Showing papers on "Integro-differential equation published in 2000"
TL;DR: Algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace transformation and procedures for generating and solving systems of differential equations in masses and momenta for master integrals are shown.
Abstract: We describe a new method of calculation of generic multiloop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace transformation. We also describe new algorithms for the identification of master integrals and the reduction of generic Feynman integrals to master integrals, and procedures for generating and solving systems of differential equations in masses and momenta for master integrals. We apply our method to the calculation of the master integrals of massive vacuum and self-energy diagrams up to three loops and of massive vertex and box diagrams up to two loops. Implementation in a computer program of our approach is described. Important features of the implementation are: the ability to deal with hundreds of master integrals and the ability to obtain very high precision results expanded at will in the number of dimensions.
1,023 citations
TL;DR: In this article, the authors show how the use of tools already employed in inclusive calculations can be suitably extended to the computation of loop integrals appearing in the virtual corrections to exclusive observables, namely two-loop four-point functions with massless propagators and up to one off-shell leg.
945 citations
TL;DR: In this article, a non-Markovian generalization of the Chapman−Kolmogorov transition equation for continuous time random processes governed by a waiting time distribution is investigated.
Abstract: A non-Markovian generalization of the Chapman−Kolmogorov transition equation for continuous time random processes governed by a waiting time distribution is investigated. It is shown under which conditions a long-tailed waiting time distribution with a diverging characteristic waiting time leads to a fractional generalization of the Klein−Kramers equation. From the latter equation a fractional Rayleigh equation and a fractional Fokker−Planck equation are deduced. These equations are characterized by a slow, nonexponential relaxation of the modes toward the Gibbs−Boltzmann and the Maxwell thermal equilibrium distributions. The derivation sheds some light on the physical origin of the generalized diffusion and friction constants appearing in the fractional Fokker−Planck equation.
99 citations
TL;DR: In this article, an n-dimensional complex Ginzburg-Landau equation is transformed to a sine-Gordon equation, sinh-Gordon equations and other equations, which depends only on one function, ζ, and can be solved.
Abstract: Applying the function transformation method, an n-dimensional complex Ginzburg–Landau equation is transformed to a sine-Gordon equation, sinh-Gordon equation and other equations, which depends only on one function, ζ, and can be solved. The general solution of the equations in ζ leads to a general soliton solution of an n-dimensional complex Ginzburg–Landau equation. It contains some interesting specific solutions such as the N multiple solitons, the propagational breathers and the quadric solitons.
96 citations
TL;DR: In this paper, a new integrable equation and the Schwarz derivative in (2+1) dimensions were introduced and the existence of the Lax pair for an equation which has the relation to the Schwarz-Korteweg-de Vries (SKdV) equation.
Abstract: In this note, we shall introduce a new integrable equation and the Schwarz derivative in (2+1) dimensions. First we show the existence of the Lax pair for an equation which has the relation to the Schwarz–Korteweg–de Vries (SKdV) equation. Next we derive a new equation in (2+1) dimensions by using a well-known higher-dimensional manner to the Lax pair for the SKdV equation. The (2+1) dimensional Schwarz derivative is defined here. Finally we briefly discuss various results which we have obtained about the new equation.
64 citations
TL;DR: In this article, it was shown that if f = f(t, x), h1 = h1(x), and h2 = h2(t) are Hölder-continuous and f(0, 0) = 0, then there is a solution such that Dα t u and D β xu are höldercontinuous as well.
Abstract: The equation D t (u− h1) +D x(u− h2) = f, 0 < α, β < 1, t, x ≥ 0, (∗) where Dα t and D β x are fractional derivatives of order α and β is studied. It is shown that if f = f(t, x), h1 = h1(x), and h2 = h2(t) are Hölder-continuous and f(0, 0) = 0, then there is a solution such that Dα t u and D β xu are Höldercontinuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to (∗). Finally the solution of (∗) with f = 1 is studied.
63 citations
01 Oct 2000
TL;DR: In this article, the authors compute the O(ϵ)-term of a particular combination of on-shell massless planar double box integrals, which appears in the tensor reduction of 2 → 2 scattering amplitudes at two loops.
Abstract: The calculation of exclusive observables beyond the one-loop level requires elaborate techniques for the computation of multi-leg two-loop integrals. We discuss how the large number of different integrals appearing in actual two-loop calculations can be reduced to a small number of master integrals. An efficient method to compute these master integrals is to derive and solve differential equations in the external invariants for them. As an application of the differential equation method, we compute the O (ϵ)-term of a particular combination of on-shell massless planar double box integrals, which appears in the tensor reduction of 2 → 2 scattering amplitudes at two loops.
46 citations
TL;DR: An algorithm based on Adomian's decomposition method is developed to approximate the solution of the nonlinear integro-differential equation.
