Showing papers in "European Journal of Applied Mathematics in 2002"
TL;DR: In this paper, an algorithm for finding local conservation laws for partial differential equations with any number of independent and dependent variables is presented, which does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations.
Abstract: An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. In the first of two papers (Part I), examples of nonlinear wave equations are used to exhibit the method. Classification results for conservation laws of these equations are obtained. In a second paper (Part II), a general treatment of the method is given.
436 citations
TL;DR: Two novel inpainting models based on the Mumford–Shah image model and its high order correction are proposed, and their efficient numerical realizationbased on the Γ-convergence approximations of Ambrosio & Tortorelli and De Giorgi are presented.
Abstract: Image inpainting is an image restoration problem, in which image models play a critical role, as demonstrated by Chan, Kang & Shen's [12] recent inpainting schemes based on the bounded variation and the elastica [11] image models. In this paper, we propose two novel inpainting models based on the Mumford–Shah image model [41], and its high order correction – the Mumford–Shah–Euler image model. We also present their efficient numerical realization based on the Γ-convergence approximations of Ambrosio & Tortorelli [2, 3] and De Giorgi [21].
401 citations
TL;DR: In this article, the authors give a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]), and apply it to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form.
Abstract: This paper gives a general treatment and proof of the direct conservation law method presented in Part I (see Anco & Bluman [3]). In particular, the treatment here applies to finding the local conservation laws of any system of one or more partial differential equations expressed in a standard Cauchy-Kovalevskaya form. A summary of the general method and its effective computational implementation is also given.
366 citations
TL;DR: In this paper, the authors compare computational aspects of four approaches to compute conservation laws of single Differential Equations (DEs) or systems of them, ODEs and PDEs.
Abstract: The paper compares computational aspects of four approaches to compute conservation laws of single Differential Equations (DEs) or systems of them, ODEs and PDEs. The only restriction, required by two of the four corresponding computer algebra programs, is that each DE has to be solvable for a leading derivative. Extra constraints for the conservation laws can be specified. Examples include new conservation laws that are non-polynomial in the functions, that have an explicit variable dependence and families of conservation laws involving arbitrary functions. The following equations are investigated in examples: Ito, Liouville, Burgers, Kadomtsev–Petviashvili, Karney–Sen–Chu–Verheest, Boussinesq, Tzetzeica, Benney.
163 citations
TL;DR: In this paper, a generalization of the Keller-Segel model for chemotactic systems is studied, where several populations interacting via several sensitivity agents in a two-dimensional domain are determined by a Fokker-Planck system of equations, coupled with a system of diffusion equations for the chemical agents.
Abstract: A generalization of the Keller–Segel model for chemotactic systems is studied. In this model there are several populations interacting via several sensitivity agents in a two-dimensional domain. The dynamics of the population is determined by a Fokker–Planck system of equations, coupled with a system of diffusion equations for the chemical agents. Conditions for global existence of solutions and equilibria are discussed, as well as the possible existence of time-periodic attractors. The analysis is based on a variational functional associated with the system.
89 citations
TL;DR: In this paper, the stability of the asymmetric k-spike patterns with respect to the large O(1) eigenvalues and the small O(e 2 ) eigen values is analyzed.
Abstract: Equilibrium solutions to the one-dimensional Gierer–Meinhardt model in the form of sequences of spikes of different heights are constructed asymptotically in the limit of small activator diffusivity e For a pattern with k spikes, the construction yields k1 spikes that have a common small amplitude and k2 = k−
k1 spikes that have a common large amplitude A k-
spike asymmetric equilibrium solution is obtained from an arbitrary ordering of the small and large spikes on the domain It is shown that such solutions exist when the inhibitor diffusivity D is less than some critical value Dm that depends upon k1, on k2, and on other parameters associated with the Gierer–Meinhardt model It is also shown that these asymmetric k-spike solutions bifurcate from the symmetric solution branch sk, for which k spikes have equal height These asymmetric solutions provide connections between the branch sk and the other symmetric branches sj , for j = 1,…, k−
1 The stability of the asymmetric k-spike patterns with respect to the large O(1) eigenvalues and the small O(e2) eigenvalues is also analyzed It is found that the asymmetric patterns are stable with respect to the large O(1) eigenvalues when D > De, where De depends on k1 and k2, on certain parameters in the model, and on the specific ordering of the small and large spikes within a given k-spike sequence Numerical values for De are obtained from numerical solutions of a matrix eigenvalue problem Another matrix eigenvalue problem that determines the small eigenvalues is derived For the examples considered, it is shown that the bifurcating asymmetric branches are all unstable with respect to these small eigenvalues
63 citations
TL;DR: In this paper, the existence of energy equilibria of the di-block copolymer problem in the unit disk was proved by solving the proper singular limit of the free energy functional.
