Relations between symmetries and conservation laws for difference systems

This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.     

Numerical solutions of distributed order fractional differential equations in the time domain using the Müntz–Legendre wavelets approach

In this paper, a numerical method is presented to obtain and analyze the behavior of numerical solutions of distributed order fractional differential equations of the general form in the time domain with the Caputo fractional derivative. The suggested method is based on the Müntz–Legendre wavelet approximation. We derive a new operational vector for the Riemann–Liouville fractional integral of the Müntz–Legendre wavelets by using the Laplace transform method. Applying this operational vector and collocation method in our approach, the problem can be reduced to a system of linear and nonlinear algebraic equations. The arising system can be solved by the Newton method. Discussion on the error bound and convergence analysis for the proposed method is presented. Finally, seven test problems are considered to compare our results with other well‐known methods used for solving these problems. The results in the tabulated tables highlighted that the proposed method is an efficient mathematical tool for analyzing distributed order fractional differential equations of the general form.     

Conservation laws for the Black–Scholes equation

In the search for solutions to the important partial differential equation due to Black, Scholes and Merton potential symmetries are very useful as new solutions of the equation can be obtained as a result. These potential symmetries require that the equation be written in conserved form, ie. we need to determine conservation laws for the equation. We calculate the conservation laws utilizing the point symmetries of the equation following the method of Kara and Mahomed [A.H. Kara, F.M. Mahomed, The relationship between symmetries and conservation laws, Int. J. Theor. Phys. 39 (2000) 23–40].     

Möbius transformation and a version of Schwarz lemma in octonionic analysis

The octonion is a generalization of complex to noncommutative and nonassociative space which has closed relation with exception geometries, wave equation, Yang‐Mills equations, black hole, string theory, and special relativity. In this paper, the Möbius transformation in this manner is first introduced, and some properties are discussed about the transformation in octonionic analysis. Some technique lemmas will be given to solve the problems caused by the weak form of associativity. These versions of Schwarz lemma and Schwarz‐Pick lemma are first studied in octonionic setting which will invoke integral representation formula for harmonic function and Möbius transformations. This will generalize the corresponding results which appear in the classical function theory to nonassociative space and may give new energy for the development of physics.     

Green's function-based integral approaches to nonlinear transient boundary-value problems (II)

Two Green's function-based numerical formulations are used to solve the time-dependent nonlinear heat conduction (diffusion) equation. These formulations, which are an extension of the first paper, utilize two fundamental solutions and the Green's second identity to achieve integral replications of the governing partial differential equation. The integral equations thus derived are discretized in space and time and aggregated in a finite element sense to give a system of nonlinear discrete equations that are solved by the Newton–Raphson algorithm. The mathematical simplicity of the Green's function of the first formulation facilitates its numerical implementation. The performance of the formulations is assessed by comparing their results with available numerical and analytical solutions. In all cases satisfactory and physically realistic results are obtained.     

On approximations of the Euler-Peano scheme for multivalued stochastic differential equations

It is known that a unique strong solution exists for multivalued stochastic differential equations under the Lipschitz continuity and linear growth conditions. In this paper we apply the Euler-Peano scheme to show that existence of weak solution and pathwise uniqueness still hold when the coefficients are random and satisfy one-sided locally Lipschitz continuous and an integral condition (i.e. Krylov's conditions put forward in On Kolmogorov's equations for finite-dimensional diffusions, Stochastic PDE's and Kolmogorov Equations in Infinite Dimensions (Cetraro, 1998), Lecture Notes in Math., 1715, Springer, Berlin, 1999, pp. 1–63). When the coefficients are nonrandom and possibly discontinuous but only satisfy some integral conditions, the sequence of solutions of the Euler-Peano scheme converges weakly, and the limit is a weak solution of the corresponding MSDE. As a particular case, we obtain a global semi-flow for stochastic differential equations reflected in closed, convex domains.     

