Let $\varOmega$ be a bounded, simply connected domain in $\mathbb{C}$ with $0\in\varOmega$ and $\partial\varOmega$ analytic. Let $S(\varOmega)$ denote the class of functions $F(z)$ which are analytic and univalent in $\varOmega$ with $F(0)=0$ and $F'(0)=1$. Let $\{\varPhi_{n}(z)\}_{n=0}^{\infty}$ be the Faber polynomials associated with $\varOmega$. If $F(z)\in S(\varOmega)$, then $F(z)$ can be expanded in a series of the form

$$ F(z)=\sum_{n=0}^{\infty}A_{n}\varPhi_{n}(z),\quad z\in\varOmega, $$

in terms of the Faber polynomials. Let

$$ E_{r}=\bigg\{(x,y)\in\mathbb{R}^{2}:\frac{x^{2}}{(1+(1/r^{2}))^{2}}+\frac{y^{2}}{(1-(1/r^{2}))^{2}}\lt1\bigg\}, $$

where $r\gt1$.

In this paper, we obtain sharp bounds for certain linear combinations of the Faber coefficients of functions $F(z)$ in $S(E_{r})$ and in certain related classes.

We study surface tension effects for two-dimensional Darcy flow with a free boundary in a corner between two non-parallel walls. The analytic solution is based on two governing expressions constructed in an auxiliary parameter domain, namely a complex velocity and a derivative of the complex potential. These expressions admit a general solution for the problem in a corner geometry for the flow generated by a source/sink at the corner vertex or at infinity. We derive an integral equation in terms of the velocity modulus and angle at the free surface, determined by the dynamic boundary condition. A numerical procedure, used to solve the obtained system of equations, and numerical results concerning the effect of surface tension on the time evolution of the free boundary, are discussed.

The propagation of a solitary wave in a horizontal fluid layer is studied. There is an interfacial free surface above and below this intrusion layer, which is moving at constant speed through a stationary density-stratified fluid system. A weakly nonlinear asymptotic theory is presented, leading to a Korteweg–de Vries equation in which the two fluid interfaces move oppositely. The intrusion layer solitary wave system thus forms a widening bulge that propagates without change of form. These results are confirmed and extended by a fully nonlinear solution, in which a boundary-integral formulation is used to solve the problem numerically. Limiting profiles are approached, for which a corner forms at the crest of the solitary wave, on one or both of the interfaces.

For the Cauchy problem for the nonlinear infiltration equation $$\left\{\begin{array}{@{}l@{\qquad}l} u_{t}=\frac{1}{m}(u^{m})_{xx},&x\in{\mathbb{R}}, t>0,m\geq{}1,\\[3pt] u|_{t=0}=u_{0}(x),&x\in{\mathbb{R}}, \end{array} \right.$$ we use its linear solution $u(x,t,1)$ to approach the nonlinear solution $u(x,t,m)$, and obtain the explicit estimate: $$\int_{0}^{T}\int_{\mathbb{R}}|u(x,t,m)-u(x,t,1)|^{2}\,dx\,dt{} \leq{}(C^{\ast}(m-1))^{2},$$ where $C^{\ast}=O(T^{\gamma})$ and $\gamma=\frac{1+m-\alpha}{2(1+m)}$ for any $0<\alpha<1$.

Sufficient conditions are obtained for the existence of positive periodic solutions of a class of neutral delay differential equations of the form \begin{equation*} \left\{\begin{array}{@{}l@{}} {\normalsize N}^{\prime}{\normalsize (t)=N(t)F[t,N(t),N(t-\tau (t,N(t))),N}^{\prime}{\normalsize (t-\gamma (t)),P(t),P(t-\mu (t))]}\\ {\normalsize P}^{\prime}{\normalsize (t)=-}e(t)P{\normalsize (t)+k(t)N(t)+h(t)N(t-\sigma (t))}\end{array}\right. \end{equation*} by using the theory of topological degree. These results extend substantially the existing relevant existence results in the literature. As a demonstration, applying the obtained analysis results to a real complex neutral Lotka-Volterra population model, the existence criterion for positive periodic solutions is easily obtained and an example is used to give an impression of how restrictive these conditions are. Especially, this method is more suitable to state-dependent delay, and which have further applications in many fields.