Abstract: We prove the existence of energy equilibria of the di-block copolymer problem in the unit disk. They consist of concentrically layered micro-domains rich in one of the two monomer building units. We construct them by solving the proper singular limit of the free energy functional. The same limit also explains how under a dynamic law of the free energy, circular interfaces of non-equilibria may move to the origin and vanish, or collapse to each other, thereby reducing the number of layers.
57 citations
TL;DR: In this paper, a general formalism is described whereby some regular singular points are effectively removed and substantial simplifications ensue for a class of Fuchsian ordinary differential equations, and related confluent equations.
Abstract: A general formalism is described whereby some regular singular points are effectively removed and substantial simplifications ensue for a class of Fuchsian ordinary differential equations, and related confluent equations. These simplifications follow provided the exponents at the singular points satisfy certain relations; explicit, illustrative examples are constructed to demonstrate the ideas.
42 citations
TL;DR: In this article, the authors consider the case of a nematic liquid crystal held between two parallel electrical conductors and show the balance between an anisotropic material trying to push the electric field away from regions of large deformation and the deformation trying to be maximum in regions of high electric field.
Abstract: The behaviour of liquid crystal materials used in display devices is discussed. The underlying continuum theory developed by Frank, Ericksen and Leslie for describing this behaviour is reviewed. Particular attention is paid to the approximations and extensions relevant to existing device technology areas where mathematical analysis would aid device development. To illustrate some of the special behaviour of liquid crystals and in order to demonstrate the techniques employed, the specific case of a nematic liquid crystal held between two parallel electrical conductors is considered. It has long been known that there is a critical voltage below which the internal elastic strength of the liquid crystal exceeds the electric forces and hence the system remains undeformed from its base state. This bifurcation behaviour is called the Freedericksz transition. Conventional analytic analysis of this problem normally considers a magnetic, rather than electric, field or a near-transition voltage since in these cases the electromagnetic field structure decouples from the rest of the problem. Here we consider more practical situations where the electromagnetic field interacts with the liquid crystal deformation. Assuming strong anchoring at surfaces and a one dimensional deformation, three nondimensional parameters are identified. These relate to the applied voltage, the anisotropy of the electrical permittivity of the liquid crystal, and to the anisotropy of the elastic stiffness of the liquid crystal. The analysis uses asymptotic methods to determine the solution in a numerous of different regimes defined by physically relevant limiting cases of the parameters. In particular, results are presented showing the delicate balance between an anisotropic material trying to push the electric field away from regions of large deformation and the deformation trying to be maximum in regions of high electric field.
40 citations
TL;DR: In this article, a method of computing periodic water waves based on solving an integral equation by means of discretization and automatically nding the mesh on which the functions to be found are approximated by the best way is presented.
Abstract: The paper presents a method of computing periodic water waves based on solving an integral equation by means of discretization and automatically nding the mesh on which the functions to be found are approximated by the best way. The power of the method to describe ‘bad functions’ well makes it possible to reproduce all the main results of asymptotic theory for the almost-highest waves (Longuet-Higgins & Fox, 1977, 1978, 1996) by a direct numerical simulation. The method is able to compute two full periods of the oscillations of wave properties for all wave height-to-length ratios. The end of the second period corresponds to the wave steepness that achieves 99.99997% of the limiting value. So, the validity of the asymptotic formulae by Longuet-Higgins & Fox is proved for the steep waves of any nite depth. The rened value of the maximum slope of the free-surfaces is found to be 30.3787.
35 citations
TL;DR: This paper addresses the question of existence of a solution and statistical quantities (e.g. mean interface width or correlation functions) and discusses the approximations of such statistical quantities by the spectral Galerkin method.
Abstract: Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of a solution and statistical quantities (e.g. mean interface width or correlation functions). Moreover, we discuss the approximations of such statistical quantities by the spectral Galerkin method. This is an important question, as the numerical computation of statistical quantities plays a key role in the verification of the models.
TL;DR: In this paper, an American call option is considered and the optimal exercise boundary for the option is analyzed in various asymptotic limits, such as situations where the interest and dividend rates are large or small compared to the volatility of the asset.
