Showing papers in "Quarterly of Applied Mathematics in 2010"
TL;DR: In this paper, the authors provide explicit solutions of linear, left-invariant diffusion equations and corresponding resolvent equations on the 2D-Euclidean motion group SE(2) = R^2 x T.
Abstract: We provide the explicit solutions of linear, left-invariant, diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2) = R^2 x T. These parabolic equations are forward Kolmogorov equations for well-known stochastic processes for contour enhancement and contour completion. The solutions are given by group convolution with the corresponding Green's functions. In earlier work we have solved the forward Kolmogorov equations (or Fokker-Planck equations) for stochastic processes on contour completion. Here we mainly focus on the forward Kolmogorov equations for contour enhancement processes which do not include convection. We derive explicit formulas for the Green's functions (i.e., the heat kernels on SE(2)) of the left-invariant partial differential equations related to the contour enhancement process. By applying a contraction we approximate the left-invariant vector fields on SE(2) by left-invariant generators of a Heisenberg group, and we derive suitable approximations of the Green's functions. The exact Green's functions are used in so-called collision distributions on SE(2), which are the product of two left-invariant resolvent diffusions given an initial distribution on SE(2). We use the left-invariant evolution processes for automated contour enhancement in noisy medical image data using a so-called orientation score, which is obtained from a grey-value image by means of a special type of unitary wavelet transformation. Here the real part of the (invertible) orientation score serves as an initial condition in the collision distribution.
140 citations
TL;DR: In this paper, a wavelet unitary transform is used to construct an orientation score from a grey-value image, which is a complex-valued function on the 2D Euclidean motion group SE(2) and gives us explicit information on the presence of local orientations.
Abstract: By means of a special type of wavelet unitary transform we construct an orientation score from a grey-value image. This orientation score is a complex-valued function on the 2D Euclidean motion group SE(2) and gives us explicit information on the presence of local orientations in an image. As the transform between image and orientation score is unitary we can relate operators on images to operators on orientation scores in a robust manner. Here we consider nonlinear adaptive diffusion equations on these invertible orientation scores. These nonlinear diffusion equations lead to clear improvements of the celebrated standard "coherence enhancing diffusion" equations on images as they can enhance images with crossing contours. Here we employ differential geometry on SE(2) to align the diffusion with optimized local coordinate systems attached to an orientation score, allowing us to include local features such as adaptive curvature in our diffusions.
98 citations
TL;DR: A multi-dimensional flocking model rigorously derived from a vector oscillatory chain model and an alternative direct approach for frequency synchronization to the Kuramoto model as an application of the flocking estimate for the Cucker-Smale model are provided.
Abstract: In this note, we present a multi-dimensional flocking model rigorously derived from a vector oscillatory chain model and study the connection between the Cucker-Smale flocking model and the Kuramoto synchronization model appearing in the statistical mechanics of nonlinear oscillators. We provide an alternative direct approach for frequency synchronization to the Kuramoto model as an application of the flocking estimate for the Cucker-Smale model.
59 citations
TL;DR: In this paper, the authors studied the initial boundary value problem of semilinear hyperbolic and parabolic equations with critical initial data and proved that there exist non-global solutions under classical conditions on f.
Abstract: We study the initial boundary value problem of semilinear hyperbolic equations u tt — Δu = f(u) and semilinear parabolic equations u t — Δu = f(u) with critical initial data E(0) = d (or J(u o ) = d), I(u 0 ) < 0, and prove that there exist non-global solutions under classical conditions on f.
51 citations
TL;DR: Achdou et al. as discussed by the authors considered a newtonian flow in domains with periodic rough boundaries and derived high order boundary layer approximations and rigorously justified their rates of convergence with respect to epsilon.
