Comparison of approximate symmetry and approximate homotopy symmetry to the Cahn–Hilliard equation

Complete infinite order approximate symmetry and approximate homotopy symmetry classifications of the Cahn–Hilliard equation are performed and the reductions are constructed by an optimal system of one-dimensional subalgebras. Zero order similarity reduced equations are nonlinear ordinary differential equations while higher order similarity solutions can be obtained by solving linear variable coefficient ordinary differential equations. The relationship between two methods for different order are studied and the results show that the approximate homotopy symmetry method is more effective to control the convergence of series solutions than the approximate symmetry one.     

Gidas-Ni-Nirenberg results for finite difference equations: Estimates of approximate symmetry

Are positive solutions of finite difference boundary value problems Δhu=f(u) in Ωh, u=0 on ∂Ωh as symmetric as the domain? To answer this question we first show by examples that almost arbitrary non-symmetric solutions can be constructed. This is in striking difference to the continuous case, where by the famous Gidas-Ni-Nirenberg theorem [B. Gidas, Wei-Ming Ni, L. Nirenberg, Symmetry and related problems via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243] positive solutions inherit the symmetry of the underlying domain. Then we prove approximate symmetry theorems for solutions on equidistantly meshed n-dimensional cubes: explicit estimates depending on the data are given which show that the solutions become more symmetric as the discretization gets finer. The quality of the estimates depends on whether or not f(0)<0. The one-dimensional case stands out in two ways: the proofs are elementary and the estimates for the defect of symmetry are O(h) compared to O(1/|log(h)|) in the higher-dimensional case.     

A new method to find series solutions of a nonlinear wave equation

We show that first-order approximate symmetries of a class of nonlinear wave equations contain Lie symmetries as particular cases. Then we present a new approach to find series solutions of the nonlinear wave equation which cannot be obtained by the standard Lie symmetry and approximate symmetry methods.     

Optimal system and approximate solutions of nonlinear dissipative media

In this paper, the problem of approximate symmetries of a class of nonlinear wave equations with a small nonlinear dissipation has been investigated. In order to compute the first-order approximate symmetry, we have applied the method that was proposed by Valenti basically based on the expansion of the dependent variables in perturbation series but removing the drawback of the impossibility to work in hierarchy in calculating symmetries. The algebraic structure of the approximate symmetries is discussed, an optimal system of one-dimensional subalgebras is defined and constructed, and, finally, some invariant solutions corresponding to the resulted symmetries are obtained.     

Second-order approximate symmetry classification and optimal system of a class of perturbed nonlinear wave equations

A complete second-order approximate symmetry classification of a class of perturbed nonlinear wave equations with a arbitrary function is performed by means of the method originated from Fushchich and Shtelen. An optimal system of one-dimensional subalgebras is given. Based on the Lie symmetry reduction method, large classes of approximate reductions and invariant solutions of the equations are constructed.     

A NEW APPROACH TO SOLVE PERTURBED NONLINEAR EVOLUTION EQUATIONS THROUGH LIE-BACKLUND SYMMETRY METHOD

A new method based on Lie-Backlund symmetry method to solve the perturbed nonlinear evolution equations is presented. New approximate solutions of perturbed nonlinear evolution equations stemming from the exact solutions of unperturbed equations are obtained. This method is a generalization of Burde＇s Lie point symmetry technique.     

An approximate analytical solution of time-fractional telegraph equation

In this article, the powerful, easy-to-use and effective approximate analytical mathematical tool like homotopy analysis method (HAM) is used to solve the telegraph equation with fractional time derivative α (1 < α ? 2). By using initial values, the explicit solutions of telegraph equation for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The method performs extremely well in terms of efficiency and simplicity to solve this historical model.     

Comparison of symmetry and homotopy solutions for nonlinear heat transfer systems

This study deals with the analytic solutions for the two nonlinear problems arising in heat transfer. These problems are due to (i) temperature distribution in lumped system of combined convection–radiation and (ii) temperature distribution in a uniformly thick rectangular fin radiation to free space. Large symmetry algebras are obtained for the nonlinear ordinary differential equations (ODEs) describing the heat transfer. We use method of canonical variables to either linearize or transform the governing equations to integrable forms. Exact solutions are constructed. Finally, a comparison is given between the homotopy and symmetry solutions.     

Conservation laws and exact solutions of the Whitham-type equations

In this paper, we study conservation laws for some partial differential equations. It is shown that interesting conserved quantities arise from multipliers by using homotopy operator that is a powerful algorithmic tool. Furthermore, the invariance properties of the conserved flows with respect to the Lie point symmetry generators are investigated via the symmetry action on the multipliers. Furthermore, the similarity reductions and some exact solutions are provided.     