Abstract: We consider an American call option and let C(S, T0) be the price of an option corresponding to asset price S at some time T0 prior to the expiration time TF . We analyze C(S, T0) in various asymptotic limits. These include situations where the interest and dividend rates are large or small, compared to the volatility of the asset. We also analyze the optimal exercise boundary for the option. We use perturbation methods to analyze either the PDE that C(S, T0) satisfies, or a nonlinear integral equation that is satisfied by the optimal exercise boundary.
TL;DR: In this paper, the authors consider the Ginzburg-Landau equation in dimension two and introduce a key notion of the vortex (interaction) energy, defined by minimizing the renormalized GLEF functional over functions with a given set of zeros of given local indices.
Abstract: We consider the Ginzburg–Landau equation in dimension two. We introduce a key notion of the vortex (interaction) energy. It is defined by minimizing the renormalized Ginzburg–Landau (free) energy functional over functions with a given set of zeros of given local indices. We find the asymptotic behaviour of the vortex energy as the inter-vortex distances grow. The leading term of the asymptotic expansion is the vortex self-energy while the next term is the classical Kirchhoff–Onsager Hamiltonian. To derive this expansion we use several novel techniques.
TL;DR: The existence of classical solutions with a free boundary changing its topology in time is proved for the multi-dimensional Hele-Shaw problem in this article, if some symmetry and monotonicity conditions on the problem data hold, then a cusp, arising at the contact between two free boundary components, disappears instantly and the free boundary becomes smooth and singly connected.
Abstract: The existence of classical solutions with a free boundary changing its topology in time is proved for the multi-dimensional Hele-Shaw problem We prove that, if some symmetry and monotonicity conditions on the problem data hold, then a cusp, arising at the contact between two free boundary components, disappears instantly and the free boundary becomes smooth and singly-connected
TL;DR: In this paper, the existence and uniqueness of a monotone increasing solution of the Painlevé 1 equation y = y2+x was proved and the monotonicity of solution was exploited to show stability of the plasma-sheath transition in a weakly ionizing plasma.
Abstract: This paper proves the existence and uniqueness of a monotone increasing solution of the Painlevé 1 equation y″ = y2+x. The monotonicity of solution is then exploited to show stability of the plasma-sheath transition in a weakly ionizing plasma.
TL;DR: In this paper, a three-phase ensemble-averaged model is developed for the flow of water and air through a deformable porous matrix, which predicts a separation of the flow into saturated and unsaturated regions.
Abstract: A three-phase ensemble-averaged model is developed for the flow of water and air through a deformable porous matrix. The model predicts a separation of the flow into saturated and unsaturated regions. The model is closed by proposing an experimentally-motivated heuristic elastic law which allows large-strain nonlinear behaviour to be treated in a relatively straightforward manner. The equations are applied to flow in the ‘nip’ area of a roll press machine whose function is to squeeze water out of wet paper as part of the manufacturing process. By exploiting the thin-layer limit suggested by the geometry of the nip, the problem is reduced to a nonlinear convection-diffusion equation with one free boundary. A numerical method is proposed for determining the flow and sample simulations are presented.
TL;DR: In this article, the probability density function (PDF) of the concentration of a scalar within a turbulent flow is studied, and the effect of molecular diffusion in reducing concentration uctuations, eventually to zero.
Abstract: The paper deals with the probability density function (PDF) of the concentration of a scalar within a turbulent flow. Following some comments about the overall structure of the PDF, and its approach to a limit at large times, attention focusses on the so-called small scale mixing term in the evolution equation for the PDF. This represents the effect of molecular diffusion in reducing concentration uctuations, eventually to zero. Arguments are presented which suggest that this quantity could, in certain circumstances, depend inversely upon the PDF, and a particular example of this leads to a new closure hypothesis. Consequences of this, especially similarity solutions, are explored for the case when the concentration field is statistically homogeneous.
TL;DR: In this article, the authors consider a one-dimensional stochastic model of sediment deposition in which the complete time history of sedimentation is the sum of a linear trend and a fractional Brownian motion wH(t) with self-similarity parameter H ∈ (0, 1).
