Showing papers on "Integro-differential equation published in 1983"
TL;DR: In this paper, it is shown that it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals, which allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.
Abstract: It is not always possible and sometimes not even advantageous to write the solutions of a system of differential equations explicitly in terms of elementary functions. Sometimes, though, it is possible to find elementary functions which are constant on solution curves, that is, elementary first integrals. These first integrals allow one to occasionally deduce properties that an explicit solution would not necessarily reveal.
279 citations
TL;DR: In this paper, the linear integral equation for the solutions of the Korteweg-de Vries (KdV) equation is derived from the direct linearization of a general nonlinear difference-difference equation.
242 citations
TL;DR: In this article, the Kadomtsev-Petviashvili equation is linearized by a suitable extension of the inverse scattering transform, and the lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t.
Abstract: The Kadomtsev-Petviashvili equation, a two-spatial-dimensional analogue of the Korteweg-deVries equation, arises in physical situations in two different forms depending on a certain sign appearing in the evolution equation. Here we investigate one of the two cases. The initial-value problem, associated with initial data decaying sufficiently rapidly at infinity, is linearized by a suitable extension of the inverse scattering transform. Essential is the formulation of a nonlocal Riemann-Hilbert problem in terms of scattering data expressible in closed form in terms of given initial data. The lump solutions, algebraically decaying solitons, are given a definite spectral characterization. Pure lump solutions are obtained by solving a linear algebraic system whose coefficients depend linearly on x, y, t. Many of the above results are also relevant to the problem of inverse scattering for the so-called time-dependent Schrodinger equation.
204 citations
TL;DR: Using Hirota's technique, a Backlund transformation for the nonlinear Schrodinger equation is obtained in this article, where the arbitrary constants which appear in the Backlund transform are found for the transformation between soliton solutions.
107 citations
01 Jul 1983
TL;DR: In this article, a vector-scalar potential approach and incorporating symmetry through group theory improves the general 3D integral equation solution, which has been particularly useful for thin sheet integral equation formulations.
Abstract: Three-dimensional (3D) interpretation of electromagnetic (EM) dsta is still in its infancy, due to a lack of practical numerical solutions for the forward problem. However, a number of algorithms for simulating the responses of simple 3D models have been developed over the last ten years, and they have provided important new insight. Integral equation methods have been more successful than differential equation methods, because they require calculating the electric field only in small anomalous regions, rather than throughout the earth. Utilizing a vector-scalar potential approach and incorporating symmetry through group theory improves the general 3D integral equation solution. Thin-sheet integral equation formulations have been particularly useful. Much recent research has focused on hybrid methods, which are finite element differential equation solutions within a mesh of limited extent, with boundary values determined by integrating over the interior fields. An elegant eigencurrent technique has been developed for calculating the transient response of a thin 3D sheet in free space, but general 3D time domain responses have only been calculated by Fourier transforming frequency domain results. Direct time domain calculations have been carried out only for 2D bodies.
80 citations
TL;DR: In this article, the integral equation technique was used to study low-frequency electromagnetic perturbation in the case of tearing modes, where the problem can be reduced to the simultaneous solution of an integral and a differential equation.
Abstract: The integral equation technique previously developed for electrostatic drift waves to study low‐frequency electromagnetic perturbation is extended. When σe≫σi (as is the case for tearing modes) the problem can be reduced to the simultaneous solution of an integral and a differential equation. Using a Fourier representation for φ(x), a differential equation is derived from Ampere’s law for a modified Green’s function that contains the magnetic effects. This equation is solved simultaneously with an integral equation (corresponding to the quasineutrality condition in k space) to obtain the eigenvalues and corresponding eigenfunctions. When applied to the study of microtearing modes this method gave, for the same values of the parameters, larger growth rates than those of the usual differential approximation.
73 citations
TL;DR: In this paper, an analytic-numeric real axis integration technique has been developed for such integrals and it is combined with piecewise sinusoidal expansions to solve the Fredholm integral equation for the unknown current density.
Abstract: Printed circuit antennas are becoming an integral part of imaging arrays in microwave, millimeter, and submillimeter wave frequencies. The electrical characteristics of such antennas can be analyzed by solving integral equations of the Fredholm first kind. The kernel involves Sommerfeld integrals which are particularly difficult to solve when source and field points lie on an electrical discontinuity, as it occurs in the determination of the characteristics of printed circuit antennas. An analytic‐numeric real axis integration technique has been developed for such integrals and it is combined with piece‐wise sinusoidal expansions to solve the Fredholm integral equation for the unknown current density.
70 citations
TL;DR: In this paper, a new approach to the theory of the sine-Gordon equation is presented, which provides a simple and unified view of the three known methods for solution, namely Hirota's method of generating N-soliton solutions, the inverse scattering transform, and Backlund transformations.
58 citations
TL;DR: In this article, a direct biharmonic boundary integral equation (BBIE) method was used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.
