IMA Journal of Applied Mathematics - recent issues

A Papkovich-Neuber-based numerical approach to cracks with contact in 3D

Hintermuller, M., Kovtunenko, V. A., Kunisch, K. — 2009-05-26

The mathematical model of a crack with non-penetration conditions is considered in the framework of 3D elasticity. The spatial crack problem is investigated with respect to its numerical realization in the context of constrained optimization. Specifically, for homogeneous isotropic solids with planar cracks, a Papkovich–Neuber-based representation is adopted. It allows to employ a primal–dual active set strategy with an unconditional global and monotone convergence property. The iterates turn out to be primally feasible. Illustrative numerical examples are presented.

Energy decay in a transmission problem in thermoelasticity of type III

Messaoudi, S. A., Said-Houari, B. — 2009-05-26

In this paper, we consider a 1D linear thermoelastic transmission problem, where the heat conduction is described by the theories of Green and Naghdi. By using the energy method, we prove that the thermal effect is strong enough to produce an exponential stability of the solution, no matter how small the action domain is.

Stability of equilibria of some switched non-linear systems with applications to control synthesis for electrohydraulic servomechanisms

Halanay, A., Ursu, I. — 2009-05-26

Stability of the zero solution is analyzed for a family of switched systems indexed by a parameter, each system having = 0 in the spectrum of the Jacobian matrix calculated in zero. It is proved that existence of a common quadratic Lyapunov function for some lower dimensional linear systems is sufficient to ensure local uniform stability of the zero solution of the switched non-linear system and a regular asymptotic behaviour. An application to control synthesis for stabilizing equilibria in a switched non-linear control system modelling an electrohydraulic servomechanism is given.

Global solution for a quasi-linear plate system with boundary memory damping

Zhang, Q. — 2009-05-26

In this work, we consider a quasi-linear plate model with boundary memory damping. We prove that this system has a unique global solution when the initial data are small enough and the non-linear coefficient function, the memory damping as well as the geometry of the domain satisfy suitable assumptions. We also prove the exponential decay of the energy of the system.

Non-existence of global solutions of a class of coupled non-linear Klein-Gordon equations with non-negative potentials and arbitrary initial energy

Wang, Y. — 2009-05-26

In the paper, we consider the non-existence of global solutions of Cauchy problem for coupled Klein–Gordon equations of the form on R x Rⁿ. First, for the case n = 2, 3, we prove the existence of ground state of the corresponding Lagrange–Euler equations of the above equations. Then, we establish a blow-up result with low initial energy, which leads to instability of standing waves of the system above. Moreover, as a byproduct we also discuss the global existence. Next, based on concavity method, we prove the blow-up result for the system with non-positive initial energy in the general case: 1 <= n < 6. Finally, when the initial energy is given arbitrarily positive, we show that if the initial datum satisfies some conditions, the corresponding solution blows up in a finite time. In other words, in this paper we establish the complete blow-up result for the Klein–Gordon equation above in the sense of the initial energy, – < E(0) < + .

An extended notion of enthalpy. Electromagnetic solids

Trimarco, C. — 2009-05-26

Some procedures are here expounded in order to introduce the physical and the material or configurational stress in deformable and moving solids. In electromagnetic materials, the electromagnetic fields are preliminarily introduced in the Lagrangian form. Then a variational approach is proposed through two different procedures. Two energy–momentum tensors are excerpted from the suggested procedures. One of them corresponds to a Cauchy-like stress and the other one to the material stress. The latter rules the balance law for the material momentum. This balance law across a first-order discontinuity surface addresses an extended notion of enthalpy and also an extension of the Maxwell condition in thermodynamical phase transitions of electromagnetic solids.

The Kelvin transformation in potential theory and Stokes flow

Dassios, G. — 2009-05-26

Kelvin's transformation is a non-linear map that, in some sense, preserves harmonicity. This property, which was the content of a letter sent by Kelvin to Liouville in 1845, provides a powerful machinery for solving particular potential problems in a very effective way. In the present work, we show that the basic theory can be extended to the biharmonic equation as well to the equations for irrotational and rotational Stokes flow. Hence, biharmonicity, stream functions and bistream functions are also preserved, in some sense, under the Kelvin transformation. We also demonstrate how the Kelvin-type theorems are interconnected with the relative Almansi-type decompositions. These results provide a way to solve analytically many problems in potential theory and Stokes flow which it is impossible to solve by the classical spectral method.

