Approximate analytical solutions of generalized linear and nonlinear reaction-diffusion equations in an infinite domain

This Letter deals with the solution of unified fractional reaction-diffusion systems in an infinite domain. The results are obtained in compact and elegant forms in terms of generalized Mittag-Leffler functions, which are suitable for numerical computation.     

非线性演化方程椭圆函数解的一种简单求法及其应用

非线性演化方程的许多行波解可以写成满足投影Riccati方程的两个基本函数的多项式形式.利用这一性质，通过建立一般的椭圆方程与投影Riccati方程解之间的关系，导出了一个构造这些解的新方法.该方法对类型Ⅰ的方程和类型Ⅱ的方程均有效，同时也回答了如何求出非线性演化方程分式形式椭圆函数解的问题. 关键词： 非线性演化方程 椭圆函数解     

A new class of gas-kinetic relaxation schemes for the compressible Euler equations

Starting from the gas-kinetic model, a new class of relaxation schemes for the Euler equations is presented. In contrast to the Riemann solver, these schemes provide a multidimensional dynamical gas evolution model, which combines both Lax-Wendroff and kinetic flux vector splitting schemes, and their coupling is based on the fact that a nonequilibrium state will evolve into an equilibrium state along with the increase of entropy. The numerical fluxes are constructed without getting into the details of the particle collisions. The results for many well-defined test cases are presented to indicate the robustness and accuracy of the current scheme.     

The influence of fractality on the time evolution of the diffusion process

In the literature, the deviations from standard behaviors of the solutions of the kinetic equation and the analogous diffusion equation are put forward by investigations which are carried out in the frame of fractional mathematics and nonextensive physics. On the other hand, the physical origins of the order of derivative namely α in fractional mathematics and the entropy index q in nonextensive physics are a topic of interest in scientific media. In this study, the solutions of the diffusion equation which have been obtained in the framework of fractional mathematics and nonextensive physics are revised. The diffusion equation is solved by the cumulative diminuation/growth method which has been developed by two of the present authors and physical nature of the parameters α and q are enlightened in connection with fractality of space and the memory effect. It has been emphasized that the mathematical basis of deviations from standard behavior in the distribution functions could be established by fractional mathematics where as the physical mechanism could be revealed using the cumulative diminuation/growth method.     

Time relaxation of the solutions of master equations for large systems

The time relaxation behavior of the solutions of certain classes of discrete master equations is studied in the limit of an infinite number of states. Depending on the range of the transition matrix, a relaxation behavior is found reaching from at ^–1/2 law for short range, over enhanced relaxation to an exponential relaxation for the extreme long-range case. The behavior in the limit of a continuous family of states is also discussed.     

Comparison of a fractal analysis and debye relaxation for the structural relaxation in amorphous metals

Abstract

Magnetic After-Effect isotherms have been measured on a metallic glass pre-annealed for different times. The results have been analyzed simultaneously by Debye relaxation functions leading to spectra of activation enthalpies and by extended exponential functions leading to a unique effective activation enthalpy. The thermodynamic activation parameters issued from both analyses compare well. The average values of the enthalpy spectra determined by assuming Debye relaxations are equal to the effective enthalpies resulting from the non-Debye relaxation.

Magnetische Nachwirkungs-Isothermen wurden an amorphen Legierungen gemessen, die mit verschiedenen Anlasszeiten vorbehandelt wurden. Die Ergebnisse wurden sowohl unter der Annahme von Debye-Relaxationsprozessen analysiert, die mit Aktivierungsspektren beschrieben wurden, als auch mit gestreckten Exponentialfunktionen, die zu einer einheitlichen effektiven Aktivierungsenthalpie führt. Die thermodynamischen Parameter von beiden Analysen sind gut vergleichbar. Die Mittelwerte der Aktivierungsenthalpie-Spektren, welche sich bei Zugrundelegung der Debye-Relaxationsprozesse ergeben, entsprechen den effektiven Enthalpien, die sich bei Anwendung der gestreckten Exponentialfunktionen ergeben.

Des isothermes de trainage magnétique ont été réalisées sur des echantillons de verres métalliques ayant subi des traitements thermiques de différentes durées. Les résultats ont été simultanément analysés par des processus de relaxation de Debye engendrant des distributions d'enthalpies d'activation, et par des exponentielles etirées possédant des enthalpies effectives. Les paramètres thermodynamiques issus des deux analyses sont cohérents et comparables. Les valeurs moyennes des distributions d'enthalpies résultant de la relaxation de Debye sont identiques aux enthalpies effectives qui découlent des exponentielles etirées.     

