Showing papers on "Integro-differential equation published in 1999"
TL;DR: In this article, the special quasiperiodic solution of the (2+1)-dimensional Kadometsev-Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the confocal involutive hierarchy associated with the nonlinearized Zakharov-Shabat eigenvalue problem.
Abstract: The special quasiperiodic solution of the (2+1)-dimensional Kadometsev–Petviashvili equation is separated into three systems of ordinary differential equations, which are the second, third, and fourth members in the well-known confocal involutive hierarchy associated with the nonlinearized Zakharov–Shabat eigenvalue problem. The explicit theta function solution of the Kadometsev–Petviashvili equation is obtained with the help of this separation technique. A generating function approach is introduced to prove the involutivity and the functional independence of the conserved integrals which are essential for the Liouville integrability.
267 citations
TL;DR: In this paper, the notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic partial differential equations with two independent variables.
Abstract: The notion of Laplace invariants is generalized to lattices and discrete equations that are difference analogues of hyperbolic partial differential equations with two independent variables. The sequence of Laplace invariants satisfies the discrete analogue of the two-dimensional Toda lattice. We prove that terminating this sequence by zeros is a necessary condition for the existence of integrals of the equation under consideration. We present formulas for the higher symmetries of equations possessing such integrals. We give examples of difference analogues of the Liouville equation.
110 citations
TL;DR: The second-order self-adjoint forms of the transport equation are the even-and odd-parity forms as mentioned in this paper, and a useful alternative to these two forms exists in the form of a second order self-a...
Abstract: The traditional second-order self-adjoint forms of the transport equation are the even- and odd-parity equations. A useful alternative to these equations exists in the form of a second-order self-a...
93 citations
TL;DR: In this paper, the existence and stability of homoclinic type solutions of a bistable integral equation were studied, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity.
53 citations
TL;DR: In this paper, the Cauchy problem for the integral equation ut = −(−∆) β 2u + h(t)u1+α in RN × (0, T ), where 0 < β ≤ 2.
Abstract: In this paper we study the Cauchy problem for the integral equation ut = −(−∆) β 2u + h(t)u1+α in RN × (0, T ), where 0 < β ≤ 2. We obtain some extension of results of Fujita who considered the case β = 2 and h ≡ 1.
52 citations
TL;DR: In this article, a new integral equation method for the numerical solution of the radial Schrodinger equation in one dimension, developed by the authors (1997, J. Comput. Phys.
27 citations
TL;DR: In this paper, a numerical solution for the singular integro-differential equation with an antisymmetric forcing function f(x) (i.e. f(- x) = -f(x)), with end conditions \phi (- 1) = \phi (1) = 0, was obtained.
Abstract: Numerical solution is obtained for the singular integro-differential equation with an antisymmetric forcing function f(x) (i.e. f(- x) = - f(x)), with end conditions \phi (- 1) = \phi (1) = 0, by three different methods, the two first of which presented here, produce the solution as accurate as the one obtained by Frankel (see [7]), recently. The convergence of the first method discussed in section 2, is also analysed.
25 citations
TL;DR: In this paper, it was shown that the hypergeometric function can be expressed as a finite expansion and that the integrand involving this series and a product of Bessel functions satisfies a linear differential equation with coefficients having a power series expansion in the reciprocal of the variable suitable for application of the nonlinear D- and D-transformations.
22 citations
TL;DR: In this article, two techniques, dual reciprocity method (DRM) and direct integral method (DIM), are developed to transform domain integrals to boundary integrals for shear deformable plate bending formulation.
Abstract: In this paper two techniques, dual reciprocity method (DRM) and direct integral method (DIM), are developed to transform domain integrals to boundary integrals for shear deformable plate bending formulation. The force term is approximated by a set of radial basis functions. To transform domain integrals to boundary integrals using the dual reciprocity method, particular solutions are employed for three radial basis functions. Direct integral method is also introduced in this paper to evaluate domain integrals. Three examples are presented to demonstrate the accuracy of the two methods. The numerical results obtained by using different particular solutions are compared with exact solutions.
22 citations
Posted Content•
TL;DR: Geometric interpretation of the Hirota equation as equation describing the Laplace sequence of two-dimensional quadrilateral lattices is presented in this paper, where different forms of the equation are given together with their geometric interpretation.