Micro/nanoparticle melting with spherical symmetry and surface tension

McCue, S. W., Wu, B., Hill, J. M. — 2009-05-26

The process of melting a small spherical particle is treated by setting up a two-phase Stefan problem. Surface tension is included through the Gibbs–Thomson condition, the effect of which is to decrease the melting temperature as the particle radius decreases. Analytical results are derived via a small-time expansion and also through large Stefan number asymptotics. Numerical solutions are computed with a front-fixing scheme, and these results suggest that the model exhibits finite-time blow-up, in the sense that both the interface speed and the temperature gradient in the solid phase (at the interface) will become unbounded at some time before complete melting. The near-blow-up behaviour appears to be similar to that encountered in the ill-posed problem of melting a superheated solid (without surface tension), and may help explain the onset of abrupt melting observed in some experiments with nanoscaled particles.

Stability of sticky particle dynamics and related scalar conservation laws

Moutsinga, O. — 2009-05-26

We show the stability of the sticky particle forward flow (x, s, t) ↦ (x, s, P_t, u_t) w.r.t. perturbations of the initial mass distribution P₀ and velocity function u₀. Then, we deduce the stability of related scalar conservation laws and pressureless gas system.

Asymptotic behaviour of ground state solutions for the Henon equation

Cao, D., Peng, S., Yan, S. — 2009-05-26

Let B₁(0) R^N be the unit ball centred at the origin, N ≥ 3. In this paper, we analyse the profile of the ground state solution of the Hénon equation – u = |x|u^p^–1 in B₁(0), u = 0 onB₁(0). We prove that for fixed p (2, 2^*), (2^* = 2N/(N – 2)), the maximum point x of the ground state solution u satisfies (1 – |x|) -> l (0, +) as -> . We also obtain the asymptotic behaviour of u, which shows that the ground state solution is non-radial. Moreover, we prove the existence of multi-peaked solutions and give their asymptotic behaviour.

A new study of the Burton and Miller method for the solution of a 3D Helmholtz problem

Chen, K., Cheng, J., Harris, P. J. — 2009-03-26

The exterior Helmholtz problem can be efficiently solved by reformulating the differential equation as an integral equation over the surface of the radiating and/or scattering object. One popular approach for overcoming either non-unique or non-existent problems which occur at certain values of the wave number is the so-called Burton and Miller method which modifies the usual integral equation into one which can be shown to have a unique solution for all real and positive wave numbers. This formulation contains an integral operator with a hypersingular kernel function and for many years, a commonly used method for overcoming this hypersingularity problem has been the collocation method with piecewise-constant polynomials. Viable high-order methods only exist for the more expensive Galerkin method. This paper proposes a new reformulation of the Burton–Miller approach and enables the more practical collocation method to be applied with any high-order piecewise polynomials. This work is expected to lead to much progress in subsequent development of fast solvers. Numerical experiments on 3D domains are included to support the proposed high-order collocation method.

Synchronization in delayed Cohen-Grossberg neural networks with bounded external inputs

Li, C.-H., Yang, S.-Y. — 2009-03-26

In this paper, we study the drive-response-type synchronization in the delayed Cohen–Grossberg neural networks with bounded external inputs. The connection time delays between neurons can be of discrete or distributed form. By using the Lyapunov functional method, we establish three criteria all independent of the time delays for ensuring the occurrence of synchronization with exponential rates. Generally speaking, we prove that the exponential synchronization occurs provided some certain weighted sum of the connection and coupling strengths with other system parameters is positive enough no matter that the connection time delay is of discrete or distributed form. Our criteria improve and extend some existing ones. Furthermore, several concrete examples are provided to show that the three criteria do not include one another. Numerical simulations are also given to demonstrate the theoretical results.

