Second order time evolution of the multigroup diffusion and P1 equations for radiation transport

An existing solution method for solving the multigroup radiation equations, linear multifrequency-grey acceleration, is here extended to be second order in time. This method works for simple diffusion and for flux-limited diffusion, with or without material conduction. A new method is developed that does not require the solution of an averaged grey transport equation. It is effective solving both the diffusion and P₁ forms of the transport equation. Two dimensional, multi-material test problems are used to compare the solution methods.     

An exact solution for the linearized multifrequency radiation diffusion equation

An exact solution, based on Fourier and Laplace (FL) transforms, is developed for a linearization of the system modeling the multifrequency radiation diffusion and matter energy balance equations. The model uses an ideal gas equation of state. Opacities are proportional to the inverse of the cube of the frequency, thereby simulating free-free transitions. The solution is obtained in terms of integrals over the FL coefficients of the initial conditions and explicit sources. Results are presented for two special cases. (1) No sources, initially cold radiation field, and a localized matter energy profile. (2) Initially cold matter and radiation fields and a source of matter energy extending over finite space and time intervals.     

Solving fractional diffusion and wave equations by modified homotopy perturbation method

This Letter applies the modified He's homotopy perturbation method (HPM) suggested by Momani and Odibat to obtaining solutions of linear and nonlinear fractional diffusion and wave equations. The fractional derivative is described in the Caputo sense. Some illustrative examples are given, revealing the effectiveness and convenience of the method.     

Application of the energy balance method to nonlinear vibrating equations

In this paper, He’s energy balance method is applied to nonlinear vibrations and oscillations. The method is applied to four nonlinear differential equations. It has indicated that by utilizing He’s energy balance method (HEBM), just one iteration leads us to high accuracy of solutions. It has illustrated that the energy balance methodology is very effective and convenient and does not require linearization or small perturbation. Contrary to the conventional methods, in energy balance method, only one iteration leads to high accuracy of the solutions. The results reveal that the energy balance method is very effective and simple. It is predicted that the energy balance method can be found wide application in engineering problems, as indicated in following examples.     

Non-Markovian diffusion equations and processes: Analysis and simulations

In this paper we introduce and analyze a class of diffusion type equations related to certain non-Markovian stochastic processes. We start from the forward drift equation which is made non-local in time by the introduction of a suitable chosen memory kernel K(t). The resulting non-Markovian equation can be interpreted in a natural way as the evolution equation of the marginal density function of a random time process l(t). We then consider the subordinated process Y(t)=X(l(t)) where X(t) is a Markovian diffusion. The corresponding time evolution of the marginal density function of Y(t) is governed by a non-Markovian Fokker-Planck equation which involves the memory kernel K(t). We develop several applications and derive the exact solutions. We consider different stochastic models for the given equations providing path simulations.     

Linearisation and potential symmetries of certain systems of diffusion equations

We consider systems of two pure one-dimensional diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries for these systems. These are determined by expressing the equations in a partially and wholly conserved form, and then by performing a potential symmetry analysis on those systems that can be linearised. We give several examples of such systems, and in a specific case we show how linearising and hodograph-type mappings can lead to new solutions of the diffusion system.     

Anomalous energy diffusion and heat conduction in one-dimensional system

We propose a new concept, the centre of energy, to study energy diffusion and heat conduction in one-dimensional hard-point model. For diatom model, we find an anomalous energy diffusion as $\langle x^2 \rangle\sim t^\beta$ with $\beta=1.33$, which is independent of initial condition and mass rate. The present model can be viewed as the model composed by independent quasi-particles, the centre of energy. In this way, heat current can be calculated. Based on theory of dynamic billiard, the divergent exponent of heat conductivity is estimated to be $\alpha=0.33$, which is confirmed by a simple numerical calculation.     