Abstract: In this work we present new wall-laws boundary conditions including microscopic oscillations. We consider a newtonian flow in domains with periodic rough boundaries that we simplify considering a Laplace operator with periodic inflow and outflow boundary conditions. Following the previous approaches, see [A. Mikelic, W. Jager, J. Diff. Eqs, 170, 96-122, (2001) ] and [Y. Achdou, O. Pironneau, F. Valentin, J. Comput. Phys, 147, 1, 187-218, (1998)], we construct high order boundary layer approximations and rigorously justify their rates of convergence with respect to epsilon (the roughness' thickness). We establish mathematically a poor convergence rate for averaged second-order wall-laws as it was illustrated numerically for instance in [Y. Achdou, O. Pironneau, F. Valentin, J. Comput. Phys, 147, 1, 187-218, (1998)]. In comparison, we establish exponential error estimates in the case of explicit multi-scale ansatz. This motivates our study to derive implicit first order multi-scale wall-laws and to show that its rate of convergence is at least of order epsilon to the three halves. We provide a numerical assessment of the claims as well as a counter-example that evidences the impossibility of an averaged second order wall-law. Our paper may be seen as the first stone to derive efficient high order wall-laws boundary conditions.
48 citations
TL;DR: In this paper, the authors mainly focused on mixed type partial differential equations and their connections with transonic flow and their typical problems in gas dynamics related to the mixed type equations are presented and analyzed.
Abstract: The paper is mainly concerned with mixed type partial differential equations and their connections with transonic flow. Some typical problems in gas dynamics related to the mixed type equations are presented and analyzed. In the meantime, the crucial points on the study of these problems, including some recent developments and new approaches, are introduced.
28 citations
TL;DR: In this article, the authors present a family of numerical implementations of Kato's ODE propagating global bases of analytically varying invariant subspaces of which the first-order version is a surprisingly simple greedy algorithm that is both stable and easy to program.
Abstract: We present a family of numerical implementations of Kato's ODE propagating global bases of analytically varying invariant subspaces of which the first-order version is a surprisingly simple "greedy algorithm" that is both stable and easy to program and the second-order version a relaxation of a first-order scheme of Brin and Zumbrun. The method has application to numerical Evans function computations used to assess stability of traveling-wave solutions of time-evolutionary PDE.
25 citations
TL;DR: In this paper, the authors deal with the artificial compressibility approximation method adapted to the incompressible Navier Stokes Fourier system, and two different types of approximations are analyzed: one for the full NSF system where viscous heating effects are considered and the other for when the dissipative function S: ∇u is omitted.
Abstract: This paper deals with the artificial compressibility approximation method adapted to the incompressible Navier Stokes Fourier system. Two different types of approximations will be analyzed: one for the full Navier Stokes Fourier system (or the so-called Rayleigh-Benard equations) where viscous heating effects are considered and the other for when the dissipative function S: ∇u is omitted. The convergence of the approximating sequences is achieved by exploiting the dispersive properties of the wave equation structure of the pressure of the approximating system.
18 citations
TL;DR: The work in Banks, Karr, Nguyen and Samuels (2008), which demonstrated a simple dynamical system framework in which to study social network behavior, is extended to include a discrete delay, which represents the time lag that is likely required for an agent to change his/her own characteristics after interacting with an agent possessing different characteristics.
Abstract: Networks are typically studied via computational models, and often investigations are restricted to the static case. Here we extend the work in Banks, Karr, Nguyen and Samuels (2008), which demonstrated a simple dynamical system framework in which to study social network behavior, to include a discrete delay. This delay represents the time lag that is likely required for an agent to change his/her own characteristics (e.g., opinions, viewpoints or behavior) after interacting with an agent possessing different characteristics. Thus this modification adds significantly to the relevance of the model in many potential applications. We have shown that the delays can be incorporated into a stochastic differential equations (SDE) framework in an efficient and computationally tractable way. Through numerical studies, we see novel outcomes when stochasticity, delay, or both are considered, demonstrating the need to include these features should they be present in the network application.
