Showing papers on "Partial differential equation published in 1984"
TL;DR: The Fokker-Planck Equation: Methods of Solution and Applications as discussed by the authors is a well-known method for solution and application in the field of Optica-Acta.
Abstract: (1984). The Fokker-Planck Equation: Methods of Solution and Applications. Optica Acta: International Journal of Optics: Vol. 31, No. 11, pp. 1206-1207.
1,541 citations
01 Jan 1984
TL;DR: In this paper, an equation for the distribution function describing Brownian motion was first derived by Fokker [11] and Planck [12] and it is shown that expectation values for nonlinear Langevin equations (367, 110) are much more difficult to obtain.
Abstract: As shown in Sects 31, 2 we can immediately obtain expectation values for processes described by the linear Langevin equations (31, 31) For nonlinear Langevin equations (367, 110) expectation values are much more difficult to obtain, so here we first try to derive an equation for the distribution function As mentioned already in the introduction, a differential equation for the distribution function describing Brownian motion was first derived by Fokker [11] and Planck [12]: many review articles and books on the Fokker-Planck equation now exist [15 – 15]
1,412 citations
TL;DR: In this paper, the authors introduced the notion of "viscosity solutions" of scalar nonlinear first order partial differential equations and proved several new facts and reproved various known results in a simpler manner.
Abstract: : Recently M. G. Crandall and P. L. Lions introduced the notion of 'viscosity solutions' of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics. The value of this concept is established by the fact that very general existence, uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application. The notion of a 'viscosity solution' admits several equivalent formulations. Here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner. Moreover, by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper. (Author)
1,243 citations
TL;DR: In this paper, a method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions using forward-time Taylor series expansions including time derivatives of second-and third-order which are evaluated from the governing partial differential equation.
Abstract: A method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions. To produce accurate temporal differencing, the method employs forward-time Taylor series expansions including time derivatives of second- and third-order which are evaluated from the governing partial differential equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The technique is illustrated first in one space dimension. With linear elements and Euler, leap-frog and Crank–Nicolson time stepping, several interesting relations with standard Galerkin and recently developed Petrov–Galerkin methods emerge and the new Taylor–Galerkin schemes are found to exhibit particularly high phase-accuracy with minimal numerical damping. The method is successively extended to deal with variable coefficient problems and multi-dimensional situations.
755 citations
TL;DR: On demontre l'existence de solutions nontriviales de −Δu−λu=u|u| 2 * −2 dans Ω∈R n, u/∂Ω=0 arbitrairement proches des valeurs propres λ k de − Δ:H 0 1,2 (Ω)→H − 1 (λ) as discussed by the authors
Abstract: On demontre l'existence de solutions nontriviales de −Δu−λu=u|u| 2 * −2 dans Ω∈R n , u/∂Ω=0 arbitrairement proches des valeurs propres λ k de −Δ:H 0 1,2 (Ω)→H −1 (Ω)
742 citations
Book•
11 Aug 1984
TL;DR: Inverse Methods for Reflector Imaging as discussed by the authors, the Dirac Delta Function, Fourier Transforms, and Asymptotics are used for direct scattering problems, and the Wave Equation in Two and Three dimensions.
Abstract: First-Order Partial Differential Equations. The Dirac Delta Function, Fourier Transforms, and Asymptotics. Second-Order Partial Differential Equations. The Wave Equation in One Space Dimension. The Wave Equation in Two and Three Dimensions. The Helmholtz Equation and Other Elliptic Equations. More on Asymptotic Techniques for Direct Scattering Problems. Inverse Methods for Reflector Imaging. Each chapter includes references. Index.
439 citations
TL;DR: On demontre la stabilite relative au systeme complet d'equations aux derivees partielles de ces ondes de propagation as discussed by the authors, a.k.a.
Abstract: On demontre la stabilite relative au systeme complet d'equations aux derivees partielles de ces ondes de propagation
345 citations
Book•
01 May 1984TL;DR: In this article, the origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed Basic Fourier, Chebyshev, and Legendre spectral concepts are demonstrated through application to simple model problems Both collocation and tau methods are considered.