Modelling thermal radiation in buoyant turbulent diffusion flames

This work focuses on the numerical modelling of radiative heat transfer in laboratory-scale buoyant turbulent diffusion flames. Spectral gas and soot radiation is modelled by using the Full-Spectrum Correlated-k (FSCK) method. Turbulence-Radiation Interactions (TRI) are taken into account by considering the Optically-Thin Fluctuation Approximation (OTFA), the resulting time-averaged Radiative Transfer Equation (RTE) being solved by the Finite Volume Method (FVM). Emission TRIs and the mean absorption coefficient are then closed by using a presumed probability density function (pdf) of the mixture fraction. The mean gas flow field is modelled by the Favre-averaged Navier–Stokes (FANS) equation set closed by a buoyancy-modified k-? model with algebraic stress/flux models (ASM/AFM), the Steady Laminar Flamelet (SLF) model coupled with a presumed pdf approach to account for Turbulence-Chemistry Interactions, and an acetylene-based semi-empirical two-equation soot model. Two sets of experimental pool fire data are used for validation: propane pool fires 0.3 m in diameter with Heat Release Rates (HRR) of 15, 22 and 37 kW and methane pool fires 0.38 m in diameter with HRRs of 34 and 176 kW. Predicted flame structures, radiant fractions, and radiative heat fluxes on surrounding surfaces are found in satisfactory agreement with available experimental data across all the flames. In addition further computations indicate that, for the present flames, the gray approximation can be applied for soot with a minor influence on the results, resulting in a substantial gain in Computer Processing Unit (CPU) time when the FSCK is used to treat gas radiation.     

一类大气等离子体反应扩散模型的解法

研究了一类大气等离子体反应扩散模型. 利用奇摄动方法得到了问题的渐近解, 并用微分不等式理论证明了解的一致有效性.     

用光电混合系统解线性方程组

本文提出一种用于迭代法求解线性方程组的光电混合系统。该系统的光学部分主要由单个全息透镜组成,它执行矩阵与矢量的乘法运算;系统的其余部分执行矢量的测量与求和,它由CCD探测器件和一台微机组成。使用这个光电混合系统,用迭代法对一个4元线性方程组求解,实验结果与理论解比较,误差约为5％。     

A model calculation of radiation damping by the energy balance method

The method of calculating radiation damping effects by means of energy balance is applied to a simple model problem. The balance equation is derived from the conservation law for energy that is a consequence of the equations of motion for the system by integrating over a four-dimensional space-time region bounded in the wave zone by two future-directed null cones and in the near zone by twot=const surfaces. It is assumed that the sources are in slow motion and the fields are calculated by the method of matched asymptotic expansions. Our principle conclusion is that even though the damping isO( ³) where is an expansion parameter the fields need only be determined toO(1) to calculate the effects of damping by this method.Work supported in part by NSF Grant No. 79-11664-01.     

New variable separation solutions for the generalized nonlinear diffusion equations

The functionally generalized variable separation of the generalized nonlinear diffusion equations ut = A(u, ux)uxx +B(u, ux) is studied by using the conditional Lie–Ba¨cklund symmetry method. The variant forms of the considered equations,which admit the corresponding conditional Lie–Ba¨cklund symmetries, are characterized. To construct functionally generalized separable solutions, several concrete examples defined on the exponential and trigonometric invariant subspaces are provided.     

扰动扩散方程初值问题的近似对称约化

本文利用近似广义条件对称方法研究一类带有源项的非线性扩散方程的初值问题.给出所研究方程的分类并将偏微分方程的初值问题约化为常微分方程的初值问题,通过求解约化后的常微分方程组可得相对应偏微分方程初值问题的近似解.     

Solving Schrodinger equations using a physically constrained neural network

Deep neural networks (DNNs) and auto differentiation have been widely used in computational physics to solve variational problems. When a DNN is used to represent the wave function and solve quantum many-body problems using variational optimization, various physical constraints have to be injected into the neural network by construction to increase the data and learning efficiency. We build the unitary constraint to the variational wave function using a monotonic neural network to represent the cumulative distribution function (CDF) begin{document}$F(x) = int_{-infty}^{x} psi^*psi {rm d}x'$end{document}. Using this constrained neural network to represent the variational wave function, we solve Schrodinger equations using auto-differentiation and stochastic gradient descent (SGD) by minimizing the violation of the trial wave function begin{document}$ psi(x) $end{document} to the Schrodinger equation. For several classical problems in quantum mechanics, we obtain their ground state wave function and energy with very low errors. The method developed in the present paper may pave a new way for solving nuclear many-body problems in the future.     

Variational iteration method for solving time-fractional diffusion equations in porous the medium

<正>The variational iteration method is successfully extended to the case of solving fractional differential equations, and the Lagrange multiplier of the method is identified in a more accurate way.Some diffusion models with fractional derivatives are investigated analytically,and the results show the efficiency of the new Lagrange multiplier for fractional differential equations of arbitrary order.     