In this paper, three iterative procedures (Landweber-Fridman, conjugate gradient and minimal error methods) for obtaining a stable solution to the Cauchy problem in slow viscous flows are presented and compared. A section is devoted to the numerical investigations of these algorithms. There, we use the boundary element method together with efficient stopping criteria for ceasing the iteration process in order to obtain stable solutions.

We study a generalized class of nonlocal evolution equations which includes those arising in the modelling of electrified film flow down an inclined plane, with applications in enhanced heat or mass transfer through interfacial turbulence. Global existence and uniqueness results are proved and refined estimates of the radius of the absorbing ball in $L^2$ are obtained in terms of the parameters of the equations (the length of the system and the dimensionless electric field-measuring parameter multiplying the nonlocal term). The established estimates are compared with numerical solutions of the equations which in turn suggest an optimal upper bound for the radius of the absorbing ball. A scaling argument is used to explain this and a general conjecture is made based on extensive computations.

We study the right-definite separated half-linear Sturm–Liouville eigenvalue problems. It is proved that the $n$th real eigenvalue of the problem depends smoothly on the equation, but may have jump discontinuities with respect to the boundary condition. Formulae are found for the derivatives of the $n$th real eigenvalue with respect to all parameters: the endpoints, the boundary condition and the coefficient functions, whenever they exist. Monotone properties and a comparison result for real eigenvalues are deduced as consequences. The generalized Prüfer transformation and the implicit function theorem in Banach spaces play key roles in the proofs.

Recent studies of lattice dynamics, with respect to the existence of global compact attractors for certain discrete evolution equations, are based on the derivation of ‘tail estimates of the solution’. We show that, due to the specific nature of the discrete nonlinear Schrödinger (DNLS) system which gives rise to a simple energy equation, the method developed by J. M. Ball is applicable, and provides an alternative proof on the existence of the compactness of the attractor for the weakly damped and driven DNLS equation considered in $\mathbb{Z}^N$, $N\geq1$, lattices, without the usage of tail estimates. The approach covers various DNLS-type equations of physical significance.

Generalized two-phase fluid flows in a Hele-Shaw cell are considered. It is assumed that the flow is driven by the fluid pressure gradient and an external potential field, for example, an electric field. Both the pressure field and the external field may have singularities in the flow domain. Therefore, combined action of these two fields brings into existence some new features, such as non-trivial equilibrium shapes of boundaries between the two fluids, which can be studied analytically. Some examples are presented. It is argued, that the approach and results may find some applications in the theory of fluids flow through porous media and microfluidic devices controlled by electric field.

This paper deals with a problem of nondestructive testing for a composite system formed by the connection of a steel beam and a reinforced concrete beam. The small vibrations of the composite beam are described by a differential system where a coupling takes place between longitudinal and bending motions. The motion is governed in space by two second order and two fourth order differential operators, which are coupled in the lower order terms by the shearing, k, and axial, μ, stiffness coefficients of the connection. The coefficients k and μ define the mechanical model of the connection between the steel beam and the concrete beam and contain direct information on the integrity of the system. In this paper we study the inverse problem of determining k and μ by mixed data. The inverse problem is transformed to a variational problem for a cost function which includes boundary measurements of Neumann data and also some interior measurements. By computing the Gateaux derivatives of the functional, an algorithm based on the projected gradient method is proposed for identifying the unknown coefficients. The results of some numerical simulations on real steel-concrete beams are presented and discussed.

It is proved that a periodically forced second-order equation with a singular nonlinearity in the origin with linear growth in infinity possesses a $T$-periodic stable solution for high values of the mean value of the forcing term. The method of proof combines a rescaling argument with the analysis of the first twist coefficient of the Birkhoff normal form for the Poincaré map.