Abstract: Origins of spectral methods, especially their relation to the Method of Weighted Residuals, are surveyed Basic Fourier, Chebyshev, and Legendre spectral concepts are reviewed, and demonstrated through application to simple model problems Both collocation and tau methods are considered These techniques are then applied to a number of difficult, nonlinear problems of hyperbolic, parabolic, elliptic, and mixed type Fluid dynamical applications are emphasized
311 citations
TL;DR: In this article, a sequential implicit time-stepping procedure is defined, in which the pressure and Darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a Galerkin finite element and the method of characteristics.
304 citations
TL;DR: On considere le probleme de Cauchy pour l'equation d'onde semi lineaire (ο 2 /∂t 2 -Δ) as discussed by the authors,
294 citations
TL;DR: The main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region to solve Laplace's and the biharmonic equations on irregular regions with smooth boundaries.
Abstract: We present fast methods for solving Laplace’s and the biharmonic equations on irregular regions with smooth boundaries. The methods used for solving both equations make use of fast Poisson solvers on a rectangular region in which the irregular region is embedded. They also both use an integral equation formulation of the problem where the integral equations are Fredholm integral equations of the second kind. The main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region. Fast solvers are then used to compute the extended solution. Aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth.
TL;DR: In this paper, the Caudrey-Dodd-Gibbon equation was found to possess the Painleve property and the Backlund transformation was employed to define a class of p.d.s that identically possesses the painleve properties.
Abstract: The Caudrey–Dodd–Gibbon equation is found to possess the Painleve property. Investigation of the Backlund transformations for this equation obtains the Kuperschmidt equation. A certain transformation between the Kuperschmidt and Caudrey–Dodd–Gibbon equation is obtained. This transformation is employed to define a class of p.d.e.’s that identically possesses the Painleve property. For equations within this class Backlund transformations and rational solutions are investigated. In particular, the sequences of higher order KdV, Caudrey–Dobb–Gibbon, and Kuperschmidt equations are shown to possess the Painleve property.
TL;DR: In this paper, the Ablowitz-Ladik scheme for the nonlinear Schrodinger equation is compared to other known numerical schemes, and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite Fourier (pseudospectral) methods.
TL;DR: In this article, it is shown that the distribution of the solution to a general Ito equation has the same regularity properties as that of a classical diffusion just so long as the coefficients of the white noise are non-degenerate.
Abstract: Publisher Summary This chapter discusses the application of Malliavin's calculus to various problems in stochastic analysis and the theory of partial differential equations. The chapter examines the regularity estimates on the distribution of functionals to which Malliavin's procedure is applicable. It is shown that solutions of Ito stochastic integral equations are smooth functions in the sense of Malliavin's calculus. The distribution of the solution to a general Ito equation has the same regularity properties as that of a classical diffusion just so long as the coefficients of the white noise are non-degenerate. Any other method, of deducing this result and believing that it is a good example to illustrate the power of Malliavin's calculus, is not known.
TL;DR: In this article, the generalized Radon transform was applied to partial differential equations with variable coefficients and a solution to the inversion problem for the attenuated and exponential Radon transforms was provided.
Abstract: We prove that under certain conditions the inversion problem for the generalized Radon transform reduces to solving a Fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation.
We consider applications of the generalized Radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential Radon transforms.
TL;DR: On considere l'existence a t grand de solutions de petite amplitude for des equations d'onde d'ordre 2 non lineaires a 4 dimensions d'espace-temps.
Abstract: On considere l'existence a t grand de solutions de petite amplitude pour des equations d'onde d'ordre 2 non lineaires a 4 dimensions d'espace-temps
TL;DR: In this paper, it was shown that some nonlinear elliptic and parabolic problems are well posed in all of ℝ====== N>>\s without prescribing the behavior at infinity.
Abstract: In this paper we establish that some nonlinear elliptic (and parabolic) problems are well posed in all of ℝ
N
without prescribing the behavior at infinity. A typical example is the following: Let 1
TL;DR: In this article, a method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred is presented, which can be regarded as arising from Bianchi identities expressing the commutativity of Backlund transformations.