Numerical simulation for the 3D seepage flow with fractional derivatives in porous media

Liu, Q., Liu, F., Turner, I., Anh, V. — 2009-03-26

In this paper, the numerical simulation of the 3D seepage flow with fractional derivatives in porous media is considered under two special cases: non-continued seepage flow in uniform media (NCSF-UM) and continued seepage flow in non-uniform media (CSF-NUM). A fractional alternating direction implicit scheme (FADIS) for the NCSF-UM and a modified Douglas scheme (MDS) for the CSF-NUM are proposed. The stability, consistency and convergence of both FADIS and MDS in a bounded domain are discussed. A method for improving the speed of convergence by Richardson extrapolation for the MDS is also presented. Finally, numerical results are presented to support our theoretical analysis.

On the temporal stability of steady-state quasi-1D bubbly cavitating nozzle flow solutions

Pasinlioglu, S., Delale, C. F., Schnerr, G. H. — 2009-03-26

Quasi-1D unsteady bubbly cavitating nozzle flows are considered by employing a homogeneous bubbly liquid flow model, where the non-linear dynamics of cavitating bubbles is described by a modified Rayleigh–Plesset equation. The various damping mechanisms are considered by a single damping coefficient lumping them together in the form of viscous dissipation and by assuming a polytropic law for the expansion and compression of the gas. The complete system of equations, by appropriate uncoupling, are then reduced to two evolution equations, one for the flow speed and the other for the bubble radius when all damping mechanisms are considered by a single damping coefficient. The evolution equations for the bubble radius and flow speed are then perturbed with respect to flow unsteadiness resulting in a coupled system of linear partial differential equations (PDEs) for the radius and flow speed perturbations. This system of coupled linear PDEs is then cast into an eigenvalue problem and the exact solution of the eigenvalue problem is found by normal mode analysis in the inlet region of the nozzle. Results show that the steady-state cavitating nozzle flow solutions are stable only for perturbations with very small wave numbers. The stable regions of the stability diagram for the inlet region of the nozzle are seen to be broadened by the effect of turbulent wall shear stress.

Thermoviscoelastic surface waves of an assigned frequency

Ivanov, Ts. P., Savova, R. — 2009-03-26

In this paper, we investigate propagation of quasi-viscoelastic and quasi-thermal surface waves of an assigned frequency on a thermoviscoelastic half-space. Their structure and mechanical characteristics are examined and compared to the properties of the classical Rayleigh waves. It is shown that a unique quasi-viscoelastic wave (surface wave of Rayleigh type) always exists at different values of the assigned frequency. A unique quasi-thermal surface wave of an assigned frequency exists only in the case of small and moderate values of the frequency. The Poynting vectors that are connected to the waves mentioned above are obtained. It is emphasized that they are not parallel to the surface of the half-space, as it is known in classical elasticity. At any given point, their directions vary with the distance from this point to the surface as well as with the assigned frequency. Some numerical results are presented when the half-space is thermally insulated.

On recovering polyhedral scatterers with acoustic far-field measurements

Liu, H. — 2009-03-26

We prove that an acoustic sound-hard scatterer, consisting of finitely many solid polyhedra in Rⁿ(n ≥ 2), is uniquely determined by the far-field patterns corresponding to n – 1 different incident waves. By suitable modifications, the method can also be used to show a similar uniqueness result in the setting without knowing the a priori physical properties of the underlying scatterer.

Wave scattering by an axisymmetric ice floe of varying thickness

Bennetts, L. G., Biggs, N. R. T., Porter, D. — 2009-03-26

The problem of water wave scattering by a circular ice floe, floating in fluid of finite depth, is formulated and solved numerically. Unlike previous investigations of such situations, here we allow the thickness of the floe (and the fluid depth) to vary axisymmetrically and also incorporate a realistic non-zero draught. A numerical approximation to the solution of this problem is obtained to an arbitrary degree of accuracy by combining a Rayleigh–Ritz approximation of the vertical motion with an appropriate variational principle. This numerical solution procedure builds upon the work of Bennets et al. (2007, J. Fluid Mech., 579, 413–443). As part of the numerical formulation, we utilize a Fourier cosine expansion of the azimuthal motion, resulting in a system of ordinary differential equations to solve in the radial coordinate for each azimuthal mode. The displayed results concentrate on the response of the floe rather than the scattered wave field and show that the effects of introducing the new features of varying floe thickness and a realistic draught are significant.