In this paper we find an explicit moving frame along curves of Lagrangian planes invariant under the action of the symplectic group. We use the moving frame to find a family of independent and generating differential invariants. We then construct geometric Hamiltonian structures in the space of differential invariants and prove that, if we restrict them to a certain Poisson submanifold, they become a set of decoupled Korteweg–de Vries (KdV) first and second Hamiltonian structures. We find an evolution of curves of Lagrangian planes that induces a system of decoupled KdV equations on their differential invariants (we call it the Lagrangian Schwarzian KdV equation). We also show that a generalized Miura transformation takes this system to a modified matrix KdV equation. In the four-dimensional case we show that there are no unrestricted compatible geometric pairs.

In certain rings containing non-central idempotents we characterize homomorphisms, derivations, and multipliers by their actions on elements satisfying some special conditions. For example, we consider the condition that an additive map $h$ between rings $\mathcal{A}$ and $\mathcal{B}$ satisfies $h(x)h(y)h(z)=0$ whenever $x,y,z\in\mathcal{A}$ are such that $xy=yz=0$. As an application, we obtain some new results on local derivations and local multipliers. In particular, we prove that if $\mathcal{A}$ is a prime ring containing a non-trivial idempotent, then every local derivation from $\mathcal{A}$ into itself is a derivation.

In this paper, we first investigate the classification of positively homogeneous equations $(\phi_p(u'))'+q(t)\phi_p(u)=0$, $u(0)=0=u(1)$, where $p>1$ is fixed, $\phi_p(u)=|u|^{p-2}u$ and $q\in L^{\infty}(0,1)$, and then discuss the existence of solutions for non-homogeneous equations. The main method of classification is by using a generalized Prufer equation

$$\theta'=|\4\cos_p\theta|^p+\frac{q(t)}{p-1}|\4\sin_p\theta|^p\quad\text{for }t\in(0,1),$$

where $\sin_p:\mathbb{R}\to[-1,1]$ is a periodic function and $\cos_pt=\mathrm{d}\sin_pt/\mathrm{d} t$ for $t\in\mathbb{R}$.

In this work we consider the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Navier-Stokes equations. It is assumed that friction forces are known on a part of the outer boundary. We first prove a uniqueness result. Then, we establish a formula for the observed friction forces, at first order, in terms of the deformation of the rigid body. In some particular situations, this provides a strategy that could be used to compute approximations to the solution of the inverse problem. In the proofs we use unique continuation and regularity results for the Navier-Stokes equations and domain variation techniques.

This paper deals with singular semilinear elliptic equations in bounded domains with Dirichlet boundary data. The elliptic operator is a second-order operator not necessarily in divergence form. We consider existence, uniqueness and linearized stability of positive solutions for a series of nonlinear eigenvalue problems.

The competition between inertia and solidification for the high-Reynolds-number flow of molten aluminium across a cool solid aluminium surface is investigated. A two-dimensional molten aluminium droplet is of finite extent and is surrounded by a passive gas. The droplet initially freezes due to rapid thermal conduction into the solid. Depending on the initial velocity of the molten aluminium, one of two situations may develop: (i) If the molten aluminium has a non-decreasing initial velocity profile, solidification continues until the passing of the trailing edge of the liquid/gas interface or the flow is engulfed; (ii) If the molten aluminium has a decreasing initial velocity profile, the droplet narrows and thickens resulting in a reduction in the heat flux and in the rate of solidification; this will eventually lead to fluid clumping and shock formation. The rate of solidification may also be reduced by increasing the ambient temperature. The results are interpreted in terms of the recast observed during the solidification phase of laser percussion drilling.

The main theorem states that a bounded linear operator $h$ from a unital $C^{\ast}$-algebra $A$ into a unital Banach algebra $B$ must be a homomorphism provided that $h(\bm{1})=\bm{1}$ and the following condition holds: if $x,y,z\in A$ are such that $xy=yz=0$, then $h(x)h(y)h(z)=0$. This theorem covers various known results; in particular it yields Johnson's theorem on local derivations.

We study homogenization processes for the heat equation in multilayers with interlayer conduction, modelled by Neumann transmission conditions. We establish the homogenized equations for three kinds of dependency of the interlayer conduction magnitude upon the interlayer distance.