Showing papers in "Siam Journal on Applied Mathematics in 1983"
TL;DR: In this paper, an analytic inversion formula allowing the reconstruction of a 3D object from x-ray cone-beams is given, for the case where the source of the beams describes a bounded curve satisfying a set of weak conditions.
Abstract: An analytic inversion formula allowing the reconstruction of a three-dimensional object from x-ray cone-beams is given. The formula is valid for the case where the source of the beams describes a bounded curve satisfying a set of weak conditions.
978 citations
TL;DR: In this article, the cooperative effect of the stochastic perturbation and periodic forcing leads to an amplification of the peak of the power spectrum, due to a mechanism that is called stochastically resonance, and a heuristic analysis of the resonance condition is presented.
Abstract: In this paper we study a one-dimensional, nonlinear stochastic differential equation when small amplitude, long-period forcing is applied. The equation arises in the theory of the climate of the earth. We find that the cooperative effect of the stochastic perturbation and periodic forcing lead to an amplification of the peak of the power spectrum, due to a mechanism that we call stochastic resonance. A heuristic analysis of the resonance condition is presented and our analytical findings are confirmed by numerical calculations.
263 citations
TL;DR: In this paper, the authors considered time-dependent potential flows of a liquid with a free surface, with surface tension the force that drives them, and the velocities were found to be proportional to the time from the start of the motion.
Abstract: Time-dependent potential flows of a liquid with a free surface are considered, with surface tension the force that drives them. Two types of configuration are analyzed, in each of which the flow and the free surface are self-similar at all times. One is a model of a breaking sheet of liquid. The other is a model of the flow near the intersection of the free surface of a liquid with a solid boundary. In both flows, the velocities are found to be proportional to $( \sigma /\rho t )^{1/3} $, where $\sigma $ is the surface tension, $\rho $ is the liquid density and t is the time from the start of the motion. Each free surface is determined by converting the problem to an integrodifferential system of equations for the free surface and the potential on it. This system is discretized and solved numerically. On the resulting surfaces there are waves, which are also calculated analytically.
255 citations
TL;DR: In this article, the authors describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities.
Abstract: We describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities. The method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail. The basic idea is to extend the Krylov–Bogolyubov–Mitropolsky method of nonlinear mechanics to bifurcation problems.
209 citations
TL;DR: In this article, the instability of the limited angle Radon transform was studied by constructing the singular value decomposition of this operator. But the instability was not studied in terms of the angular range of projections.
Abstract: The instability of inverting the limited angle Radon transform is studied by constructing the singular value decomposition of this operator. Connections are established with previously known series expansions and with the discrete prolate spheroidal wave functions. Mollification methods are shown to delay, but not prevent, the onset of instability as the angular range of projections is decreased.
204 citations
TL;DR: In this paper, a new set of differential methods for solving the inverse scattering problem associated to the propagation of waves in an inhomogeneous medium is discussed, where the medium equations in the form of a two-component system describing the interaction of rightward and leftward propagating waves are used to identify the medium layer by layer.
Abstract: This paper discusses a new set of differential methods for solving the inverse scattering problem associated to the propagation of waves in an inhomogeneous medium. By writing the medium equations in the form of a two-component system describing the interaction of rightward and leftward propagating waves, the causality of the propagation phenomena is exploited in order to identify the medium layer by layer. The recursive procedure that we obtain constitutes a continuous version of an algorithm first derived by Schur in order to test for the boundedness of functions analytic inside the unit circle. It recovers the local reflectivity function of the medium. Using similar ideas, some other differential methods can also be derived to reconstruct alternative parametrizations of the layered medium in terms of the local impedance or of the potential function.The differential inverse scattering methods turn out to be very efficient since, in some sense, they let the medium perform the inversion by itself and thus...
142 citations
TL;DR: In this article, a model for flame propagation though a condensed combustible mixture in which the limiting component of the mixture melts during the reaction process is presented, and an asymptotic analysis, valid for large activation energies, is employed to derive a two-term expansion for the steady, planar adiabatic flame speed.
Abstract: A model is presented for flame propagation though a condensed combustible mixture in which the limiting component of the mixture melts during the reaction process. An asymptotic analysis, valid for large activation energies, is employed to derive a two-term expansion for the steady, planar adiabatic flame speed. A linear stability analysis is then used to show that for sufficiently large values of the activation energy and/or a special group of melting parameters, the steady, planar solution loses stability to various types of planar and nonplanar pulsating modes. The effect of melting is found to be destabilizing in the sense that these pulsating modes occur for lower values of the activation energy than would be the case for strictly solid fuel combustion.