15 citations
TL;DR: In this paper, a perfectly conductive ellipsoid under magnetic dipolar excitation at low frequency is used to obtain an analytical solution of this scattering problem in order to have a fast numerical estimation of the scattered field that can be useful for real data inversion.
Abstract: This paper deals with the scattering by a perfectly conductive ellipsoid under magnetic dipolar excitation at low frequency. The source and the ellipsoid are embedded in an infinite homogeneous conducting ground. The main idea is to obtain an analytical solution of this scattering problem in order to have a fast numerical estimation of the scattered field that can be useful for real data inversion. Maxwell equations and boundary conditions, describing the problem, are firstly expanded using low-frequency expansion of the fields up to order three. It will be shown that fields have to be found incrementally. The static one (term of order zero) satisfies the Laplace equation. The next non-zero term (term of order two) is more complicated and satisfies the Poisson equation. The order-three term is independent of the previous ones and is described by the Laplace equation. They constitute three different scattering problems that are solved using the separated variables method in the ellipsoidal coordinate system. Solutions are written as expansions on the few analytically known scalar ellipsoidal harmonics. Details are given to explain how those solutions are achieved with an example of numerical results.
14 citations
TL;DR: In this paper, a reaction-diffusion equation whose diffusion term is a time convolution of the Laplace operator against a nonin-creasing summable memory kernel is considered.
Abstract: In a two-dimensional space domain, we consider a reaction-diffusion equation whose diffusion term is a time convolution of the Laplace operator against a nonin-creasing summable memory kernel κ. This equation models several phenomena arising from many different areas. After rescaling κ by a relaxation time e > 0, we formulate a Cauchy-Dirichlet problem, which is rigorously translated into a similar problem for a semilinear hyperbolic integro-differential equation with nonlinear damping, for a particular choice of the initial data. Using the past history approach, we show that the hyperbolic equation generates a dynamical system which is dissipative provided that e is small enough, namely, when the equation is sufficiently "close" to the standard reaction-diffusion equation formally obtained by replacing κ with the Dirac mass at 0. Then, we provide an estimate of the difference between e-trajectories and 0-trajectories, and we construct a family of regular exponential attractors which is robust with respect to the singular limit e → 0. In particular, this yields the existence of a regular global attractor of finite fractal dimension. Convergence to equilibria is also examined. Finally, all the results are reinterpreted within the original framework.
TL;DR: In this article, the authors study the flow of a piezo-viscous fluid down an incline in various flow regimes, using the lubrication theory as revised by Rajagopal and Szeri.
Abstract: Stokes was the first to recognize that the viscosity of many fluids varies significantly with pressure. Later, several experimental studies showed that such a variation may be exponential. Here, by using the lubrication theory as revised by Rajagopal and Szeri, we study the flow of a piezo-viscous fluid down an incline in various flow regimes.
TL;DR: In this paper, the existence and uniqueness of a weak solution of a nonhomogeneous stationary Oseen flow around a rotating body in an exterior domain D was established and the localization procedure was combined with classical results in an appropriate bounded domain.
Abstract: We establish the existence and uniqueness of a weak solution of the three-dimensional nonhomogeneous stationary Oseen flow around a rotating body in an exterior domain D. We mainly use the localization procedure (see Kozono and Sohr (1991)) to combine our previous results (see Kracmar, Necasova, and Penel (2007, 2008)) with classical results in an appropriate bounded domain. We study the case of a nonintegrable right-hand side, where f is given in (W ―1,q (D)) 3 for certain values of q.
TL;DR: In this article, a limit analysis for normal materials based on energy minimization is presented, which includes no-tension materials and materials with bounded compressive strength; it also includes the Hencky plastic materials.