Some recent methods for partial differential equations of divergence form

Some recent methods for solving second-order nonlinear partial differential equations of divergence form and related nonlinear problems are surveyed. These methods include entropy methods via the theory of divergence-measure fields for hyperbolic conservation laws, kinetic methods via kinetic formulations for degenerate parabolichyperbolic equations, and free-boundary methods via free-boundary iterations for multidimensional transonic shocks for nonlinear equation of mixed elliptic-hyperbolic type. Some recent trends in this direction are also discussed.Dedicated to IMPA on the occasion of its 50^th anniversary     

A conservative formulation for plasticity

In this paper we propose a fully conservative form for the continuum equations governing rate-dependent and rate-independent plastic flow in metals. The conservation laws are valid for discontinuous as well as smooth solutions. In the rate-dependent case, the evolution equations are in divergence form, with the plastic strain being passively convected and augmented by source terms. In the rate-independent case, the conservation laws involve a Lagrange multiplier that is determined by a set of constraints; we show that Riemann problems for this system admit scale-invariant solutions.     

Appell’s lemma and conservation laws of KdV equation

An alternative method to construct a class of conservation laws of the KdV equation based on the classical Appell’s lemma and the trace formulas of Deift-Trubowitz type is studied. A new type of infinite sequence of conservation laws whose local densities cannot be expressed in terms of differential polynomials is constructed.     

New exact solutions of a generalised Boussinesq equation with damping term and a system of variant Boussinesq equations via double reduction theory

The conservation laws of a generalised Boussinesq (GB) equation with damping term are derived via the partial Noether approach. The derived conserved vectors are adjusted to satisfy the divergence condition. We use the definition of the association of symmetries of partial differential equations with conservation laws and the relationship between symmetries and conservation laws to find a double reduction of the equation. As a result, several new exact solutions are obtained. A similar analysis is performed for a system of variant Boussinesq (VB) equations.     

Symmetries,Conservation Laws,and Noether's Theorem for Differential‐Difference Equations

This paper mainly contributes to the extension of Noether's theorem to differential‐difference equations. For this purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation possible. The relations of symmetries, conservation laws, and the Fréchet derivative are also investigated. For nonvariational equations, because Noether's theorem is now available, the self‐adjointness method is adapted to the computation of conservation laws for differential‐difference equations. Several differential‐difference equations are investigated as illustrative examples, including the Toda lattice and semidiscretizations of the Korteweg–de Vries (KdV) equation. In particular, the Volterra equation is taken as a running example.     

Microlocal defect measures

In order to study weak continuity of quadratic forms on spaces of L² solutions of systems of partial differential equations, we define defect measures on the space of positions and frequencies.A systematic use of these measures leads in particular to a compensated compactness theorem, generalizing MURAT"TARTAR's compensated compactness to variable coefficients and GOLSE"LIONS"PERTHAME"SENTIS's averaging lemma. We also obtain results on homogenization for differential operators of order I with oscillating coefficients.     

扩充偏微分方程(组)守恒律和对称的辅助方程方法及微分形式吴方法的应用

首先,我们给出了引入伴随方程(组)扩充原方程(组)的策略使给定偏微分方程(组)的扩充方程组具有对应泛瓯即,成为Lagrange系统的方法,以此为基础提出了作为偏微分方程(组)传统守恒律和对称概念的一种推广-偏微分方程(组)扩充守恒律和扩充对称的概念;其次,以得到的Lagrange系统为基础给定了确定原方程(组)扩充守恒律和扩充对称的方法,从而达到扩充给定偏微分方程(组)的首恒律和对称的目的;第三,提出了适用于一般形式微分方程(组)的计算固有守恒律的方法;第四,实现以上算法过程中,我们先把计算(扩充)守恒律和对称问题均归结为求解超定线性齐次偏微分方程组(确定方程组)的问题.然后,对此关键问题我们提出了用微分形式吴方法处理的有效算法;最后,作为方法的应用我们计算确定了非线性电报方程组在内的五个发展方程(组)的新守恒律和对称,同时也说明了方法的有效性.     