103 citations
TL;DR: In this article, the authors considered the stability of the displacement of a viscous fluid in a porous medium by a less viscous liquid containing a solute, and formulated a theory for the optimal injection policy, i.e., the policy which minimizes the growth constant of the Saffman-Taylor-Chouke instability.
Abstract: We consider the stability of the displacement of a viscous fluid in a porous medium by a less viscous fluid containing a solute. For the case in which the total amount of injected solute is fixed, we formulate a theory for the optimal injection policy, i.e., the policy which minimizes the growth constant of the Saffman–Taylor–Chouke instability. The result is a nonlinear constrained eigenvalue problem with the viscosity ratio, $\alpha $, of the two fluids and the total solute, N, as parameters. We develop perturbation solutions for large and small N and numerical solutions for all N. Several features of the optimal policy which have been noted or demonstrated empirically by previous workers are proven, and new features of the solution are discussed.
98 citations
TL;DR: In this article, the Frank-Kamenetskii approximation for high-activation energy exothermic chemical reaction was studied, and it was shown that for many such systems there is finite time blow-up of the solution whenever the Frank Kamaneti parameter is greater than the upper bound of the spectrum of the corresponding steady state.
Abstract: Parabolic initial value problems, including the Frank–Kamenetskii approximation for a high-activation energy exothermic chemical reaction, are studied. It is found that for many such systems there is finite time blow-up of the solution whenever the Frank–Kamenetskii parameter $\delta $, or equivalent, is greater than the upper bound $\delta ^ * $ to the spectrum of the corresponding steady state. When the upper bound lies in the spectrum the blow-up time is found to increase as $O( {\delta - \delta ^ * } )^{ - 1 /2} $ as $\delta $ approaches $\delta ^ * $ from above. For $\delta < \delta ^ * $ lower bounds on the initial data are determined above which thermal runaway must also occur.
94 citations
TL;DR: It is shown that if $\lambda $ is an eigenvalue (different from zero) of the generator of the kth reduced chain with multiplicity p then $A( \varepsilon )$ admits p eigenvalues $\lambda ( \varpsilon ).
Abstract: Given a perturbed Markov chain with generator $A( \varepsilon ) = A_0 + \varepsilon A_1 + \varepsilon ^2 A_2 + \cdots $ it is shown that to each time scale $( t,t /\varepsilon , t/\varepsilon ^2 , \cdots )$ is associated a reduced Markov chain. This process of reduction has only finitely many steps. It is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage. It is shown that if $\lambda $ is an eigenvalue (different from zero) of the generator of the kth reduced chain with multiplicity p then $A( \varepsilon )$ admits p eigenvalues $\lambda ( \varepsilon )$ such that $\lambda ( \varepsilon ) = \varepsilon ^k \lambda + o( {\varepsilon ^k } )$.A basis of $\ker A_0 $ (resp. $\ker A_0^ * $) which is robust to the perturbation is constructed.
90 citations
TL;DR: In this paper, a simplified asymptotic model for studying nonlinear detonation waves in chemically reacting fluids which propagate with wave speed close to the acoustical sound speed was developed.
Abstract: The authors develop a simplified asymptotic model for studying nonlinear detonation waves in chemically reacting fluids which propagate with wave speed close to the acoustical sound speed. In this regime the fluid mechanical and chemical phenomena interact substantially with each other. The model provides simplified equations for describing this interaction. When the model is specialized to unidirectional combustion waves advancing into a region of uniform flow, the travelling waves of this model system moving with positive speed coincide with the travelling waves in a qualitative model for such effects introduced previously by the second author. Some interesting new combustion waves with partial burning are also analyzed. The two main assumptions in deriving the asymptotic model are weak nonlinearity and a sufficiently high activation energy for the chemical kinetics.
TL;DR: In this paper, the linear stability of a premixed flame attached to a porous plug burner is examined using activation energy asymptotics, and a dispersion relation is obtained which defines the stability boundaries in the wave-, Lewis-number plane, and the movement of these boundaries is followed as the mass flux is reduced below the adiabatic value.