Abstract: Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible Newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green’s Theorem is used to reformulate the differential equation as a pair of coupled integral equations which are applied only on the boundary of the solution domain.
48 citations
34 citations
TL;DR: In this paper, a least-squares method of solving the first-order Boltzmann equation is given, and it is used to derive directly a maximum principle which is shown to be equivalent to the well-known maximum principles for the second-order even-and odd-parity BoltZmann equations.
TL;DR: In this paper, the authors prove the existence of a set of initial data to which correspond solutions of the nonlinear Klein-Gordon equation with a polynomial nonlinear term, which converge asymptotically, when t→+∞, to solutions of a linear Klein Gordon equation.
Abstract: We prove the existence of a set of initial data to which correspond solutions of the nonlinear Klein-Gordon equation with a polynomial nonlinear term, which converge asymptotically, when t→+∞, to solutions of the linear Klein-Gordon equation.
TL;DR: In this article, a new linear integral equation is proposed, yielding an exact linearization of the integrable discrete versions of e.g. the nonlinear Schrodinger equation, the isotropic Heisenberg spin chain and the complex sine-Gordon equation.
TL;DR: In this paper, the initial value problem associated with the Benjamin-Ono equation, a physically significant nonlinear singular-integrodifferential equation, is linearized and an alternative method which is called the direct linearizing transform is introduced.
TL;DR: In this article, it was shown that eigenvalue estimation techniques based on an integral equation formulation are more effective than variational methods for finding upper and lower bounds for the natural frequencies of vibration of a circular membrane with stepped radial density.
TL;DR: In this article, a new representation for the Schrodinger equation for the three-body problem in the molecular state approach is built, which “kills” cross derivatives in the Schröter equation, and this new representation has a simple form and good asymptotic properties.
TL;DR: In this article, the solution of Fokker-planck equation using Trotter's formula is discussed and the method yields an integral representation amenable to approximations.
Abstract: Solution of Fokker-Planck equation using Trotter's formula is discussed The method is illustrated on the linear Fokker-Planck equation and the Ornstein-Uhlenbeck solution is obtained For the case of a general nonlinear Fokker-Planck equation the method yields an integral representation amenable to approximations In the lowest order approximation Suzuki's scaling result emerges Physical interpretation and limitations of the approximations are also discussed
Dissertation•
01 Jan 1983
TL;DR: In this article, it was shown that the integral equation can be solved for ℎ in the space ℒ μ, p of [3] for 1 ≤ p 0, and that for these spaces, which include L 2 (0, ∘), f is given by the simpler formula
Abstract: The integral equation of the title is It was studied in [4], though h(x) was written as x -1 g(x -1 ) there, and using a method involving orthogonal Watson transformations, it was shown there that if h ∈ L 2 (0, ∞), then the equation has a solution f ∈ L 2 (0, ∞), and that / is given by In this paper, using the techniques of [3], we shall show that the equation can be solved for ℎ in the space ℒ μ, p of [3] for 1 ≤ p 0, and that for these spaces, which include L 2 (0, ∘), f is given by the simpler formula We shall further show that these results can be extended to the spaces ℒ w, μ, p of [3]. This forms the content of our theorem below.
TL;DR: In this paper, a direct approach to the solution of singular integral equations with Cauchy principal value integrals is proposed, which replaces the integrals by a quadrature formula, and satisfies the resulting equation at a discrete set of collocation points.
Abstract: A direct approach to the solution of singular integral equations with Cauchy principal value integrals is to replace the integrals by a quadrature formula, and to satisfy the resulting equation at a discrete set of collocation points. For the canonical equation, i.e. the integral equation of the first kind with only the principal part, interpolatory polynomials are explicitly constructed from the discrete values of the solution obtained from Gauss-Chebyshev and Lobatto-Chebyshev formulae, and are shown to converge to the analytic solution under fairly reasonable conditions.
TL;DR: In this paper, a form of the chiral equation for which first integrals can be written explicitly is considered, and a symplectic structure, the Lagrangian and first integral in involution, is found.
Abstract: We deal with a form of the chiral equation, for which first integrals can be written explicitly. For these equations, we find a symplectic structure, the Lagrangian and first integrals in involution.
TL;DR: In this paper, the pressure for systems with small z, whose interaction is stable and rapidly decreasing, is written as a functional of the correlation functions, and the differential equation of state is obtained for such systems.
Abstract: The pressure for systems with small z, whose interaction is stable and rapidly decreasing, is written as a functional of the correlation functions. The differential equation of state is obtained for such systems. The integration of this equation leads to the generalized virial expansion.
TL;DR: On considere l'existence, l'unicite et le caractere finie des solutions aleatoires d'une equation integrale mixte non lineaire aleatoire de type Voltena-Fredholm.
Abstract: On considere l'existence, l'unicite et le caractere finie des solutions aleatoires d'une equation integrale mixte non lineaire aleatoire de type Voltena-Fredholm