Abstract: Geometric interpretation of the Hirota equation is presented as equation describing the Laplace sequence of two-dimensional quadrilateral lattices Different forms of the equation are given together with their geometric interpretation: (i) the discrete coupled Volterra system for the coefficients of the Laplace equation, (ii) the gauge invariant form of the Hirota equation for projective invariants of the Laplace sequence, (iii) the discrete Toda system for the rotation coefficients, (iv) the original form of the Hirota equation for the tau-function of the quadrilateral lattice
01 Jan 1999
TL;DR: In this article, the regularity of solutions of the equation (u - u 0)) (t, x) + c(t,x)u x = f( t,x), t, x > 0, where denotes the fractional derivative, is studied.
Abstract: The regularity of solutions of the equation (u - u 0)) (t,x) + c(t,x)u x (t,x) = f(t,x), t, x > 0, where denotes the fractional derivative, is studied. It is shown that if c and f are Holder continuous in either t or x and c is strictly positive, then both u x and (u — u 0) are Holder continuous as well (in either t or x).
TL;DR: In this article, the radiosity equation over the boundary of a polyhedral domain is considered and the solution of the second kind integral equation with non-compact integral operator is piecewise continuous.
Abstract: In the present paper we consider the radiosity equation over the boundary of a polyhedral domain Similarly to corresponding results on the double-layer potential equation, the solution of the second kind integral equation with non-compact integral operator is piecewise continuous The partial derivatives, however, are not bounded In the present paper we derive the first term in the asymptotic expansion of the solution in the vicinity of an edge Note that, knowing this term, optimal mesh gradings can be designed for the numerical solution of this equation
TL;DR: In this paper, the authors used the two-dimensional Green's function for crack problems in potential theory for application to the steady-state electromagnetic problem in three dimensions and validated the results using boundary element modeling of the Mode III, pure (antiplane) shear fracture mechanics problem.
Abstract: 1 Introduction The two-dimensional Green's function for crack problems in potential theory is developed for application to the steady-state electromagnetic problem in three dimensions. The Green's function formulation is for a single, straight crack contained within or intersecting a regular boundary. The Green's function formulation for potential theory is validated using boundary element modeling of the Mode III, pure (antiplane) shear fracture mechanics problem. The Green's function formulation is then used to model the three-dimensional magnetic ®eld for the steady-state current ¯ow through a two-dimensional plate with a plane crack. The BIE formulation is used to compute the normal component of the magnetic ®eld for an arbitrary remote sensing location. This component of the magnetic ®eld can be detected by a superconducting quantum interference device (SQUID) which is under research development for detecting cracks in aerostructures. The required boundary integrals involve the tangential current ¯ow on all boundaries including that for the crack. The explicit form for each of the two crack tip singularity terms is derived. The problem is validated with experimental data taken by a research SQUID system for a plate containing a single, thin slot representing the crack. The use of special Green's functions for two dimensional cracks in elasticity is well established. The ®rst application for straight cracks in anisotropic elastic plates was given by Snyder and Cruse (1975). The Green's function was derived at the same time for micromechanics applications by Sinclair and Hirth (1975). The use of the Green's function to obtain crack tip stress intensity factor path integrals was demonstrated by Snyder and Cruse (1975) and by Stern et al. (1976); a later application of this path integral was given by Kim (1985). Applications of the Green's function method have been made to curved cracks, branched cracks, and multiple cracks, as in Rudolphi and Koo (1985) and Ang (1986). A Green's function for a crack interacting with an inclusion has also been developed by Li and Chudnovsky (1994). A summary of these and other BIE fracture mechanics advances is given in Cruse (1996). 2 The two dimensional electromagnetic potential theory problem The target problem is the prediction of the gradient of the three dimensional magnetic ®eld created by steady current ¯ow through a two dimensional plate containing one or two cracks coming from a hole. The electrical ®eld problem in two dimensions is solved using the Green's function formulation for the two-dimensional crack …
TL;DR: In this paper, the time-dependent density equation has been shown to be equivalent to the timedependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable.