Abstract: We examine the linear stability of a premixed flame attached to a porous plug burner, using activation energy asymptotics. Limit function-expansions are not an appropriate mathematical framework for this problem, and are avoided. A dispersion relation is obtained which defines the stability boundaries in the wave-, Lewis-number plane, and the movement of these boundaries is followed as the mass flux is reduced below the adiabatic value and the flame moves towards the burner from infinity. Cellular instability is suppressed by the burner, but the pulsating instability usually associated with Lewis numbers greater than 1 is, at first, enhanced. For some parameter values the flame is never stable for all wavenumbers; the Lewis number stability band that exists for the unbounded flame disappears. For sufficiently small values of the stand-off distance the pulsating instability is suppressed.
TL;DR: In this article, a reconstruction method for the attenuation coefficient in a 2D cross section of a 3D object from tomographic data which does not require knowledge of the positions or the intensity of the sources, and the sources may even be inside the object.
Abstract: We give a reconstruction method for the attenuation coefficient in a 2D cross section of a 3D object from tomographic data which does not require knowledge of the positions or the intensity of the sources, and the sources may even be inside the object. Such a situation arises in emission computerized tomography. The method is based on consistency conditions in the range of the relevant integral transforms. The paper contains a detailed description of the algorithm and numerical results for computer generated data.
TL;DR: In this article, a nonlinear system that depends on a number of parametric functions of the spatial concentration of the particles is considered and its stability, its asymptotic behavior under different scalings and its validity from more microscopic description is analyzed.
Abstract: We consider a suspension of particles in a fluid settling under the influence of gravity and dispersing by Brownian motion. A mathematical description is provided by the Stokes equations and a Fokker–Planck equation for the one-particle phase space density. This is a nonlinear system that depends on a number of parametric functions of the spatial concentration of the particles. These functions are known only empirically or for dilute suspensions. We analyze the system, its stability, its asymptotic behavior under different scalings and its validity from more microscopic description. We summarize our conclusions at the end.
TL;DR: In this paper, a theory to explain the spontaneous formation of Mach stems in reacting shock fronts is developed, where a linearized mechanism of instability through radiating boundary waves is analyzed, and an integro-differential scalar conservation law is derived.
Abstract: A theory to explain the experimentally observed spontaneous formation of Mach stems in reacting shock fronts is developed. A linearized mechanism of instability through radiating boundary waves is analyzed. Then through weakly nonlinear asymptotics, an integro-differential scalar conservation law is derived. The asymptotic expansion relates the breakdown of solutions for this scalar equation with the spontaneous formation of the complete triple-shock, slip-line Mach stem configuration.
TL;DR: In this article, a spatially aggregating population model is considered, which provides the homogenizing process due to density-dependent diffusion and the dehomogenizing one due to a certain long-range transport.
Abstract: We consider a spatially aggregating population model which provides the homogenizing process due to density-dependent diffusion and the dehomogenizing one due to a certain long-range transport. The result asserts that, by a balance between two processes, an initial distribution of populations forms itself into a traveling solitary wave pattern for large time, which exhibits phenomenologically a kind of aggregation of a species.
TL;DR: In this article, boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third-order boundary value problems which govern the motion of thin liquid films subject to viscous, capillary and gravitational forces are derived.
Abstract: This paper studies boundary and interior layer phenomena exhibited by solutions of certain singularly perturbed third-order boundary value problems which govern the motion of thin liquid films subject to viscous, capillary and gravitational forces. Precise conditions specifying where and when the third-order derivative terms in the differential equations can be neglected are derived, and improved estimates for the actual solutions in terms of solutions of the lower-order models are constructed. The paper also contains a technique for replacing a third-order problem with an asymptotically equivalent second-order one that may have wider applicability.
TL;DR: In this paper, the effect of small white noise perturbations on a deterministic dynamical system in the plane with (i) an asymptotically stable equilibrium point or limit cycle and (ii) an equilibrium point surrounded by closed trajectories is considered.
