic functional integral—the exponential of the effective action after integrating out fermions—is often defined only up to a phase without an additional choice. We term this choice ``setting the quantum integrand''. In the low-energy approximation to M-theory the E 8-model for the C-field allows us to set the quantum integrand using geometric index theory. We derive mathematical results of independent interest about pfaffians of Dirac operators in 8k+3 dimensions, both on closed manifolds and manifolds with boundary. These theorems are used to set the quantum integrand of M-theory for closed manifolds and for compact manifolds with either temporal (global) or spatial (local) boundary conditions. In particular, we show that M-theory makes sense on arbitrary 11-manifolds with spatial boundary, generalizing the construction of heterotic M-theory on cylinders.

Thermal barrier coatings (TBCs) have been investigated both experimentally and through simulation for mixing controlled combustion (MCC) concepts as a method for reducing heat transfer losses and increasing cycle efficiency, but it is still a very active research area. Early studies were inconclusive, with different groups discovering obstacles to realizing the theoretical potential. Nuanced papers have shown that coating material properties, thickness, microstructure, and surface morphology/roughness all can impact the efficacy of the thermal barrier coating and must be accounted for. Adding to the complexities, a strong spatial and temporal heat flux inhomogeneity exists for mixing controlled combustion (diesel) imposed onto the surfaces from the impinging flame jets. In support of the United States Department of Energy SuperTruck II program goal to achieve 55% brake thermal efficiency on a heavy-duty diesel engines, this study sought to develop a deeper insight into the inhomogeneous heat flux from mixing controlled combustion on thermal barrier coatings and to infer concrete guidance for designing coatings. To that end, a co-simulation approach was developed that couples high-fidelity computational fluid dynamics (CFD) modeling of in-cylinder processes and combustion, and finite element analysis (FEA) modeling of the thermal barrier-coated and metal engine components to resolve spatial and temporal thermal boundary conditions. The models interface at the surface of the combustion chamber; FEA modeling predicts the spatially resolved surface temperature profile, while CFD develops insights into the effect of the thermal barrier coating on the combustion process and the boundary conditions on the gas side. The paper demonstrates the capability of the framework to estimate cycle impacts of the temperature swing at the surface, as well as identify critical locations on the piston/thermal barrier coating that exhibit the highest charge temperature and highest heat fluxes. In addition, the FEA results include predictions of thermal stresses, thus enabling insight into factors affecting coating durability. An example of the capability of the framework is provided to illustrate its use for investigating novel coatings and provide deeper insights to guide future coating design.

There is no unique stationary scattering theory. We distinguish the main versions by analyzing the different idealizations (spatial or temporal boundary conditions - SBC and TBC, respectively) implied by different orders of taking the limits of infinite volume and of sharp initial conditions. The relations of the different representations are discussed in detail. The equivalence of the Schrödinger and Heisenberg dynamical pictures, as well as the hermiticity of the Hamiltonian fail for SBC but can be applied in TBC. However, when using TBC the scattering wave does not behave as an eigenfunction of the Hamiltonian. The evaluation of transition rates and rates of change of observables is described for the different approaches. We also distinguish the meaning of two types of invariants of the motion by resolving the temporal variation of the observables as an "internal" production rate plus a term representing the flux across the boundary of the scattering region.

The corner problem is a century-old canonical scattering problem describing wave diffraction at a quarter-plane spatial discontinuity. Here, we study its space-time analog: wave scattering at a corner in space-time, arising at a time-switched spatial interface. We highlight and resolve an inconsistency between spatial and temporal boundary conditions arising in this problem, and analytically demonstrate the emergence of shock waves launched by the scattering process. After numerically verifying our theory, we realize and experimentally probe the scattering at a space-time corner arising at the edge of a time-switched waveguide. Our results unveil and efficiently model the unusual phenomena arising at the spatial interface between time-modulated and static media, of great relevance for the growing field of spatiotemporal metamaterials.

Presented in this paper is a device to solve diffusion problems. From the view point of probability diffusion problems are defined as an investigation of Markov process under some initial and boundary conditions. It is proved that the solution of the given diffusion differential equation is the transition probability of the process generated under some conditions required by the diffusion problems. This method of analysis is also applicable to the nonstationary problems. The generator of Markov process with any condition is presented and by this generator the O-U process, a typical process of the diffusion problems, is generated. Its transition probabilities are measured and compared with the theoretical solution of the corresponding diffusion differential equation. Experiment shows that this analyzer is useful even in case the solution can not be obtained analytically.

Operator-splitting methods for the 2D convective Cahn–Hilliard equation

Solitary wave interactions of the GRLW equation

Finite element analyses of multi-dimensional, partial differential equations require accurate and complete boundary condition assignments. However, continuous temporal and spatial values along physical boundaries are rarely available when solving realistic problems based on field data. To mitigate this situation a neural network system was developed and coupled with an interpolation routine. This innovative computational utility was used to provide continuous boundary condition information in time and space along physical boundaries of interest. The neural network was trained to predict the time response at discrete boundary locations based on field measurements at those locations. Once trained the neural network system was capable of providing a continuous time history for those locations. This system was then linked to an interpolation routine which handled the spatial component of the boundary condition specifications. These coupled routines facilitated rapid deployment and testing of various finite element representations. Further, the neural network system captured the transient physics of the situation more accurately than interpolation routines used previously.