Sturm‐Picone comparison theorems for nonlinear impulsive differential equations with discontinuous solutions

The nonlinear versions of Sturm‐Picone comparison theorem as well as Leighton's variational lemma and Leighton's theorem for regular and singular nonlinear impulsive differential equations with discontinuous solutions having fixed moments of impulse actions are established. Although discontinuity of the solutions causes some difficulties, these new comparison theorems cover the old ones where impulse effects are dropped.     

Weak convergence of infinite-dimensional diffusions 1

Continuous dependence - in the sense of weak convergence of laws - of martingale solutions to stochastic partial differential equations on coefficients is studied, the results obtained being applicable to equations with rapidly oscillating coefficients. In the proofs, Gatarek's and Goldys’ recent approach to martingale solutions is substantially used     

Weak Convergence of Laws on ℝ<Superscript><Emphasis Type="Italic">K</Emphasis></Superscript> with Common Marginals

We present a result on topologically equivalent integral metrics as reported by Rachev (Probability Metrics and the Stability of Stochastic Models, Wiley, Chichester, 1991) and Müller (J. Appl. Probab. 29, 429–443, 1997) that metrize weak convergence of laws with common marginals. This result is relevant for applications, as shown in a few simple examples.     

Variational Characterizations of Weak Solutions to the Dirichlet Problem for Mixed‐Type Equations

For linear and nonlinear second‐order partial differential equations of mixed elliptic‐hyperbolic type, we prove that weak solutions to the Dirichlet problem are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functional restricted to suitable infinite‐dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights associated to the linear operator. This spectral theory has been recently developed by the authors, which in turn exploits weak well‐posedness results obtained by Morawetz and the authors. © 2015 Wiley Periodicals, Inc.     

Non-classical laws with conserved quantities from the arbitrary functions included in general solution to elasticity and application

The general solution to static and/or dynamic linear elasticity is a transformation between the displacements and new arbitrary functions, whose conservativeness depends on some independent partial differential equations (PDEs) satisfied by the new arbitrary functions. Zhang's general solutions are mathematically appropriate since the displacements are expressed in terms of two new arbitrary functions, and the sum of the highest order derivative added together from the independent PDEs satisfied by the two new arbitrary functions is the same as that of Navier–Cauchy equations. Therefore, the following points should be emphasized: (i) the independent PDEs come from the Laplace and D'Alembert operators acting on the two new arbitrary functions in static and dynamic general solutions, respectively, and it is found that the two new arbitrary functions are related to the rotations, first strain invariant and distortion; (ii) especially, conservation laws constructed from the equations satisfied by the spatial integrals of functions hold true, although some arbitrary functions of the spatial integrals have been canceled. Based on these facts, since Noether's identity not only can be applied to a Lagrangian but also can be used to construct a functional for widespread PDEs, the functionals relating to the rotations, first strain invariant and distortion are constructed with arbitrary integer order spatial derivative or integral, and the conservation laws follow. This kind of non-classical conservation laws does not come from the Lagrangian density of an elastic body and belongs to the deep-level natures of symmetries of elastic field derived by standard techniques. Availability is shown by two examples, from which the field intensity of a vertical load applied to the surface of an elastic half-space and the path-independent integrals in a coordinate system moving with Galilean transformation are presented for comparison.     

Kinetic decomposition for singularly perturbed higher order partial differential equations

In this paper, we consider singularly perturbed higher order partial differential equations. We establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in Hwang and Tzavaras (Comm. Partial Differential Equations 27 (2002) 1229). First, we obtain the approximate transport equation for the given dispersive equations. Then using the averaging lemma, we obtain the convergence.     

Interpreting weak Kőnig's lemma in theories of nonstandard arithmetic

We show how to interpret weak Kőnig's lemma in some recently defined theories of nonstandard arithmetic in all finite types. Two types of interpretations are described, with very different verifications. The celebrated conservation result of Friedman's about weak Kőnig's lemma can be proved using these interpretations. We also address some issues concerning the collecting of witnesses in herbrandized functional interpretations.