Conditional Lie-Bäcklund symmetries and solutions of inhomogeneous nonlinear diffusion equations

The second-order conditional Lie-Bäcklund symmetries of nonlinear diffusion equations with variable coefficients are studied. A number of examples are considered and some exact solutions are constructed via the compatibility of conditional Lie-Bäcklund symmetries and the governing equations. These solutions possess the extended forms of the separation of variables, including the extensions of the instantaneous source solutions of the porous medium equations.     

A nonextensive maximum entropy approach to a family of nonlinear reaction–diffusion equations

We consider a monoparametric family of reaction–diffusion equations endowed with both a nonlinear diffusion term and a nonlinear reaction one that possess exact time-dependent particular solutions of the Tsallis’ maximum entropy (MaxEnt) form. The evolution of these solutions is governed by a system of three coupled nonlinear ordinary differential equations that are integrated numerically. A simple population dynamics interpretation provides a qualitative understanding of the behaviour of the q-MaxEnt solutions. When the reaction term vanishes the time-dependent distributions studied here reduce to the previously known Tsallis’ MaxEnt solutions for the nonlinear diffusion equation.     

Spectral spin diffusion and magnetic dipolar energy in the NMR of 13CH3 compounds

Spin diffusion between ¹³CH₃ groups in solids is studied both theoretically and experimentally. It is shown to be dominated by mutual spin flip–flops of protons belonging to neighbouring methyl groups. Also nonmethyl protons may contribute significantly if present in the sample. The spin–rotational ground state of ¹³CH₃ consists of 16 sublevels. When their populations are used to describe spin diffusion, eight population combinations are shown to be important, two of them corresponding to the ¹³C–proton and proton–proton intra-methyl magnetic dipolar energies, D_c and D_p, respectively. Spin-diffusion transitions modulate these combinations so that a further reduction to two sets of four combinations is possible, with no coupling between the sets. Coupled differential equations are derived to describe the time dependence of the combinations in each set. They are solved numerically and compared with experimental results on a single crystal of aspirin with ¹³C-labelled methyl groups at the carbon resonance. The ¹³C NMR induction signal was observed as a function of time after the preparation either at the carbon resonance (a two-pulse sequence) or at the proton resonance (proton saturation). Usually carbon spectra were computed first and then three of the mentioned population combinations were obtained from the individual spectral components. Some results on the time dependence of D_c were also obtained directly from the amplitude of the out-of-phase induction signal. Theoretical predictions are found to describe semiquantitatively the overall time dependence of these three combinations and especially their variation with different initial conditions, which are discussed in detail. Also the partial transfer of the magnetic dipolar energy between D_c and D_p is nicely explained. Reasons for discrepancies are discussed.     

Simultaneous tracer diffusion and interdiffusion in a sandwich-type configuration to provide the composition dependence of the tracer diffusion coefficients

In this paper, a new formalism of a combined tracer and interdiffusion experiment for a binary interdiffusion couple is developed. The analysis requires an interdiffusion couple that initially contains a thin layer of tracers of one or both of the constituent elements at the original interface of the couple (sandwich interdiffusion experiment). This type of interdiffusion experiment was first performed in 1958 by J.R. Manning. The theoretical analysis presented in this paper is based on a newly developed phenomenological theory of isotopic interdiffusion combined with the Boltzmann–Matano formalism. This new analysis now provides the means to obtain the composition dependent interdiffusion coefficient and tracer diffusion coefficients simultaneously from analysis of the interdiffusion and tracer profiles in a single sandwich interdiffusion experiment. The new analysis is successfully applied to the results of Manning’s original ‘sandwich interdiffusion’ experiment in the Ag–Cd system (six couples in total) and is validated with an independent determination of the Ag and Cd tracer diffusion coefficients by Schoen using the conventional thin film technique. Suggestions for further development of the sandwich interdiffusion experiment and analysis to the case of multicomponent alloys are provided.     

Generalized heat diffusion equations with variable coefficients and their fractalization from the Black-Scholes equation

In this study, we prove that modified diffusion equations, including the generalized Burgers' equation with variable coefficients, can be derived from the Black-Scholes equation with a time-dependent parameter based on the propagator method known in quantum and statistical physics.The extension for the case of a local fractal derivative is also addressed and analyzed.