Abstract: This note presents a limit analysis for normal materials based on energy minimization. The class of normal materials includes some of those used to model masonry structures, namely, no-tension materials and materials with bounded compressive strength; it also includes the Hencky plastic materials. Considering loads L(λ) that depend affinely on the loading multiplier λ ∈ R, we examine the infimum I 0 (λ) of the potential energy I(u,λ) over the set of all admissible displacements u. Since I 0 (λ) is a concave function of λ, the set A of all λ with I 0 (λ) > -∞ is an interval. Each finite endpoint λ c ∈ ℝ of A is called a collapse multiplier, and we interpret the loads corresponding to λ c as the loads at which the collapse of the structure occurs. We show that the standard definition of collapse based on the collapse mechanism does not capture all situations: the collapse mechanism is sufficient but not necessary for the collapse. We then examine the validity of the static and kinematic theorems of limit analysis under the present definition. We show that the static theorem holds unconditionally while the kinematic theorem holds for Hencky plastic materials and materials with bounded compressive strength. For no-tension materials it generally does not hold; a weaker version is given for this class of materials.
TL;DR: In this article, the authors generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor.
Abstract: The reaction-diffusion systems which are based on an isothermal autocatalytic chemical reaction involving both an autocatalytic step of the (m + 1)th order (A + mB → (m + 1)B) and a decay step of the same order (B → C) have very rich and interesting dynamics. Previous studies in the literature indicate that traveling waves play a key role in understanding these interesting dynamical phenomena. However, there is a lack of rigorous proof of the existence of traveling waves to this system. Here we generalize this isothermal autocatalytic chemical reaction model and provide a rigorous proof of the existence of traveling waves for the resulting reaction-diffusion system which also includes the systems arising from epidemiology and the microbial growth in a flow reactor.
TL;DR: In this article, the authors considered an oscillation model to a plate comprised of two different thermoelastic materials, and proved that the corresponding semigroup associated to this problem is of analytic type.
Abstract: In this paper we consider an oscillation model to a plate comprised of two different thermoelastic materials; that is, we study a transmission problem to thermoelastic plates. Our main result is to prove that the corresponding semigroup associated to this problem is of analytic type.
TL;DR: Fokas and Kapaev as discussed by the authors showed that for simple polygons and simple boundary conditions, the Laplace equation can be mapped to the solution of a Dirichlet problem formulated in the interior of a convex polygon formed by three sides.
Abstract: A general method for studying boundary value problems for linear and for integrable nonlinear partial differential equations in two dimensions was introduced in Fokas, 1997. For linear equations in a convex polygon (Fokas and Kapaev (2000) and (2003), and Fokas (2001)), this method: (a) expresses the solution q(x, y) in the form of an integral (generalized inverse Fourier transform) in the complex κ-plane involving a certain function q(κ) (generalized direct Fourier transform) that is defined as an integral along the boundary of the polygon, and (b) characterizes a generalized Dirichlet-to-Neumann map by analyzing the so-called global relation. For simple polygons and simple boundary conditions, this characterization is explicit. Here, we extend the above method to the case of elliptic partial differential equations in the exterior of a convex polygon and we illustrate the main ideas by studying the Laplace equation in the exterior of an equilateral triangle. Regarding (a), we show that whereas q(κ) is identical with that of the interior problem, the contour of integration in the complex κ-plane appearing in the formula for q(x, y) depends on (x, y). Regarding (b), we show that the global relation is now replaced by a set of appropriate relations which, in addition to the boundary values, also involve certain additional unknown functions. In spite of this significant complication we show that, for certain simple boundary conditions, the exterior problem for the Laplace equation can be mapped to the solution of a Dirichlet problem formulated in the interior of a convex polygon formed by three sides.
TL;DR: In this article, the authors considered a nonlinear system of hyperbolic thermoelasticity in two or three dimensions with DIRICHLET boundary conditions in the case of radial symmetry.
Abstract: In this paper we consider a nonlinear system of hyperbolic thermoelasticity in two or three dimensions with DIRICHLET boundary conditions in the case of radial symmetry. We prove the global existence of small, smooth solutions and the exponential stability.