Abstract: In this paper we present a systematic method to obtain various integrable nonlinear difference-difference equations and the associated linear integral equations from which their solutions can be inferred. It is argued that these difference-difference equations can be regarded as arising from Bianchi identities expressing the commutativity of Backlund transformations. Applying an appropriate continuum limit we first obtain integrable nonlinear differential-difference equations together with the associated linear integral equations and after a second continuum limit we can obtain the corresponding integrable nonlinear partial differential equations and their linear integral equations. As special cases we treat the difference-difference versions and the differential-difference versions of the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrodinger equation, the isotropic classical Heisenberg spin chain, and the complex and real sine-Gordon equation.
TL;DR: In this paper, the authors introduce a class of models for the motion of a boundary between time-dependent phase domains in which the interface itself satisfies an equation of motion, and formulate local equations of motion as tractable simplifications of the complex nonlocal dynamics that govern moving-interface problems.
Abstract: We introduce a class of models for the motion of a boundary between time-dependent phase domains in which the interface itself satisfies an equation of motion. The intended application is to systems for which competing stabilizing and destabilizing forces act on the phase boundary to produce irregular or patterned structures, such as those which occur in solidification. We discuss the kinematics of moving interfaces in two or more dimensions in terms of their intrinsic geometric properties. We formulate local equations of motion as tractable simplifications of the complex nonlocal dynamics that govern moving-interface problems. Special solutions for dendritic crystal growth and their stability are analyzed in some detail.
TL;DR: Soit H:R n → R, on considere l'equation de Hamilton-Jacobi u t +H(Du)=0 dans R + n+1 ≡ R n ×(0, ∞) as mentioned in this paper.
Abstract: Soit H:R n →R, on considere l'equation de Hamilton-Jacobi u t +H(Du)=0 dans R + n+1 ≡R n ×(0, ∞). On etudie les proprietes des formules de Hopf pour les solutions
Journal Article•
TL;DR: On etudie l'unicite des solutions viscosite non bornees du probleme stationnaire u+H(x,Du)=0 dans R N et du problemsme de Cauchy u t +H(t,x, Du) = 0 dans (O,T)×R N, u(o,x)=u o (x) pour x∈R N as discussed by the authors.
Abstract: On etudie l'unicite des solutions viscosite non bornees du probleme stationnaire u+H(x,Du)=0 dans R N et du probleme de Cauchy u t +H(t,x,Du)=0 dans (O,T)×R N , u(o,x)=u o (x) pour x∈R N
TL;DR: In this article, the sensitivity of inclusive observables in heavy ion collisions to the nuclear equation of state can be tested with the Boltzmann equation, including mean field and Pauli blocking effects, by a method that follows closely the cascade model.
Abstract: The sensitivity of inclusive observables in heavy ion collisions to the nuclear equation of state can be tested with the Boltzmann equation. We solve the Boltzmann equation, including mean field and Pauli blocking effects, by a method that follows closely the cascade model. We find that the inclusive pion production is insensitive to the nuclear equation of state, contrary to recent claims.
TL;DR: In this paper, a pedagogical introduction to relaxation methods for the numerical solution of elliptic partial differential equations is given, with particular emphasis on treating nonlinear problems with delta-function source terms and axial symmetry, which arise in the context of Lagrangian approximations to the dynamics of quantized gauge fields.
Abstract: This article gives a pedagogical introduction to relaxation methods for the numerical solution of elliptic partial differential equations, with particular emphasis on treating nonlinear problems with delta-function source terms and axial symmetry, which arise in the context of effective Lagrangian approximations to the dynamics of quantized gauge fields. The authors present a detailed theoretical analysis of three models which are used as numerical examples: the classical Abelian Higgs model (illustrating charge screening), the semiclassical leading logarithm model (illustrating flux confinement within a free boundary or ''bag''), and the axially symmetric Bogomol'nyi-Prasad-Sommerfield monopoles (illustrating the occurrence of p topological quantum numbers in non-Abelian gauge fields). They then proceed to a self-contained introduction to the theory of relaxation methods and allied iterative numerical methods and to the practical aspects of their implementation, with attention to general issues which arise in the three examples. The authors conclude with a brief discussion of details of the numerical solution of the models, presenting sample numerical results.