Abstract: We consider a one-dimensional stochastic model of sediment deposition in which the complete time history of sedimentation is the sum of a linear trend and a fractional Brownian motion wH(t) with self-similarity parameter H ∈ (0, 1). The thickness of the sedimentary layer as a function of time, d(t), looks like the Cantor staircase. The Hausdorff dimension of the points of growth of d(t) is found. We obtain the statistical distribution of gaps in the sedimentary record, periods of time during which the sediments have been eroded. These gaps define sedimentary unconformities. In the case H = 1/2 we obtain the statistical distribution of layer thicknesses between unconformities and investigate the multifractality of d(t). We show that the multifractal structures of d(t) and the local time function of Brownian motion are identical; hence d(t) is not a standard multifractal object. It follows that natural statistics based on local estimates of the sedimentation rate produce contradictory estimates of the range of local dimension for d(t). The physical object d(t) is interesting in that it involves the above anomalies, and also in its mechanism of fractality generation, which is different from the traditional multiplicative process.
TL;DR: In this article, the authors considered the one-dimensional and two-dimensional filtration-absorption equation and constructed a family of axisymmetric solutions in 2D. The main analytical result is the linear stability of the above self-similar solutions both in the 1D case and in the 2D case.
Abstract: We consider the one-dimensional and two-dimensional filtration-absorption equation ut = uΔu−(c−1)(∇u)2. The one-dimensional case was considered previously by Barenblatt et al. [4], where a special class of self-similar solutions was introduced. By the analogy with the 1D case we construct a family of axisymmetric solutions in 2D. We demonstrate numerically that the self-similar solutions obtained attract the solutions of non-self-similar Cauchy problems having the initial condition of compact support. The main analytical result we provide is the linear stability of the above self-similar solutions both in the 1D case and in the 2D case.
TL;DR: In this paper, an initial boundary value problem for the non-local equation, where u t = u xx +λ f ( u )/(∫ 1 -1 f( u ) dx ) 2, with Robin boundary conditions was considered.
Abstract: We consider an initial boundary value problem for the non-local equation, u t = u xx +λ f ( u )/(∫ 1 -1 f ( u ) dx ) 2 , with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and
the solution u ( x , t ) blows up globally in finite time t *, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t *, by using comparison methods. For f ( u ) = e − u , we give an asymptotic estimate: t * ∼ t u (λ−λ*) −1/2 for 0 < (λ−λ*) [Lt ] 1, where t u is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.
TL;DR: In this article, a macroscopic model for the mushy region is proposed, where a fine mixture of two distinct phases separated by free boundaries is assumed to consist of a mixture of different phases.
Abstract: A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. A method of multiple scales, with restrictions on the form of the microscopic free boundaries, is used to derive a macroscopic model for the mushy region. The final model depends both on the microscopic structure and on how the free-boundary temperature varies with curvature (Gibbs–Thomson effect), kinetic undercooling, or, for an alloy, composition.
TL;DR: In this article, the authors studied the unique solvability of the reduced Navier-Stokes problem globally in time and the collapse of the solution in finite time in a 3D nonstationary flow of viscous incompressible liquid.
Abstract: Three-dimensional nonstationary flow of a viscous incompressible liquid is investigated in a layer, driven by a nonuniform distribution of temperature on its free boundaries. If the temperature given on the layer boundaries is quadratically dependent on horizontal coordinates, external mass forces are absent, and the motion starts from rest then the free boundary problem for the Navier–Stokes equations has an ‘exact’ solution in terms of two independent variables. Here the free boundaries of the layer remain parallel planes and the distance between them must be also determined. In present paper, we formulate conditions for both the unique solvability of the reduced problem globally in time and the collapse of the solution in finite time. We further study qualitative properties of the solution such as its behaviour for large time (in the case of global solvability of the problem), and the asymptotics of the solution near the collapse moment in the opposite case.
TL;DR: In this paper, symmetry and computer algebra can assist in solving the Cauchy problem for Pfaffian systems, and the results on integrating Frobenius integrable distributions via solvable symmetry structures are used to develop two techniques that when used in conjunction with symmetry determination software DIMSYM, allow the authors to solve the problem for the special situation when there exists a one-dimensional cauchy characteristic space.
Abstract: We examine how symmetry and computer algebra can assist in solving the Cauchy problem for Pfaffian systems. We use recent results on integrating Frobenius integrable distributions via solvable symmetry structures to develop two techniques that when used in conjunction with symmetry determination software DIMSYM, allow us to solve the Cauchy problem for the special situation when there exists a one-dimensional Cauchy characteristic space. We also illustrate how our work can assist in extracting local solutions of a certain class of first and second order non-linear partial differential equations.