Discrete stochastic evolution rules and continuum evolution equations

The continuum evolution equations are derived from updating rules for three classes of stochastic models. The first class corresponds to models whose stochastic continuum equations are of the Langevin type obtained after carrying out a “local average” known as coarse-graining. The second class consists of a hierarchy of continuum equations for the correlations of the dynamical variables obtained after making an average over realizations. This average generates a hierarchy of deterministic partial differential equations except when the dynamical variables do not depend on the values of the neighboring dynamical variables, in which case a hierarchy of ordinary differential equations is obtained. The third class of evolution equations for the correlations of the dynamical variable constitutes another hierarchy after calculating an average over both realizations and all the sites of the lattice. This double average generates a hierarchy of deterministic ordinary differential equations. The second and third classes of equations are truncated using a mean field (m,n)-closure approximation in order to obtain a finite set of equations. Illustrative examples of every class are given.     

A theorem allowing to derive deterministic evolution equations from stochastic evolution equations

The deterministic evolution equations of classical as well as quantum mechanical models are derived from a set of stochastic evolution equations after taking an average over realizations using a theorem. Examples are given that show that deterministic quantum mechanical evolution equations, obtained initially by R.P. Feynman and subsequently studied by Boghosian and Taylor IV [B.M. Boghosian, W. Taylor IV, Phys. Rev. E 57 (1998) 54. See also arXiv:quant-ph/9904035] and Meyer [D.A. Meyer, Phys. Rev. E 55 (1997) 5261], among others, are derived from a set of stochastic evolution equations. In addition, a deterministic classical evolution equation for the diffusion of monomers, similar to the second Fick law, is also obtained.     

The Wronskian technique for nonlinear evolution equations

The investigation of the exact traveling wave solutions to the nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. To understand the mechanisms of those physical phenomena, it is necessary to explore their solutions and properties. The Wronskian technique is a powerful tool to construct multi-soliton solutions for many nonlinear evolution equations possessing Hirota bilinear forms. In the process of utilizing the Wronskian technique,the main difficulty lies in the construction of a system of linear differential conditions, which is not unique. In this paper,we give a universal method to construct a system of linear differential conditions.     

The derivative-dependent functional variable separation for the evolution equations

This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation u_tt=Au,u_xu_xx+Bu,u_x,u_t which admits the derivative-dependent functional separable solutions DDFSSs). We also extend the concept of the DDFSS to cover other variable separation approaches.     

射频场照射下扩展的Solomon方程及射频场的照射对异核体系弛豫速率与NOE的影响

根据WBR理论,采用“改进的矩阵计算方法”,推导出了射频场照射下扩展的Solomon方程,并据此研究了射频场的照射对异核体系各种弛豫速率与NOE的影响,得出了如下结论:1)给出的方程比Boulat和Bodenhausen的方程更准确、更具普遍性,能够具体描述射频场对各种弛豫速率与NOE的影响.2)射频场的照射对纵向与横向弛豫速率的影响甚微,可以忽略,而交叉弛豫速率在射频场的照射下则有一定程度的降低.3)射频场的照射将使NOE变弱,且射频场越强,NOE越弱. 关键词： 核磁共振 Solomon方程 弛豫 射频场     

New solitary wave solutions for nonlinear evolution equations

Three important nonlinear evolution equations are solved with the aid of the symbolic manipulation system.Maple,using the direct algebraic method proposed recently,We explicitly obtain several new solutions of physical interest in addition to rederiving all the known solutions.     

On the equivalence of the parallel channel and the correlated cluster relaxation models

The question of the origins of nonexponential relaxation is addressed in terms of the probabilistic approach to relaxation. The interconnection between two differently rooted probabilistic models, i.e., between the parallel channel and the correlated cluster models, is presented. We show that clearly different probabilistic origins yield in both approaches a well-defined class of universally valid two-power-law responses with the stretched-exponential and exponential decay laws as special cases. The equivalence of both models indicates that variations in the local environment of the relaxing configurational units (parallel channel relaxation) can provide a basis for self-similar relaxation dynamics without the need for hierarchically constrained dynamics (correlated clusters relaxation).     

A deep learning method for solving third-order nonlinear evolution equations

It has still been difficult to solve nonlinear evolution equations analytically. In this paper, we present a deep learning method for recovering the intrinsic nonlinear dynamics from spatiotemporal data directly. Specifically, the model uses a deep neural network constrained with given governing equations to try to learn all optimal parameters. In particular, numerical experiments on several third-order nonlinear evolution equations, including the Korteweg–de Vries (KdV) equation, modified KdV equation, KdV–Burgers equation and Sharma–Tasso–Olver equation, demonstrate that the presented method is able to uncover the solitons and their interaction behaviors fairly well.     

Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear partial differential evolution equations of dynamical systems

Using functional derivative technique in quantum field theory, the algebraic dynamics approach for solution of ordinary differential evolution equations was generalized to treat partial differential evolution equations. The partial differential evolution equations were lifted to the corresponding functional partial differential equations in functional space by introducing the time translation operator. The functional partial differential evolution equations were solved by algebraic dynamics. The algebraic dynamics solutions are analytical in Taylor series in terms of both initial functions and time. Based on the exact analytical solutions, a new numerical algorithm—algebraic dynamics algorithm was proposed for partial differential evolution equations. The difficulty of and the way out for the algorithm were discussed. The application of the approach to and computer numerical experiments on the nonlinear Burgers equation and meteorological advection equation indicate that the algebraic dynamics approach and algebraic dynamics algorithm are effective to the solution of nonlinear partial differential evolution equations both analytically and numerically. Supported by the National Natural Science Foundation of China (Grant Nos. 10375039, 10775100 and 90503008), the Doctoral Program Foundation of the Ministry of Education of China, and the Center of Nuclear Physics of HIRFL of China     

一类高维耦合的非线性演化方程的简单求解

利用一个简单的变换，一类高维耦合的非线性演化方程可以被约化为一低维的简单方程，将已有的求解法应用于简单方程，十分简捷的获得了原方程大量的精确解. 关键词： 非线性耦合方程 精确解 tanh函数方法     

非线性发展方程的Wronskian解及Young图证明

给出一般非线性发展方程构造Wronskian解的间接法. 根据Young图运算的性质给出了文中命题的证明, 并讨论了置换群特征标与Young图表达式系数间的关系. 关键词： 非线性发展方程 Wronskian解 Young图 特征标     

构造孤子方程的Weierstrass椭圆函数解的一个新方法

利用具有Weierstrass椭圆函数解的方程，首先获得了投影Riccati方程的两组新解.由于投影Riccati方程可用于多种具孤子解的非线性演化方程的求解，因而得到了一个可以构造这些方程的Weierstrass椭圆函数解的新方法. 关键词： Weierstrass椭圆函数解 投影Riccati方程 非线性演化方程     

Ionic conduction and dielectric relaxation in Ag/Na ion-exchanged aluminosilicate glasses: mixed mobile ion effect and KWW relaxation

Ag⁺/Na⁺ ion-exchanged aluminosilicate glasses with uniform concentration profiles were prepared, and their electrical conductivities were investigated as functions of the ion-exchange ratio and the initial glass compositions. In the case of the ion-exchanged glasses of x20Ag₂O–(1−x)20Na₂O–10Al₂O₃–70SiO₂ in mol%, the conductivity, σ, and its activation energy, E_σ, showed a minimum and a maximum at the same ion-exchange ratio x=0.3, respectively, and the mixed mobile ion effect (MMIE) was observed. The fully ion-exchanged sample attained σ=3.5×10⁻⁵ S/cm at 200 °C, which was 1.5 orders of magnitude larger than that of initial glass. In the case of x25Ag₂O–(1−x)25Na₂O–25Al₂O₃–50SiO₂, the mixed mobile ion effect was also observed at x=0.5. The maximum conductivity of 2×10⁻⁴ S/cm at 200 °C was obtained in the fully ion-exchanged glass sample.

The electric relaxation analysis was also conducted on both systems, and Kohlrausch–Williams–Watts (KWW) fractional exponent β was obtained as a function of x. The decrease of β was observed near x≈0.3 in the former system, while that of the later system was independent of the ion-exchange ratio. Based on the structural analysis results, the observed behaviors were investigated from the point of view of the occupation of Ag⁺ ions on the non-bridging oxygen-site (NBO-site) and the charge compensation-site (CC-site) of AlO₄ tetrahedral unit.     

Application of radial basis functions to evolution equations arising in image segmentation

In this paper, radial basis functions are used to obtain the solution of evolution equations which appear in variational level set method based image segmentation. In this method, radial basis functions are used to interpolate the implicit level set function of the evolution equation with a high level of accuracy and smoothness. Then, the original initial value problem is discretized into an interpolation problem. Accordingly, the evolution equation is converted into a set of coupled ordinary differential equations, and a smooth evolution can be retained. Compared with finite difference scheme based level set approaches, the complex and costly re-initialization procedure is unnecessary. Numerical examples are also given to show the efficiency of the method.