Abstract: We consider the problems of the effect of small white noise perturbations on a deterministic dynamical system in the plane with (i) an asymptotically stable equilibrium point or limit cycle and (ii) an equilibrium point surrounded by closed trajectories The mean exit time and the distribution of exit points for each problem is determined by solving singularly perturbed elliptic boundary value problems in domains with closed characteristic boundaries with critical points Uniformly valid asymptotic solutions are constructed for each of the problems For the asymptotically stable equilibrium point, the method of matched asymptotic expansions with the integral condition of Matkowsky and Schuss is employed A method of averaging combined with boundary layer analysis is used for the problem of an equilibrium point surrounded by closed trajectories The influence on the solutions, of the critical points on the boundary, is exhibited and explained An application to the physical pendulum is given Finally our r
TL;DR: In this paper, a method for the fast, accurate and efficient calculation of large classes of seismic rays joining two points x+s and x_R in very general two-dimensional configurations is presented.
Abstract: New methods for the fast, accurate and efficient calculation of large classes of seismic rays joining two points x+s and x_R in very general two-dimensional configurations are presented. The medium is piecewise homogeneous with arbitrary interfaces separating regions of different elastic properties (i.e., differing wave speeds c_P and c_S). In general there are 2^(N+1) rays joining x_S to x_R while making contact with N interfaces. Our methods find essentially all such rays for a given N by using continuation or homotopy methods on the wave speeds to solve the ray equations determined by Snell’s law. In addition travel times, ray amplitudes and caustic locations are obtained. When several receiver positions x^(j)_(R) are to be included, as in a gather, our techniques easily yield all the rays for the entire gather by employing continuation in the receiver location. The applications, mainly to geophysical inverse problems, are reported elsewhere.
TL;DR: In this article, the authors showed that additional lower order terms in the nonlinear heat conduction equation can lead to essentially new behavior of the solutions, and they showed that when the support of the solution is bounded in space and when the solution "dies" in finite time.
Abstract: Additional lower order terms in the nonlinear heat conduction equation can lead to essentially new behavior of the solutions. The statements answer the questions: when the support of the solution is bounded in space and when the solution “dies” in finite time.
TL;DR: In this paper, the authors present a singular perturbation analysis of the fundamental semiconductor device equations which form a system of three second order elliptic differential equations subject to mixed Neumann-Dirichlet boundary conditions.
Abstract: In this paper we present a singular perturbation analysis of the fundamental semiconductor device equations which form a system of three second order elliptic differential equations subject to mixed Neumann–Dirichlet boundary conditions. The system consists of Poisson’s equation and the continuity equations and describes potential and carrier distributions in an arbitrary emiconductor device.The singular perturbation parameter is the minimal normed Debye length of the device under consideration.Using matched asymptotic expansions we demonstrate the occurrence of internal layers at surfaces across which the impurity distribution (appearing as an inhomogeneity of Poisson’s equation) has a jump discontinuity (these surfaces are called “junctions”) and the occurrence of boundary layers at semiconductor-oxide interfaces. We derive the layer-equations and the reduced problem (charge-neutral-approximation) and give existence proofs for these problems. The layer solutions which characterize the solutions of the s...
TL;DR: In this paper, a constructive algorithm is developed to determine whether a given linear pde can be mapped into a constant pde with constant coefficients based on analyzing the infinitesimals of the Lie group of transformations leaving invariant the given pde.
Abstract: A constructive algorithm is developed to determine whether or not a given linear pde can be mapped into a linear pde with constant coefficients The algorithm is based on analyzing the infinitesimals of the Lie group of transformations leaving invariant the given pde As consequences, in two dimensions, necessary and sufficient conditions are given for mapping: (1) a parabolic pde into the heat equation; (2) a hyperbolic pde into the wave equation; (3) an elliptic pde into Laplace’s equation or the Helmholtz equationThe corresponding mappings are given explicitly
TL;DR: In this article, a branch of stable oscillatory solutions of a two predator-one prey competition model is found in the positive octant by a multiparameter bifurcation analysis.
Abstract: A nontrivial branch of stable oscillatory solutions of a two predator–one prey competition model is found in the positive octant by a multiparameter bifurcation analysis. The branch of solutions is “global” in that it is a continuous branch connecting the oscillations in each of the one predator–one prey planes. This result gives analytical verification of the numerical observations of other authors that a branch of nonplanar oscillations crosses the positive octant in certain parameter ranges.
TL;DR: In this article, an inverse problem is considered in which an unknown source term S in the wave equation is to be identified from boundary information, and a number of properties of the solution of an associated initial-boundary value problem are derived and used to develop an existence result for the identification of the unknown source.