Inhomogeneous spatial patterns for predator-prey models: bifurcation at a higher eigenvalue

Zou, H. — 2009-03-26

In this paper, we study a Rosenzweig–MacArthur predator–prey (steady-state) model with diffusion and subject to homogeneous Neumann boundary conditions. Namely, we consider the following system of elliptic equations: where Rⁿ (n ≥ 2) is a bounded and smooth domain. We call a pair of smooth (say, C²()) functions u(x) and v(x) which are strictly positive and non-constant in the region and satisfy the above system a spatially inhomogeneous pattern. Employing global bifurcation theory and exploring the global structure of the system, we obtain new existence results of spatially inhomogeneous patterns of large amplitude and global nature.

Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

Hasanov, A. — 2009-01-23

The problem of determining the pair w:={F(x, t);f(t)} of source terms in the hyperbolic equation u_tt = (k(x)u_x)_x + F(x, t) and in the Neumann boundary condition k(0)u_x(0, t) = f(t) from the measured data µ(x):=u(x, T) and/or (x):=u_t(x, t) at the final time t = T is formulated. It is proved that both components of the Fréchet gradient of the cost functionals J₁(w) = ||u(x, t;w) – µ(x)||₀² and J₂(w) = ||u_t(x, T;w) – (x)||₀² can be found via the solutions of corresponding adjoint hyperbolic problems. Lipschitz continuity of the gradient is derived. Unicity of the solution and ill-conditionedness of the inverse problem are analysed. The obtained results permit one to construct a monotone iteration process, as well as to prove the existence of a quasi-solution.

Analysis of the Rayleigh-Plesset equation with chirp excitation

Barlow, E., Mulholland, A. J., Gachagan, A., Nordon, A., MacPherson, K. — 2009-01-23

The case of a bubble being insonified by an ultrasonic excitation in the form of a linear chirp is considered here. The dynamical equation of the bubble's motion is solved analytically and compared to a numerical solution. The analytical solution is then used to investigate the problem of maximizing the amplitude of the second harmonic with respect to the various system and signal parameters.

A singular perturbation problem with discontinuous data in a cuboid

Lopez, J. L., Sinusia, E. P. — 2009-01-23

We analyse the asymptotic behaviour of the solution of a 3D singularly perturbed convection–diffusion problem with discontinuous Dirichlet boundary data defined in a cuboid. We write the solution in terms of a double series and we obtain an asymptotic approximation of the solution when the singular parameter -> 0. This approximation is given in terms of a finite combination of products of error functions and characterizes the effect of the discontinuities on the small -behaviour of the solution in the singular layers.

Symbolic-computation study of integrable properties for the (2 + 1)-dimensional Gardner equation with the two-singular manifold method

Zhang, H.-Q., Tian, B., Li, J., Xu, T., Zhang, Y.-X. — 2009-01-23

The singular manifold method from the Painlevé analysis can be used to investigate many important integrable properties for the non-linear partial differential equations. In this paper, the two-singular manifold method is applied to the (2 + 1)-dimensional Gardner equation with two Painlevé expansion branches to determine the Hirota bilinear form, Bäcklund transformation, Lax pairs and Darboux transformation. Based on the obtained Lax pairs, the binary Darboux transformation is constructed and the N x N Grammian solution is also derived by performing the iterative algorithm N times with symbolic computation.

An alternating method for Cauchy problems for Helmholtz-type operators in non-homogeneous medium

Johansson, B. T., Kozlov, V. A. — 2009-01-23

Kozlov & Maz'ya (1989, Algebra Anal., 1, 144–170) proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems. However, in many applied problems, operators appear that do not satisfy these requirements, e.g. Helmholtz-type operators. Therefore, in this study, an alternating procedure for solving Cauchy problems for self-adjoint non-coercive elliptic operators of second order is presented. A convergence proof of this procedure is given.