TL;DR: In this article, the global existence for the following thermoelastic model of type II was studied, and the existence of the model was shown to be stable in the real world.
Abstract: In this chapter, we shall study the global existence for the following thermoelastic model of type II.
TL;DR: In this article, the authors studied the Hilbert boundary-value problem of the theory of analytic functions for an (N + 1)-connected circular domain and derived an exact series-form solution in terms of two analogues of the Cauchy kernel.
Abstract: In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an (N + 1)-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetric Schottky group. We prove the existence theorem for the second (quasimultiplicative) kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem, which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with N + 1 circular holes with electrodes and dielectrics on the walls when the conductor is placed at a right angle to the applied magnetic field.
TL;DR: In this paper, the Glimm estimate for a strictly hyperbolic system of two conservation laws is violated, which is the only known example of such a theorem being violated.
Abstract: We present an example in which the Glimm estimate for a strictly hyperbolic system of two conservation laws is violated.
TL;DR: In this article, the authors studied the Poisson problem and the Helmholtz problem in bounded domains with angular corners in the plane and u = 0 on the boundary, and formulated these as variational problems in weighted Sobolev spaces and proved existence and uniqueness of solutions in what would be weighted counterparts of H 2 ∩ H 1 0.
Abstract: We study the Poisson problem ―Δu = f and the Helmholtz problem ―Δu + λu = f in bounded domains with angular corners in the plane and u = 0 on the boundary. On non-convex domains of this type, the solutions are in the Sobolev space H 1 but not in H 2 in general, even though f may be very regular. We formulate these as variational problems in weighted Sobolev spaces and prove existence and uniqueness of solutions in what would be weighted counterparts of H 2 ∩ H 1 0 . The specific forms of our variational formulations are motivated by, and are particularly suited to, applying a finite element scheme for solving the time-dependent Navier-Stokes equations of fluid mechanics.
TL;DR: In this article, the global existence and uniform stability estimate of mild solutions to the inelastic Boltzmann system for gas mixtures, when initial data are small and decay exponentially in phase space, was studied.
Abstract: We study the global existence and uniform stability estimate of mild solutions to the inelastic Boltzmann system for gas mixtures, when initial data are small and decay exponentially in phase space, and we also provide a general multi-dimensional Bony-type potential which yields a priori weighted two-point correlation estimates in phase space to the mild solutions with finite mass and energy without any smallness restriction.
TL;DR: In this paper, the structure of Riemann solutions for certain systems of conservation laws can be so complicated that the classical constructions are unable to establish global existence and stability, and classically the local solution is found by intersecting two wave curves specified by the riemann data.
Abstract: The structure of Riemann solutions for certain systems of conservation laws can be so complicated that the classical constructions are unable to establish global existence and stability. For systems of two conservation laws, classically the local solution is found by intersecting two wave curves specified by the Riemann data. The intersection point 2000 Mathematics Subject Classification. 35L65.
TL;DR: In this article, the viscoplastic model and the thermoviscous model of rate-dependent non-homogeneous materials with non-oscillating strain-rate sensitivity submitted to simple quasistatic shearing are studied.
Abstract: In this paper we study two models, the viscoplastic model and the thermoviscous model, of rate-dependent non-homogeneous materials with non-oscillating strain-rate sensitivity submitted to simple quasistatic shearing. We prove that the two models are stable by homogenization, i.e. that the equations in both the heterogeneous problems and the homogenized one have the same form, and we give explicit formulas for the homogenized (effective) coefficients. These formulas depend on the initial conditions, but not on the boundary conditions. Our theoretical results are illustrated by a numerical example.
TL;DR: In this paper, a numerical approach to solve the stability equations using a finite difference scheme is presented and analyzed, and an upper bound on the growth rate is derived from numerical analysis of the discrete system which also shows the diffusive slowdown of instabilities.