Abstract: An inverse problem is considered in which an unknown source term S in the wave equation \[ u_{tt} - u_{xx} = S( u ),\quad x > 0,\quad t > 0 \], is to be identified from boundary information, \[ u( {0,t} ) = f( t ),\qquad - u_x ( {0,t} ) = g( t )\quad t > 0 \]. A number of properties of the solution of an associated initial-boundary value problem are derived and used to develop an existence result for the identification of the unknown source. Results of some primitive numerical experiments are presented to show that the identification is numerically feasible.
TL;DR: In this article, the problem of optimal switching control of a diffusion process with costly switchings leads to a system of fully nonlinear elliptic partial differential equations with implicit obstacles, and the existence, uniqueness and regularity of the solution of this system are obtained.
Abstract: The problem of optimal switching control of a diffusion process with costly switchings leads to a system of fully nonlinear elliptic partial differential equations with implicit obstacles. We obtain results on the existence, uniqueness and regularity of the solution of this system.
TL;DR: This work derives analytic bifurcation formulae and evaluates these for the case of a cubic nonlinearity to determine regions in parameter space where the steady solution is unstable and interprets physiologically the effect of stimulus form.
Abstract: Various nerve axons and other excitable systems exhibit repetitive activity (e.g., trains of propagated impulses) in response to a spatially localized, time-independent stimulus. If the stimulus is too weak, or in some cases too strong, one finds rather a spatially nonuniform, steady response which attenuates with distance from the input site. We consider these stimulus-response properties for a qualitative model of nerve conduction, the FitzHugh–Nagumo (parabolic partial differential) equations. For each stimulus amplitude there is a unique steady state solution. At critical stimulus values this steady solution loses stability and a branch of time periodic (spatially nonuniform) solutions appears via Hopf bifurcation. We associate this with the onset of repetitive activity. We derive analytic bifurcation formulae and evaluate these for the case of a cubic nonlinearity to determine regions in parameter space where the steady solution is unstable. We interpret physiologically the effect of stimulus form; e...
TL;DR: In this article, the problems of diffraction of an E-polarized plane wave by an inhomogeneous imperfectly conducting dielectric wedge or a homogeneous deformation-free wedge with vanishing conductivity are formulated as integral equations and conditions for existence and uniqueness of the solutions of the integral equations are found.
Abstract: The problems of diffraction of an E-polarized plane wave by an inhomogeneous imperfectly conducting dielectric wedge or a homogeneous dielectric wedge with vanishing conductivity are formulated as integral equations. Conditions for existence and uniqueness of the solutions of the integral equations are found. The integral equations are solved by iterative methods. An explicit expression for the far field is found. The series expansion of the far field near the edge of a homogeneous wedge converges very fast for relative permittivities in the interval from one to ten. The physical optical approximation of the far field is poor.
TL;DR: In this paper, the authors display and discuss these weighting functions for the most common reconstruction methods, namely nearest neighbor, linear and cubic spline interpolation, and show that the use of a prefilter prior to sampling or resampling leads to reduced mean square error after reconstruction.
Abstract: The use of a prefilter prior to sampling or resampling can lead to reduced mean square error after reconstruction. This means that the samples should be obtained from the original data by convolution with an optimal local weighting function, whose form depends upon the reconstruction method to be used. We display and discuss these weighting functions for the most common reconstruction methods, namely nearest neighbor, linear and cubic spline interpolation.
TL;DR: The integral equation governing the skin friction in the neighborhood of an incipient separation point on a body in laminar flow at high Reynolds number and finite Mach number is further investigated in this paper.
Abstract: The integral equation governing the skin friction in the neighborhood of an incipient separation point on a body in laminar flow at high Reynolds number and finite Mach number is further investigated For certain values of the parameter, related to the incidence of the body, an additional nonuniqueness is encountered Analytic solutions are presented for the limiting situation in which solutions come to an end A brief study of a related problem, applicable to reattachment, is also made
TL;DR: In this article, a half space of combustible material is subjected to an arbitrary energy flux at the boundary where convection heat loss is also allowed, and an asymptotic analysis of the temperature growth reveals two conditions necessary for ignition to occur.
Abstract: A half space of combustible material is subjected to an arbitrary energy flux at the boundary where convection heat loss is also allowed. An asymptotic analysis of the temperature growth reveals two conditions necessary for ignition to occur. Cases of both large and order unity Lewis number are shown to lead to a nonlinear integral equation governing the thermal runaway. Some global and asymptotic properties of the integral equation are obtained.