The residual velocity method applied to a steady free boundary-value problem of vector Laplacian type

Chen, W., Wetton, B. — 2009-01-23

We consider a free boundary-value problem based on a simplified model of two-phase flow in porous media. The model has two independent variables on each side of the free interface. At the interface at steady state, five mixed Dirichlet and Neumann conditions are given. The movement of the interface in time-dependent situations can be reduced to a normal motion proportional to the residual in one of the steady-state interface conditions (the elliptic interior problems and the other interface conditions are satisfied at each time). Following previous work, we consider the use of other residuals for the normal velocity that have superior numerical properties. The well-posedness criteria for this vector example are particularly clear. The advantages of the correctly chosen, non-physical residual velocities are demonstrated in numerical computations. Although the finite-difference implementation in this work is not applicable to general problems, it has superior performance to previous implementations.

Non-linear responses of a one-sided constrained beam with base excitation

Qian, Q., Wang, L. — 2009-01-23

The paper discussed the non-linear responses of a buckled beam under base excitation and constrained by a one-sided motion restraint. The geometric non-linearity due to axial extension is taken into account. We apply the Galerkin method to the governing partial differential equation of the transverse motion to obtain a general model of n degrees of freedom (nDOF). The results of dynamic response for the 1DOF mode of a pinned-pinned beam are presented for the system both with and without one-sided motion restraint. The regular and irregular motions of the 1DOF model for the beam are represented in the forms of time trace, phase plot, bifurcation diagram and power spectra. An in-depth study based on an energy approach is done to illustrate the non-linear responses resulting from the multiplicity of resonant solutions. It is shown that the effect of the motion restraint on the dynamics of the system is significant.

A higher-order Boussinesq equation in locally non-linear theory of one-dimensional non-local elasticity

Duruk, N., Erkip, A., Erbay, H. A. — 2009-01-23

In one space dimension, a non-local elastic model is based on a single integral law, giving the stress when the strain is known at all spatial points. In this study, we first derive a higher-order Boussinesq equation using locally non-linear theory of 1D non-local elasticity and then we are able to show that under certain conditions the Cauchy problem is globally well-posed.

On travelling-wave solutions for a moving boundary problem of Hele-Shaw type

Gunther, M., Prokert, G. — 2009-01-23

We discuss a 2D moving boundary problem for the Laplacian with Robin boundary conditions in an exterior domain. It arises as a model for Hele–Shaw flow of a bubble with kinetic undercooling regularization and is also discussed in the context of models for electrical streamer discharges. The corresponding evolution equation is given by a degenerate, non-linear transport problem with non-local lower-order dependence. We identify the local structure of the set of travelling-wave solutions in the vicinity of trivial (circular) ones. We find that there is a unique non-trivial travelling wave for each velocity near the trivial one. Therefore, the trivial solutions are unstable in a comoving frame. The degeneracy of our problem is reflected in a loss of regularity in the estimates for the linearization. Moreover, there is an upper bound for the regularity of its solutions. To prove our results, we use a quasi-linearization by differentiation, index results for degenerate ordinary differential operators on the circle and perturbation arguments for unbounded Fredholm operators.

Generalized Calderon-Ryaben'kii's potentials

Utyuzhnikov, S. V. — 2009-01-23

Calderón–Ryaben'kii potentials provide the foundation for the difference potential method, which is an efficient way for solving boundary-value problems (BVPs) in arbitrary domains. This method allows us to reduce a uniquely solvable and well-posed BVP to a pseudo-differential boundary equation. The general theory of Calderón–Ryaben'kii potentials is considered via the theory of distributions. The definition of Calderón–Ryaben'kii potentials is based on the notion of a clear trace. The criterion of the clear trace is formulated. Partial differential equations of the first order and the second order are considered as particular examples. On the basis of the Calderón–Ryaben'kii potential theory, a solution of the active sound control problem is obtained in a general formulation. For the first time, the solution of the problem takes into account the feedback of the active shielding sources on the input (measurement) data. The exact transfer of the boundary conditions from the original boundary to an artificial boundary is also considered.

Solvability of a Reissner-Mindlin-Timoshenko plate-beam vibration model

van Rensburg, N. F. J., Zietsman, L., van der Merwe, A. J. — 2009-01-23

In this paper, we consider the vibration of a plate–beam system consisting of a Reissner–Mindlin plate and a Timoshenko beam. A variational form is derived directly from the equations of motion and the constitutive equations. We show how an existence result for a general linear vibration problem in variational form may be applied to the weak variational problem for this system.