TL;DR: In this article, the free boundary model of diusion-induced grain boundary motion is extended to account for bidirectional motion, together with the appearance of S-shapes and double seam congurations.
Abstract: The free boundary model of diusion-induced grain boundary motion derived in Cahn et al [3], Fife et al [6] and Cahn & Penrose [4] is extended, in the case of thin metallic lms, to account for bidirectional motion, together with the appearance of S-shapes and double seam congurations These are often observed in the laboratory Computer simulations based on the extended model are given to illustrate these and other features of bidirectional motion More generally, the extension accounts for the motion of grain boundaries whose traces on the lm’s surface are curved The new free boundary model is one of forced motion by curvature, the forcing term possibly changing sign due to the bidirectionality The thin lm model is derived systematically under explicit assumptions, and an adjustment for grooving is included
TL;DR: In this article, an eigenvalue problem of 3D elasticity for a multi-structure consisting of a finite three-dimensional solid linked with a thin-walled elastic cylinder was considered.
Abstract: We consider an eigenvalue problem of three-dimensional elasticity for a multi-structure consisting of a finite three-dimensional solid linked with a thin-walled elastic cylinder. An asymptotic method is used to derive the junction conditions and to obtain the skeleton model for the multi-structure. Explicit asymptotic formulae have been obtained for the first six eigen-frequencies.
TL;DR: In this article, the authors show how the formal construction of the Ginzburg-Landau system gives a natural way for constructing a subsolution for the half-space version of the superheating field.
Abstract: Dorsey et al. [8] have constructed formal solutions for the half-space Ginzburg–Landau model, when κ is small. Dorsey et al. deduce a formal expansion for the superheating field in powers of κ½ up to order 4. In this paper, we show how the formal construction gives a natural way for constructing a subsolution for the Ginzburg–Landau system. We improve the result obtained by Bolley & Helffer [2], and take a step in the proof of the Parr Formula [13], getting two terms in the lower bound for the superheating field as κ → 0.
TL;DR: In this article, the authors analyzed the front structures evolving under the difference-differential equation ∂tCj = −Cj+C2j−1 from initial conditions 0 [les] Cj(0) [les ] 1 such that Cj (0) → 1 as j → ∞ suffciently fast.
Abstract: We analyze the front structures evolving under the difference-differential equation ∂tCj = −Cj+C2j−1 from initial conditions 0 [les ] Cj(0) [les ] 1 such that Cj(0) → 1 as j → ∞ suffciently fast. We show that the velocity v(t) of the front converges to a constant value v* according to v(t) = v*−3/(2λ*t)+(3√π/2) Dλ*/(λ*2Dt)3/2+[Oscr ](1/t2). Here v*, λ* and D are determined by the properties of the equation linearized around Cj = 1. The same asymptotic expression is valid for fronts in the nonlinear diffusion equation, where the values of the parameters λ*, v* and D are specific to the equation. The identity of methods and results for both equations is due to a common propagation mechanism of these so-called pulled fronts. This gives reasons to believe that this universal algebraic convergence actually occurs in an even larger class of equations.
TL;DR: In this article, the authors considered the two-dimensional Rayleigh-Taylor problem for the dynamics of the free interface between two layers of immiscible viscous liquids and proved that the increase in time t is bounded by an exponential function with exponent independent of the interface.
Abstract: We consider the two-dimensional Rayleigh–Taylor problem for the dynamics of the free interface Γ between two layers of immiscible viscous liquids. For a slow flow model (which corresponds to the case of a small relative jump of density) and under sufficiently wide assumptions on the geometry of Γ, we analyze the time dynamics of Γ. In particular, we prove that its increase in time t is bounded by an exponential function with exponent independent of Γ.
TL;DR: In this paper, the annihilation dynamics arising in the KPP-Fisher equation was studied via a comparison theorem, where the comparison functions satisfy equations which are linearizable to the heat equation.
Abstract: We study the annihilation dynamics arising in the KPP-Fisher equation, proposed by Fisher in 1936 to model the propagation of a mutant gene and subsequently studied rigorously in the seminal work of Kolmogorov, Petrovskii and Piskunov. The approach is via a comparison theorem, where the comparison functions satisfy equations which are linearizable to the heat equation. In some sense, we have obtained a ‘linearization’ of the KPP-Fisher equation.
TL;DR: In this paper, a nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. to model the heat and mass exchange in stably stratified turbulent shear flow.
Abstract: A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.