Young‐measure solutions to a generalized Benjamin–Bona–Mahony equation

We consider the evolution of microstructure under the dynamics of the generalized Benjamin–Bona–Mahony equation (1) with u: ?² → ?. If we model the initial microstructure by a sequence of spatially faster and faster oscillating classical initial data vⁿ, we obtain a sequence of spatially highly oscillatory classical solutions uⁿ. By considering the Young measures (YMs) ν and µ generated by the sequences vⁿ and uⁿ, respectively, as n → ∞, we derive a macroscopic evolution equation for the YM solution µ, and show exemplarily how such a measure‐valued equation can be exploited in order to obtain classical evolution equations for effective (macroscopic) quantities of the microstructure for suitable initial data vⁿ and non‐linearities f. Copyright © 2005 John Wiley & Sons, Ltd.     

Ritz–Galerkin method for solving a parabolic equation with non‐local and time‐dependent boundary conditions

The paper is devoted to the investigation of a parabolic partial differential equation with non‐local and time‐dependent boundary conditions arising from ductal carcinoma in situ model. Approximation solution of the present problem is implemented by the Ritz–Galerkin method, which is a first attempt at tackling parabolic equation with such non‐classical boundary conditions. In the process of dealing with the difficulty caused by integral term in non‐local boundary condition, we use a trick of introducing the transition function G(x,t) to convert non‐local boundary to another non‐classical boundary, which can be handled with the Ritz–Galerkin method. Illustrative examples are included to demonstrate the validity and applicability of the technique in this paper. Copyright © 2015 John Wiley & Sons, Ltd.     

A New Continuation Method for the Study of Nonlinear Equations at Resonance

A new continuation theorem for the existence of solutions to an equation Lu = N(u), where N is a nonlinear continuous operator and L a linear Fredholm noninvertible one, is proved. The continuation which makes N collapse is replaced by a deformation of L to an invertible linear operator. This implies results concerning sublinear N, N having a linear growth at infinity and superlinear N. These generalize the classical theorems on the solvability of semilinear elliptic BVP′s at resonance. The periodic solutions of Liénard equations are studied.     

On semi‐classical d‐orthogonal polynomials

In this paper a general theory of semi‐classical d‐orthogonal polynomials is developed. We define the semi‐classical linear functionals by means of a distributional equation , where Φ and Ψ are matrix polynomials. Several characterizations for these semi‐classical functionals are given in terms of the corresponding d‐orthogonal polynomials sequence. They involve a quasi‐orthogonality property for their derivatives and some finite‐type relations.     

Séparations et transferts dans la hiérarchie polynomiale des groupes abéliens infinis

We prove some results on the polynomial hierarchies of infinite abelian groups. In particular, we show the polynomial hierarchy over an infinite group of exponent a prime number p collapse if and only if it is the case in the classical theory.     

The Self-Focusing Singularity in the Nonlinear Schrödinger Equation

Under certain circumstances, solutions of the cylindrically symmetric nonlinear Schrödinger equation collapse to a singularity in a finite time. An asymptotic series for the solution near the singularity is derived here. At leading order, the central amplitude of the spike grows like[(log Δt)/Δt]^1/2, where Δt is the time remaining to the appearance of the singularity.     

New exact solutions for the classical Drinfel’d–Sokolov–Wilson equation

In this paper, we investigate the classical Drinfel’d–Sokolov–Wilson equation (DSWE)

where p, q, r, s are some nonzero parameters. Some explicit expressions of solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain solitary wave solutions, blow-up solutions, periodic solutions, periodic blow-up solutions and kink-shaped solutions. Some previous results are extended.     

W2, p estimates of the heat equation in Ω⊂ℝn and the restrictions on ∂Ω

This is an alternative approach of finding the W^{2, p} estimates of the heat equation in a domain, Ω??ⁿ. Methods used in (Acta Math. Sin. 2003; 19 (2):381–396) are expanded to the case of a bounded domain. As a result, milder restrictions are applied to ?Ω than previously required by using the classical singular integral approach. Copyright © 2005 John Wiley & Sons, Ltd.     