Abstract: In a recently published article of Daripa and Pasa [Transp. Porous Media (2007) 70:11-23], the stabilizing effect of diffusion in three-layer Hele-Shaw flows was proved using an exact analysis of normal modes. In particular, this was established from an upper bound on the growth rate of instabilities which was derived from analyzing stability equations. However, the method used there is not constructive in the sense that the upper bound derived from actual numerical discretization of the problem could be significantly different from the exact one reported depending on the scheme used. In this paper, a numerical approach to solve the stability equations using a finite difference scheme is presented and analyzed. An upper bound on the growth rate is derived from numerical analysis of the discrete system which also shows the diffusive slowdown of instabilities. Upper bounds obtained by this numerical approach and by the analytical approach are compared. The present approach is constructive and directly leads to the implementation of the numerical approach to obtain approximate solutions in the presence of diffusion. The contributions of the paper are the novelty of the approach and a bound on the growth rates that does not depend on the solution itself.
TL;DR: In this article, the authors consider some expressions for the free energy, already proposed and studied for viscoelastic solids, and adapt them to incompressible viscous fluids, and evaluate the internal dissipation corresponding to each of these various forms of free energy.
Abstract: In this work we consider some expressions for the free energy, already proposed and studied for viscoelastic solids, and adapt them to incompressible viscoelastic fluids. The internal dissipation corresponding to each of these various forms of the free energy is also evaluated. In particular, the form of the minimum free energy for the discrete spectrum model is also considered in order to show its equivalence to some classical free energies.
TL;DR: In this article, a collocation method for solving numerically a singular integral equation with Cauchy and Volterra operators, associated with a proper constraint condition, is described, and optimal convergence rates for the collocation and discrete collocation methods are given in suitable weighted Sobolev spaces.
Abstract: This paper describes a collocation method for solving numerically a singular integral equation with Cauchy and Volterra operators, associated with a proper constraint condition. The numerical method is based on the transformation of the given integral problem into a hypersingular integral equation and then applying a collocation method to solve the latter equation. Convergence of the resulting method is then discussed, and optimal convergence rates for the collocation and discrete collocation methods are given in suitable weighted Sobolev spaces. Numerical examples are solved using the proposed numerical technique.
TL;DR: In this paper, the authors derive various partial spherical means formulas for the 3+1 wave equation, which are appropriate for faces, edges, and corners of "computational domains".
Abstract: We derive various partial spherical means formulas for the 3+1 wave equation. Such formulas, considered earlier by both Weston and Teng, involve only partial integration over a solid angle in addition to history-dependent boundary terms, and are appropriate for faces, edges, and corners of “computational domains”. For example, a hemispherical means formula corresponds to a face (plane boundary). Exploiting the theory of wave front sets for linear operators developed by Hörmander, Warchall has proved theorems which suggest the existence of “one-sided update formulas” for wave equations. We attempt to realize such an update formula via an explicit construction based on our hemispherical means formula. We focus on face points and plane boundaries, but also introduce one-fourth and one-eighth spherical means formulas with most of our arguments going through for a point located on either a domain edge or a corner. Throughout our analysis we encounter a number of, we believe, heretofore unknown identities for classical solutions to the wave equation.
TL;DR: In this article, the authors apply a recently introduced asymptotic method for Mellin convolution integrals to derive three expansions of Appell's hypergeometric function F 2 in decreasing powers of x and y with x/y bounded.
Abstract: The second Appell's hypergeometric function F 2 (a,b,b',c,c';x,y) has a Mellin convolution integral representation in the region ℜ(x + y) 0. We apply a recently introduced asymptotic method for Mellin convolution integrals to derive three asymptotic expansions of F 2 (a, b, b', c, c'; x, y) in decreasing powers of x and y with x/y bounded. For certain values of the real parameters a, b, b', c and c', two of these expansions involve logarithmic terms in the asymptotic variables x and y. Some coefficients of these expansions are given in terms of the Gauss hypergeometric function 3 F 2 and its derivatives.