37 citations
TL;DR: In this paper, the authors reduce the problem to the computation of 1-dimensional integrals and present an algorithm for computing these integrals, and consider appropriate adaptive approximations and study the induced error.
29 citations
TL;DR: In this article, an integro-differential equation of Prandtl's type and a collocation method for its approximate solution are studied in weighted spaces of continuous functions. But the authors focus on the problem of finding the optimal solution.
Abstract: An integro-differential equation of Prandtl's type and a collocation method as well as a collocation-quadrature method for its approximate solution is studied in weighted spaces of continuous functions.
26 citations
01 Jan 2000
TL;DR: In this article, the existence and stability of a semilinear parabolic equation, a semi-linear functional equation, and a nonlinear singular version of the Gronwall and Bihari inequalities were proved.
Abstract: We prove global existence and stability results for a semilinear parabolic equation, a semilinear functional equation and a semilinear integral equation using an inequality which may be viewed as a nonlinear singular version of the well known Gronwall and Bihari inequalities.
TL;DR: In this article, an attempt has been made to carry over known results for the finite element Galerkin method for a time dependent parabolic equation with nonsmooth initial data to an integro-differential equation of parabolic type.
Abstract: In this paper, an attempt has been made to carry over known results for the finite element Galerkin method for a time dependent parabolic equation with nonsmooth initial data to an integro-differential equation of parabolic type. More precisely, for the homogeneous problem a standard energy technique in conjunction with a duality argument is used to obtain an L2-error estimate of order \(\) for the semidiscrete solution when the given initial function is only in L2. Further, for the nonhomogeneous case with zero initial condition, an error estimate of order \(\) uniformly in time is proved, provided that the nonhomogeneous term is in L∞(L2). The present paper provides a complete answer to an open problem posed on p. 106 of the book Finite Element Methods for Integro-differential Equations by Chen and Shih.
01 Jan 2000
TL;DR: In this paper, the inverse scattering transform method for the nonlinear evolution equation (Ut + 'Uux)x + U = 0 (the Vakhnenko equation) is discussed.
Abstract: The formulation of the inverse scattering transform method is discussed for the nonlinear evolution equation (Ut + 'Uux)x + U = 0 (the Vakhnenko equation). It is shown that the equation system for the inverse scattering problem associated with the Vakhnenko equation can not contain the isospectral Schrodinger equation. The exact two soliton solutions are obtained by means of the use of elements of the inverse scattering problem for the KdV equation.
01 Jan 2000
TL;DR: In this paper, the authors give some properties of solutions of the iterative functional f n+1 (x)+⋯+a 0 x, equation considering its characteristic equation, and auseful method to discuss the general case is detailed described for the case n = 2.
Abstract: in this paper we give some properties of solutions of the iterative functional f n+1 (x)+⋯+a 0 x, equation considering its characteristic equation. Auseful method to discuss the general case is detailed described for the case n = 2.
TL;DR: In this paper, it was shown that the Lame's equation can be reduced to the hypergeometric equation, where the characteristic exponents of this equation are expressed in terms of elementary functions of the parameter $\kappa, and an analytical condition for parametric amplification is obtained.
Abstract: It is shown that the Lame's equation ${d^{2}\over d z^{2}}X+\kappa^{2} cn^{2}(z, {1\over \sqrt 2})X=0$ can be reduced to the hyper-geometric equation. The characteristic exponents of this equation are expressed in terms of elementary functions of the parameter $\kappa$. An analytical condition for parametric amplification is obtained.
TL;DR: In this article, the authors presented a solution methodology that utilizes symbolic computations to obtain analytic solutions of some nonlinear evolution equations by balancing the nonlinear and the dispersive effects, and demonstrated that the solution method is demonstrated by obtaining solutions to Burgers' equation, nonlinear heat equation, the modified KdV equation, and the Kuramoto-Sivashinsky equation.
Abstract: In this paper, we present a solution methodology that utilizes symbolic computations to obtain analytic solutions of some nonlinear evolution equations by balancing the nonlinear and the dispersive effects. The solution method is demonstrated by obtaining solutions to Burgers' equation, the nonlinear heat equation, the modified KdV equation, and the Kuramoto-Sivashinsky equation.
TL;DR: It is found here that the simple hypernetted chain approximation when applied at the three- particle level can yield a correct semiquantitative description of both the pair and triplet structure of the supercooled state, capturing features which escape more sophisticated closures implemented on the two-particle level.
Abstract: A simple model of glass-forming liquid modeled via Dzugutov's pair potential is studied by means of the triplet hypernetted chain approximation as formulated by Attard [P. Attard, J. Chem. Phys. 93, 7301 (1990)]. This system, which is known to be mostly dominated by microscopic icosahedral ordering, eludes a correct description in terms of classical two-body integral equation theories. However, it is found here that the simple hypernetted chain approximation when applied at the three-particle level can yield a correct semiquantitative description of both the pair and triplet structure of the supercooled state, capturing features which escape more sophisticated closures implemented on the two-particle level.