Mixed finite element methods for linear elasticity with weakly imposed symmetry

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approximations to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

     

Estimates of approximate solutions and well-posedness for variational inequalities

The purpose of this paper is to estimate the approximate solutions for variational inequalities. In terms of estimate functions, we establish some estimates of the sizes of the approximate solutions from outside and inside respectively. By considering the behaviors of estimate functions, we give some characterizations of the well-posedness for variational inequalities. This work was partially supported by the Basic and Applied Research Projection of Sichuan Province (05JY029-009-1) and the National Natural Science Foundation of China (10671135).     

Exact solutions using symmetry methods and conservation laws for the viscous flow through expanding–contracting channels

We present a range of definitions and methods dealing with the reduction of partial differential equations on the basis of the underlying symmetry structure, conservation laws and a combination of these. The method is used to reduce a complex system to an easy-to-handle second-order ordinary differential equation system independent of restrictions on any physical parameters. In particular, we construct exact solutions of a system modelling viscous flow between slowly expanding and contracting walls.     

Analytic approximate solutions of parameterized unperturbed and singularly perturbed boundary value problems

A novel approach is presented in this paper for approximate solution of parameterized unperturbed and singularly perturbed two-point boundary value problems. The problem is first separated into a simultaneous system regarding the unknown function and the parameter, and then a methodology based on the powerful homotopy analysis technique is proposed for the approximate analytic series solutions, whose convergence is guaranteed by optimally chosen convergence control parameters via square residual error. A convergence theorem is also provided. Several nonlinear problems are treated to validate the applicability, efficiency and accuracy of the method. Vicinity of the boundary layer is shown to be adequately treated and satisfactorily resolved by the method. Advantages of the method over the recently proposed conventional finite-difference or Runga–Kutta methods are also discussed.     

Comparison of two reliable analytical methods based on the solutions of fractional coupled Klein–Gordon–Zakharov equations in plasma physics

In this paper, homotopy perturbation transform method and modified homotopy analysis method have been applied to obtain the approximate solutions of the time fractional coupled Klein–Gordon–Zakharov equations. We consider fractional coupled Klein–Gordon–Zakharov equation with appropriate initial values using homotopy perturbation transform method and modified homotopy analysis method. Here we obtain the solution of fractional coupled Klein–Gordon–Zakharov equation, which is obtained by replacing the time derivatives with a fractional derivatives of order α ∈ (1, 2], β ∈ (1, 2]. Through error analysis and numerical simulation, we have compared approximate solutions obtained by two present methods homotopy perturbation transform method and modified homotopy analysis method. The fractional derivatives here are described in Caputo sense.     

On symmetry properties of parabolic equations in bounded domains

We study symmetry properties of nonnegative bounded solutions of fully nonlinear parabolic equations on bounded domains with Dirichlet boundary conditions. We propose sufficient conditions on the equation and domain, which guarantee asymptotic symmetry of solutions.     

Nonclassical symmetry solutions for reaction–diffusion equations with explicit spatial dependence

Nonclassical symmetry methods are used to study the linear diffusion equation with a nonlinear source term which includes explicit spatial dependence. Mathematical forms for the spatial dependence are found which enable strictly nonclassical symmetries to be admitted when the nonlinearity is cubic. A number of new exact solutions are constructed, and an application of one of these solutions to diploid population genetics is discussed.     

Spin parity effect resulting from time-inversion symmetry

The spin parity effect resulting from time-inversion symmetry is derived by generalizing the pure quantum theory of the effect resulting from finite-fold axial-rotation symmetry. The quantum system concerned may consist of either one spin or many spins with arbitrary spin quantum numbers and the states involved may be any pair of eigenstates of spin operator projected on arbitrary axis with their eigenvalues having opposite signs. The two kinds of spin parity effect, resulting from time-inversion symmetry and from axial-rotation symmetry, are in general not equivalent but complementary to each other. Project supported by the National Natural Science Foundation of China (Grant No. 19677101).     

Characterizing properties of approximate solutions for optimization problems

Approximate solutions for optimization problems become of interest if the ‘true’ optimum cannot be found: this may happen for the simple reason that an optimum does not exist or because of the ‘bounded rationality’ (or bounded accuracy) of the optimizer. This paper characterizes several approximate solutions by means of consistency and additional requirements. In particular we consider invariance properties. We prove that, where the domain contains optimization problems without maximum, there is no non-trivial consistent solution satisfying non-emptiness, translation and multiplication invariance. Moreover, we show that the class of ‘satisficing’ solutions is obtained, if the invariance axioms are replaced with Chernoff’s Choice Axiom.     

Modulation spaces with scaling symmetry

We indicate how to construct a family of modulation spaces that have a scaling symmetry. We also illustrate the behavior of the Schrödinger multiplier on such function spaces.