Abstract: The density equation proposed previously for the direct determination of the density matrix, i.e. for the wave mechanics without wave, is extended to a time-dependent case. The time-dependent density equation has been shown to be equivalent to the time-dependent Schrodinger equation so long as the density matrix, included as a self-contained variable, is N-representable. Formally, it is obtainable from the previous time-independent equation by replacing the energy E with . The perturbation theory formulas for the density equation have also been given for both the time-dependent and time-independent cases.
TL;DR: In this paper, the convergence of a difference solution of the Dirichlet problem for the Laplace equation on a rectangle was studied and the authors proved the uniform convergence of its first and purely second divided differences over the whole grid domain to the corresponding derivatives of the sought solution with rate /ι.
Abstract: Lebedev [1] established the convergence of a difference solution of the Laplace equation in metrics of increased smoothness rigorously in the interior of a given domain. The author [3] obtained by the difference method an approximation in Cn of the solution of the Dirichlet problem for the Poisson equation on a closed domain with smooth boundary, which has a second-order convergence with respect to the grid step Λ. In this paper we study the convergence of a difference solution of the Dirichlet problem for the Laplace equation on a rectangle. We prove the uniform convergence of its first and purely second divided differences over the whole grid domain to the corresponding derivatives of the sought solution with rate /ι. We find the effect of angles on the convergence of a mixed difference to a mixed second derivative. The essence of this effect is that there is a weight in the error estimate, which is unremovable in the general case and contains a logarithm of distance to the vertices of the rectangle. We give a special condition under which the logarithmic weight is absent. We say that the function / £ (^(Ω) if / is fc-times differentiable on Ω and all of its derivatives of order k satisfy on Ω the Holder condition with index λ. The norm Ι Ι / Ι Ι < ? * , λ ( Ω ) k determined as usual (see Section 2 in [2]). We denote by c,co,ci,... the constants independent of the postmultipliers. For simplicity we use the same notation for different constants. Suppose that R = { ( x , y ) : 0 < ζ < α, 0 < t/ < 6} is a rectangle, a/6 is rational, 7; are rectangle sides (including the end points), which are numbered counterclockwise starting from the left side (70 = 74), s is the arc length along the boundary 7 = 7iU.. .074 in a counterclockwise direction, s; is the value of s corresponding to the beginning of 7r We consider the Dirichlet problem Au = 0 on R, u = (fj on 7,·, j = 1,2,3,4 (1) where _9^_ &_ dx + dy (Pj are the given functions of 5, and ,eC4,A(7,), 0 < A < 1 , j = 1,2,3,4. (2) Besides, the consistency conditions at the vertices of the rectangle R are satisfied: φ^(3} + 0) = (-1) V?:ta 0), r = 0,1,2. (3) Lemma 1. Under the conditions (2), (3) the classical solution of the problem (1) is u E C4,\\(R), and •V.A. Steklov Institute of Mathematics of the Russian Academy of Sciences, Moscow 117966, Russia The work was supported by the Russian Foundation for the Basic Research (99-01-00045).
TL;DR: In this paper, it was proved that by using Darboux transformation, from a known solution of the sinh-Laplace equation, new solutions of sine-laplace equation (resp. sinh Laplace equation) are obtained explicitly.
Abstract: In this Letter, it is proved that by using Darboux transformation, from a known solution of the sinh-Laplace equation (resp. sine-Laplace equation), new solutions of sine-Laplace equation (resp. sinh-Laplace equation) are obtained explicitly. The corresponding geometrical configuration is the space-like pseudo-spherical line congruence with two focal surfaces of negative constant curvature being space-like and time-like, respectively.
TL;DR: In this article, a general method for the solution of the Landau-Lifshitz equation is presented and applied to the motion of a particle in a uniform electromagnetic field, where the particle is assumed to be a Gaussian.
Abstract: The Landau–Lifshitz equation is derived as a first-order iteration of the Lorentz–Dirac equation for the charged particle. In those cases with null electromagnetic field’s gradient, the Landau-Lifshitz gives the so named Herrera equation. A general method for the solution of the latter equation is presented and applied to the motion of a particle in a uniform electromagnetic field.
TL;DR: In this paper, an express method to approximate trajectories of spaceindependent kinetic equations is developed, which involves a two-step treatment of relaxation through a quasiequilibria located on a line emerging from the initial state in the direction prescribed by the kinetic equation.