TL;DR: In this article, the Dirac equation is studied for a sufficiently large class of Lemaįtre-Tolman-Bondi cosmological models and the separation of spatial and temporal dependence is studied.
Abstract: The Dirac equation is studied for a sufficiently large class of Lemaįtre—Tolman—Bondi cosmological models. While the angular equation (whose solution is known) separates directly, the spatial and temporal dependence de-couples only after a suitable separation procedure. The separated time equation is integrated by series. The separated spatial equation still depends on an arbitrary function relative to the integration of the cosmological model.
TL;DR: In this paper, the long-time dynamics of a system of reaction-diffusion equations arising from the viscous forced Burgers equation (u + uu x -uuxx F) is studied.
Abstract: We study numerically the long-time dynamics of a system of reaction-diffusion equations that arise from the viscous forced Burgers equation (u + uu x -uuxx F) A nonlinear transformation introduced by Kwak is used to embed the scalar Burgers equation into a system of reaction diffusion equations The Kwak transformation is used to determine the existence of an inertial manifold for the 2-D Navier-Stokes equation We show analytically as well as numerically that the two systems have a similar,
TL;DR: In this article, a generalization of a one-dimensional fractional kinetic equation with a multivariate Levy noise has been considered, where the Levy noise is assumed to have independent components.
Abstract: Starting from the integral Chapman-Kolmogorov equation and the Langevin equations with a multivariate Levy noise, we consider possible generalization of a one-dimensional fractional kinetic equations. It is assumed that the Levy noise has independent components. As the result, we get fractional Fokker-Planck equation for the probability density function in a phase space and fractional Einstein-Smoluchowski equation for the probability density function in a real space. We consider force-free superdiffusion and relaxation described by fractional equations and carry out numerical simulation of these processes by solving the corresponding Langevin equations. Analytical and numerical results agree quantitatively.
TL;DR: Using new inverses and other techniques from Laplace transform theory, a new class of three-parameter definite integrals, that yield to exact evaluation, is generated and it is shown that these integrals evaluate to simple closed-form expressions.
TL;DR: A general form of the Laplace equation using non-diagonal tensors for pressure and surface tension typical of non-spherical interfaces in external fields was derived in this paper.
01 Jan 2000
TL;DR: In this paper, the authors proved the existence of period ic traveling wave solutions to the forced generalized nearly concentric Korteweg-de Vries equation with periodic boundary conditions.
Abstract: This paper is concernedwith period ic traveling wave solutions of the forced generalizednearly concentric Korteweg-d e Vries equation in the form of (uη + u/(2η) + (f (u))ξ + uξξξ)ξ + uθθ/η 2 = h0. The authors first convert this equation into a forced generalizedKad omtsev-Petviashvili equation, (ut + (f (u))x + uxxx)x + uyy = h0, and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is establishedby using the Green's function method . The integral representations generate compact operators in a Banach space of real-valued continuous functions. The Schaud er's fixedpoint theorem is then usedto prove the ex- istence of nonconstant solutions to the integral equations. Therefore, the existence of period ic traveling wave solutions to the forcedgeneralizedKP equation, andhence the nearly concentric KdV equation, is proved. Keyword s andphrases. Existence theorem, traveling wave solution, forcedgeneralized nearly concentric Korteweg-de Vries equation.
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TL;DR: The partial differential equation (PDE) of the Marquis de Laplace (1749-1827) is one of the most important equations in mathematics which has wide applications to a number of topics relevant to mathematical physics and engineering as discussed by the authors.
Abstract: The partial differential equation which is identified with the name of Pierre Simon Marquis de Laplace (1749–1827) is one of the most important equations in mathematics which has wide applications to a number of topics relevant to mathematical physics and engineering. In the area of mathematical physics, Laplace’s equation can be used to formulate the general class of problems associated with the theory of gravitation, electrostatics, dielectrics and magnetostatics. In all these applications, the associated fields can be expressed as a gradient of a potential. For example, in the theory of gravitation, the force of attraction associated with matter is expressed as the gradient of a gravitational potential; in electrostatics, the electric vector is expressed as the gradient of an electrostatic potential; in magnetostatics, the magnetic vector is expressed in terms of a magnetostatic potential; in electricity, the conduction current vector is derived from the gradient of a potential function. This ability to express fundamental vector fields of interest to particular areas is an underlying theme in all applications which culminate in the development of Laplace’s equation.
TL;DR: In this paper, an explicit representation of the solution of the two-dimensional heat equation through solutions of boundary integral equations is given, using this representation we get a semi-discretization, in time, where a sequence of boundary-integral equations must be solved.
Abstract: An explicit representation of the solution of the two-dimensional heat equation through solutions of boundary integral equations is given. Using this representation we get a semi-discretization, in time, where a sequence of boundary integral equations must be solved. For the last ones the collocation quadrature method can be used.