The Numerical Solution of Integral Equations on the Half-Line

Surface effects on a diffusion process governed by a fractional diffusion equation in a confined region with spatial and time dependent boundary conditions are investigated. First, we consider the one-dimensional case with the boundary conditions rho(0,t)=Phi0(t) and rho(a,t)=Phia(t). Subsequently, the two-dimensional case in the cylindrical symmetry with rho(a,theta,t)=Phia(theta,t) and rho(b,theta,t)=Phib(theta,t) is investigated. For these cases, we also obtain exact solutions for an arbitrary initial condition by using the Green's function approach.

Tropical cyclone (TC) track predictions from the operational regional nonhydrostatic TC forecast system of the Taiwanese Central Weather Bureau (CWB) are examined for their sensitivities to initial and lateral boundary conditions. Five experiments are designed and discussed, each using a combination of different initial and lateral boundary conditions coming either from the CWB or the National Centers for Environmental Prediction (NCEP) global forecast system. Eight typhoons in the western Pacific Ocean with 51 cases in 2004 and 2005 are tested with the five designed experiments for the 3-day forecast. The average track forecasts are the best when both the initial and lateral boundary conditions are from the NCEP global forecast system. This reflects the generally superior performance of the NCEP global forecast system relative to that of the CWB. Using different lateral boundary conditions has a greater impact on the track than using different initial conditions. Diagnostics using piecewise inversion of potential vorticity perturbations are carried out to identify synoptic features surrounding the featured typhoon that impact the track the most in each experiment. For the two cases demonstrated with the largest track improvement using NCEP global fields, the diagnostics indicate that the prediction of the strength and extent of the subtropical high in the western Pacific plays the major role in affecting these storm tracks. Using the analysis and predictions of the CWB global forecast system as the initial and lateral boundary conditions produces an overintensified subtropical ridge in the regional TC forecast model. Because most of the typhoons studied are located in the southwestern peripheral of the western Pacific subtropical high, the stronger steering from the more intense and extended high system is the main cause of the poleward bias in the predicted typhoon tracks in the operational run, which uses the CWB global forecast fields. The study suggests that, when efforts are made to improve a regional TC forecast model, it is also critically important to improve the global forecast system that provides the lateral boundary and initial conditions to the regional system.

The kinetics of capillary condensation into an initially dry pore is treated as an unsteady diffusion problem with a moving boundary at which the vapor concentration satisfies the Kelvin equation. The (justifiable) approximation that the phenomenon is radial leads to important simplifications. An exact similarity solution is found which is valid for the early stages of the condensation process. An approximate method of analysis is developed, using the class of similarity solutions of the two-dimensional radial diffusion problem with a constant concentration condition at an appropriately moving boundary. The method has some analogies to the Pohlhausen method in fluid mechanics in that the initial and boundary conditions are satisfied exactly, and the diffusion equation is satisfied in an integral sense. The method depends on numerical methods of integration, but approximate analytical solutions are available when the time is sufficiently large. Numerical solutions are given for the capillary condensation of water. The characteristic equilibration time is highly sensitive to h0, the relative saturation of the atmosphere to which the dry pore is exposed, varying approximately as (1—h0)—3. The initial condensation rate is a constant, largely independent of h0 when this is close to one. The kinetics of capillary condensation is highly sensitive to temperature, equilibration times for water at 80°C being only 1/233 as great as at 0°C. The major influence of temperature on kinetics is due to the temperature dependence of the saturated vapor density. A limitation to this approach is that the treatment is essentially macroscopic, although the early stages of condensation may involve relatively few molecules.

While Vilhelm Bjerknes and his team were developing their synoptic models in Bergen, a radically different approach to forecasting was being pursued by Lewis Fry Richardson. Richardson’s starting point was the system of fundamental physical principles governing atmospheric motion. He assembled the set of mathematical equations that represent these principles and formulated an approximate algebraic method of calculating their solution. Starting from the state of the atmosphere at a given time—the initial conditions—the method could be used to work out its future evolution.

This article discusses the case of immersion burning in the presence of an inert compound. Topics considered include the hydrodynamic problem, the diffusion problem, the thermal problem, the determination of boundary values, flame shape and length, and the solution sequence. The flow of oxidant contains inert components that substantially influence the heat and mass transfer. The temperature and concentration fields are determined for the flame axis, the flame front, and the free surface, which appear in the solutions to the diffusion and thermal problems. When the concentration has been determined, the Clausius-Clapeyron equation is used to find the corresponding equilibrium temperature. It is concluded that the resulting boundary values for temperature and concentrations uniquely define the distributions of the thermal and diffusion quantities in a laminar immersion flame.

Physically guided deep learning solver for time-dependent Fokker–Planck equation