On the asymptotic solution of the Graetz‐Nusselt problem for short x → 0 and large x → ∞ with partial usage of finite differences

The principal goal of this article is to present two asymptotic solutions for the classical Graetz‐Nusselt problem. The method of lines (MOL) has been adopted for solving the governing partial differential energy equation in two independent variables in an asymptotic manner. Two temperature subfields are determined semianalytically: one for small x (x → 0) and the other for large x (x → ∞). Later, the two asymptotic mean Nusselt number subdistributions, Nu _X→0(x) and Nu _X→∞(x), blend themselves into a generalized correlation equation for the mean Nusselt number distribution Nu (x) covering the entire x‐domain. The simplicity of the MOL procedure, combined with the high quality asymptotic mean Nusselt number subdistributions, provides an alternative methodology for solving the Graetz‐Nusselt problem without using higher level mathematics. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion

We prove that the Hamilton–Jacobi equation for an arbitrary Hamiltonian H (locally Lipschitz but not necessarily convex) and fractional diffusion of order one (critical) has classical C1,α solutions. The proof is achieved using a new Hölder estimate for solutions of advection–diffusion equations of order one with bounded vector fields that are not necessarily divergence free.     

Analysis of the operator splitting scheme for the Cahn‐Hilliard equation with a viscosity term

In this paper, we consider a second‐order fast explicit operator splitting method for the viscous Cahn‐Hilliard equation, which includes a viscosity term αΔu_t (α ∈ (0, 1)) described the influences of internal micro‐forces. The choice α = 0 corresponds to the classical Cahn‐Hilliard equation whilst the choice α = 1 recovers the nonlocal Allen‐Cahn equation. The fundamental idea of our method is to split the original problem into linear and nonlinear parts. The linear subproblem is numerically solved using a pseudo‐spectral method, and thus an ordinary differential equation is obtained. The nonlinear one is solved via TVD‐RK method. The stability and convergence are discussed in L²‐norm. Numerical experiments are performed to validate the accuracy and efficiency of the proposed method. Besides, a detailed comparison is made for the dynamics and the coarsening process of the metastable pattern for various values of α. Moreover, energy degradation and mass conservation are also verified.     

On closed boundary value problems for equations of mixed elliptic‐hyperbolic type

For partial differential equations of mixed elliptic‐hyperbolic type we prove results on existence and existence with uniqueness of weak solutions for closed boundary value problems of Dirichlet and mixed Dirichlet‐conormal types. Such problems are of interest for applications to transonic flow and are overdetermined for solutions with classical regularity. The method employed consists in variants of the a ? b ? c integral method of Friedrichs in Sobolev spaces with suitable weights. Particular attention is paid to the problem of attaining results with a minimum of restrictions on the boundary geometry and the form of the type change function. In addition, interior regularity results are also given in the important special case of the Tricomi equation. © 2006 Wiley Periodicals, Inc.     

A Γ‐convergence result for the two‐gradient theory of phase transitions

The generalization to gradient vector fields of the classical double‐well, singularly perturbed functionals, where W(ξ) = 0 if and only if ξ = A or ξ = B, and A ? B is a rank‐1 matrix, is considered. Under suitable constitutive and growth hypotheses on W, it is shown that I_ε Γ‐converge to where K^* is the (constant) interfacial energy per unit area. © 2002 Wiley Periodicals, Inc.     

Well‐posedness and asymptotic analysis for a Penrose–Fife type phase field system

In this paper, an asymptotic analysis of the (non‐conserved) Penrose–Fife phase field system for two vanishing time relaxation parameters ε and δ is developed, in analogy with the similar analyses for the phase field model proposed by G. Caginalp (Arch. Rational Mech. Anal. 1986; 92 :205–245), which were carried out by Rossi and Stoth (Adv. Math. Sci. Appl. 2003; 13 :249–271; Quart. Appl. Math. 1995; 53 :695–700). Although formally the singular limits for ε ↓ 0 and for ε and δ ↓ 0 are, respectively, the viscous Cahn–Hilliard equation and the Cahn–Hilliard equation, it turns out that the Penrose–Fife system is indeed a bad approximation for these equations. Therefore, we consider an alternative approximating phase field system, which could be viewed as a generalization of the classical Penrose–Fife phase field system, featuring a double non‐linearity given by two maximal monotone graphs. A well‐posedness result is proved for such a system, and it is shown that the solutions converge to the unique solution of the viscous Cahn–Hilliard equation as ε ↓ 0, and of the Cahn–Hilliard equation as ε ↓ 0 and δ ↓ 0. Copyright © 2004 John Wiley & Sons, Ltd.     