TL;DR: In this article, the existence, uniqueness, regularity, and continuous dependence on the data are proved for the solution of the Euler equation for incompressible fluids in a bounded domain in R m.
01 Jan 1984
TL;DR: In this article, the ADHM construction of all instantons, i.e. of all gauge potentials in R 4 with self-dual and square integrable field strengths, is presented.
Abstract: The study of nonlinear partial differential equations remained outside the mainstream of mathematics, because their solution spaces seemed to be rather arbitrary and complicated. But physicists discovered that some of those equations occur naturally, and a closer study by both physicists and mathematicians reveals more and more beautiful structures. Among them, the Yang-Mills equations in four dimensions are today the most outstanding ones. Results on general solutions are still scanty, but quite a lot is known about the more specialized solutions of the self-duality equation for YangMills fields on euclidean four-manifolds. Indeed, this equation already became a valuable tool in the study of differentiable four-manifolds. A basic step in the investigation of this equation was the ADHM construction I) of all instantons, i.e. of all gauge potentials in R 4 with self-dual and square integrable field strengths. The construction uses the cohomology of certain sheaves over the twistdr space, which does not yet belong to the tool kit of many physicists. Thus we shall give an elementary modification of it, which also has the advantage of being easily geoeralizable to self-dual monopoles and calorons. We only consider the gauge group SU(n), but it is easy to specialize to the other classical Lie groups. For the space coordinates and the covariant derivatives we use the standard quaternionic notation
TL;DR: The authors propose an equation d'onde non lineaire permettant des tourbillons topologiques which is analogue a 2 dimensions d'espace de la chaine de spin de Heisenberg isotrope.
Abstract: On propose une equation d'onde non lineaire permettant des tourbillons topologiques qui est analogue a 2 dimensions d'espace de la chaine de spin de Heisenberg isotrope continue
TL;DR: In this article, it is shown how the method of singular manifold analysis obtains the Backlund transform and the Lax pair for the sine-Gordon equation in one space-one time dimension.
Abstract: The sine–Gordon equation in one space‐one time dimension is known to possess the Painleve property and to be completely integrable. It is shown how the method of ‘‘singular manifold’’ analysis obtains the Backlund transform and the Lax pair for this equation. A connection with the sequence of higher‐order KdV equations is found. The ‘‘modified’’ sine–Gordon equations are defined in terms of the singular manifold. These equations are shown to be identically Painleve. Also, certain ‘‘rational’’ solutions are constructed iteratively. The double sine–Gordon equation is shown not to possess the Painleve property. However, if the singular manifold defines an ‘‘affine minimal surface,’’ then the equation has integrable solutions. This restriction is termed ‘‘partial integrability.’’ The sine–Gordon equation in (N+1) variables (N space, 1 time) where N is greater than one is shown not to possess the Painleve property. The condition of partial integrability requires the singular manifold to be an ‘‘Einstein space with null scalar curvature.’’ The known integrable solutions satisfy this constraint in a trivial manner. Finally, the coupled KdV, or Hirota–Satsuma, equations possess the Painleve property. The associated ‘‘modified’’ equations are derived and from these the Lax pair is found.
TL;DR: In this paper, le systeme parabolique δu i /δt-div(|⊇u| p−2 ⊧u i ) = 0 (≤i≤m), avec p>1, ou u i =u i (x,t), ⊇=grad x et x∈Ω⊂R N ouvert.
Abstract: Soit le systeme parabolique δu i /δt-div(|⊇u| p−2 ⊇u i )=0 (≤i≤m), avec p>1, ou u i =u i (x,t), ⊇=grad x et x∈Ω⊂R N ouvert. Pour toute solution de ce systeme avec max{1, 2N/N+2}