Abstract: An express method to approximate trajectories of spaceindependent kinetic equations is developed. It involves a two-step treatment of relaxation through a quasiequilibria located on a line emerging from the initial state in the direction prescribed by the kinetic equation. A test for the Boltzmann equation shows the validity of the method.
TL;DR: In this article, an important new case in the classical problem of the classification of nonlinear hyperbolic equations possessing integrals was discovered and a formula describing the splitting of the right-hand side of such equations with respect to the first derivatives was given.
Abstract: We discover an important new case in the classical problem of the classification of nonlinear hyperbolic equations possessing integrals. In the general (least degenerate) case, in addition, we obtain a formula describing the splitting of the right-hand side of such equations with respect to the first derivatives.
Journal Article•
TL;DR: In this article, it is shown that the geometrical optics analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way.
Abstract: According to Schrödinger’s ideas, classical dynamics of point particles should correspond to the « geometrical optics » limit of a linear wave equation, in just the same way as ray optics is the limit of wave optics. It is shown that the « geometrical optics » analogy leads to the correspondence between a classical Hamiltonian H and a linear wave equation in a natural and general way. In particular, the correspondence is unambiguous also in the case where H contains mixed terms involving momentum and position. This is obtained through a theory of the dispersion relations, which leads to properly define the group velocity and to enlighten its role. Using this latter notion, it is shown that, for a quite general class of potentials, « momentum states » can be defined in a physically more satisfying way than by assuming plane waves. These momentum states are solutions of the time-dependent Schrödinger equation and their amplitude functions move rigidly on a well-defined trajectory. In the case of a spatially uniform force field, such momentum states must have a singularity and the trajectory is defined by Newton’s second law in the given force field. PACS 03.40.Kf – Waves and wave propagation: general mathematical aspects PACS 03.65.–w – Quantum mechanics
01 Nov 1999
TL;DR: The integro-differential equation for diffusive waves is used to develop a 3D-slice imaging algorithm based the on angular spectrum representation in the parallel plate geometry that may be useful for near infrared optical imaging of breast tissue and is applicable to other diagnostics such as ultrasound and microwave imaging.
Abstract: A new integro-differential equation for diffuse photon density waves (DPDW) is derived within the diffusion approximation. The new equation applies to inhomogeneous bounded turbid media. Interestingly, it does not contain any terms involving gradients of the light diffusion coefficient. The integro-differential equation for diffusive waves is used to develop a 3D-slice imaging algorithm based on the angular spectrum representation in the parallel plate geometry. The algorithm may be useful for near infrared optical imaging of breast tissue, and is applicable to other diagnostics such as ultrasound and microwave imaging.
TL;DR: In this paper, an integral transformation T defined by -W f(y)dy, in order to do harmonic analysis on the space of C°° solutions of heat equation is introduced.
Abstract: We introduce an integral transformation T defined by -W f(y)dy, in order to do harmonic analysis on the space of C°° solutions of heat equation. First, the Paley-Wiener type theorem for the transformation T will be given for the C°° functions and distributions with compact support. Secondly, as an application of the transformation the solutions of heat equation given on the torus J" will be characterized. Finally, we represent solutions of heat equation as an infinite series of Hermite temperatures, which are to be defined as the images of Hermite polynomials under the transformation T. §
01 May 1999
TL;DR: In this paper, the authors give sufficient conditions for the existence and uniqueness of a continuous solution of the Urysohn-Volterra equation with hysteresis, which is the case of the continuous solution presented in this paper.
Abstract: In this paper we give some sufficient conditions for the existence and uniqueness of a continuous solution of the system of Urysohn-Volterra equation with hysteresis.
01 Jan 1999
TL;DR: The method of determination of steady-state processes in circuits of voltage converters, which operate on periodical load, is considered in this article, based on using Lyapunov transform and, introducing additional independent variables of time.
Abstract: The method of determination of steady-state processes in circuits of voltage converters, which are operating on periodical load, is considered This method is based on using Lyapunov transform and, introducing additional independent variables of time At first the initial differential equation with periodical coefficients is transformed into the equation with constant coefficients Then, this equation is expanding to the equation in partial derivatives variables with several independent variables of time In the obtained equation the forcing function is a periodical function with these variables The solution of equation in partial derivatives is produced with the help of multidimensional Laplace transform