Quantum repulsive Nonlinear Schrödinger models and their ‘Superconductivity’

The fundamental role played by the quantum repulsive Nonlinear Schrödinger (NLS) equation in the evolution of our understanding of the phenomenon of superconductivity in appropriate metals at very low temperatures is surveyed. The first major work was that in 1947 by N. N. Bogoliubov, who studied the very physical 3-space-dimensions problem and super fluidity; and the survey takes the form of an actual dedication to that outstanding scientist who died four years ago. The 3-space-dimensions NLS equation is not integrable either classically or quantum mechanically. But a number of recently discovered closely related lattices in one space dimension (one space plus one time dimension) are integrable as both classical lattices and quantum lattices while their continuum limits are the now well-known fundamental and integrable system the quantum ‘Bose gas’. These models are all examined in this paper in a physical application of recent so-called ‘quantum groups’ theory, itself fundamental to integrability theory. The ‘superfluid’ phase transitions shown by these lattices, as well as by the bose gas, all at zero temperature in 1 + 1 dimensions, are analysed in terms of the behaviour of certain lattice correlation functions which are either quantum or, in the case of the so-called XY-model, classical correlation functions. Although the repulsive NLS models in 1 + 1 are integrable, they do not have actual soliton solutions. Nevertheless the material as surveyed here is a fundamental application of soliton-theory in the broader context of integrability or near-integrability which has had profound effects in the evolution of current understandings in all of modern theoretical physics.     

Absence of collapse in a parabolic chemotaxis system with signal‐dependent sensitivity

We consider the chemotaxis system under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ? ?ⁿ. The chemotactic sensitivity function is assumed to generalize the prototype It is proved that no chemotactic collapse occurs in the sense that for any choice of nonnegative initial data (with some r > n), the corresponding initial‐boundary value problem possesses a unique global solution that is uniformly bounded (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear Elliptical Equations on the Sierpiński Gasket

This paper investigates properties of certain nonlinear PDEs on fractal sets. With an appropriately defined Laplacian, we obtain a number of results on the existence of non-trivial solutions of the semilinear elliptic equation with zero Dirichlet boundary conditions, where u is defined on the Sierpiński gasket. We use the mountain pass theorem and the saddle point theorem to study such equations for different classes of a and f. A strong Sobolev-type inequality leads to properties that contrast with those for classical domains.     

Continuous pseudospectral methods for the neutron diffusion equation in 1D geometries

The P_L equations are classical approximations to the neutron transport equation that admit a diffusive form. The diffusive form of the P₁ approximation is known as the neutron diffusion equation. Different methods based on the expansion of the neutron flux in terms of a continuous basis of polynomials have been developed for the neutron diffusion equation and tested using two 1D benchmark problems.     

Unification of Stieltjes‐Calogero type relations for the zeros of classical orthogonal polynomials

The classical orthogonal polynomials (COPs) satisfy a second‐order differential equation of the form σ(x)y^′′+τ(x)y^′+λy = 0, which is called the equation of hypergeometric type (EHT). It is shown that two numerical methods provide equivalent schemes for the discrete representation of the EHT. Thus, they lead to the same matrix eigenvalue problem. In both cases, explicit closed‐form expressions for the matrix elements have been derived in terms only of the zeros of the COPs. On using the equality of the entries of the resulting matrices in the two discretizations, unified identities related to the zeros of the COPs are then introduced. Hence, most of the formulas in the literature known for the roots of Hermite, Laguerre and Jacobi polynomials are recovered as the particular cases of our more general and unified relationships. Furthermore, we present some novel results that were not reported previously. Copyright © 2014 John Wiley & Sons, Ltd.     

The Fischer decomposition for Hodge‐de Rham systems in Euclidean spaces

The classical Fischer decomposition of spinor‐valued polynomials is a key result on solutions of the Dirac equation in the Euclidean space . As is well‐known, it can be understood as an irreducible decomposition with respect to the so‐called L‐action of the Pin group Pin(m). But, in Clifford algebra valued polynomials, we can consider also the H‐action of Pin(m). In this paper, the corresponding Fischer decomposition for the H‐action is obtained. It turns out that, in this case, basic building blocks are the spaces of homogeneous solutions to the Hodge‐de Rham system. Moreover, it is shown that the Fischer decomposition for the H‐action can be viewed even as a refinement of the classical one. Copyright © 2011 John Wiley & Sons, Ltd.