Showing papers on "Integro-differential equation published in 1972"
71 citations
TL;DR: In this article, Frechet differentials are introduced to decide when a classical variational principle exists for a given nonlinear differential equation, and they are applied to the steady-state Navier-Stokes equation and the continuity equation.
Abstract: Frechet differentials are introduced to decide when a classical variational principle exists for a given nonlinear differential equation. The formalism is applied to the steady‐state Navier‐Stokes equation and the continuity equation, and no variational principle exists unless u × (∇ × u) = 0 or u· ∇u = 0. The concept of an adjoint equation is extended to nonlinear equations and a variational principle is derived for the Navier‐Stokes equation and its adjoint.
69 citations
TL;DR: In this paper, the authors derived the smallest solution of Bellman's differential equation in the payoff function v(x) for a broad class of cases, and proved that v (x) is the smallest possible solution of this equation.
Abstract: This paper is devoted to the derivation of Bellman's differential equation in the payoff function v(x) for a broad class of cases (Theorems 1 and 2). We prove that v(x) is the smallest solution of this equation (Theorem 3).
26 citations
TL;DR: In this article, a method for the evaluation of molecular multi-centre spin-orbit integrals of Coulomb, hybrid and exchange types using cartesian gaussian functions is presented.
Abstract: A method is presented for the evaluation of molecular multi-centre spin-orbit integrals of Coulomb, hybrid and exchange types using cartesian gaussian functions. It is shown that if gaussian bases are employed the spin-same-orbit integrals reduce to one-electron field integrals and the spin-other-orbit integrals to two-electron field integrals. The final formulae for these integrals are found to be convenient for computer programming. Some relationships among the nuclear attraction, the field and the field-gradient integrals as well as among the electronic repulsion, the two-electron field, and the zero-field splitting integrals are discussed in connection with the spin-orbit integrals. Some calculations are carried out for the radicals CH2 and NH.
21 citations
14 citations
01 Jan 1972
10 citations
TL;DR: In this article, the generalized Langevin equation introduced by Kubo and Mori is formulated as a random integral equation and the existence and uniqueness of the solution, the moments of solution process, a comparison theorem for solution processes, and the Cauchy polygonal approximation to the solution.
Abstract: In this paper, the generalized Langevin equation introduced by Kubo and Mori is formulated as a random integral equation. We consider (1) the existence and uniqueness of the solution, (2) moments of the solution process, (3) a comparison theorem for solution processes, and (4) the Cauchy polygonal approximation to the solution.
10 citations
TL;DR: In this paper, the projection operator was used to obtain the N-body kinetic equation for the one-particle distribution function with the assumption of short-range interaction and low density.
9 citations
01 Jan 1972
9 citations
TL;DR: In the general case in which the potential is non-local, bound-state solutions to the time-independent Schroedinger equation can be obtained numerically by approximating the integrodifferential equation by a matrix equation as mentioned in this paper.
9 citations
TL;DR: In this article, the identity of the kernel in equation (1) was used to solve the problem of equation (2) using Copson's identity and the technique of interchanging the orders of integration.
Abstract: In a recent paper Cooke [1] obtained a solution of the integral equation by using the identity and the technique, first used by Copson, of interchanging the orders of integration and hence reducing the problem to that of the successive solution of two Abel integral equations. It is also shown in [1] that the above identity can also be used to solve the dual series equations The kernel in equation (1) is a particular member of a general class of kernels which the author [6] has shown to be such that the resulting integral equation is directly soluble by using Copson's technique. The particular example of equation (1) is given in [6] and the identity of equation (2) was used by the author [7] to obtain the solution of equation (3).
TL;DR: In this paper, the moment equation method for solving the Boltzmann equation in a Knudsen layer is considered and the calculation of one of the moments of the collision integral is presented.
Abstract: We consider the moment equation method for solving the Boltzmann equation in a Knudsen layer; the calculation of one of the moments of the collision integral is presented.
TL;DR: In this paper, a new method is presented for deriving necessary molecular integrals of spin-orbit coupling and zero-field splitting integrals in benzene, showing that the contribution of the delta function integrals is large.
TL;DR: In this paper, the boundary value problems for the Falkner-skan Equation when β is a small positive number were studied, and the boundary conditions which were considered were and.
Abstract: We shall be concerned with two boundary value problems for the Falkner-Skan Equation when –β is a small positive number. Our interest is in solutions of (1) which exhibit “reversed flow”; that is, solutions f such that f ′( x ) x . The boundary conditions which we wish to consider are and
TL;DR: In this article, the N -body kinetic equation for a classical gas with central-force interaction was extended to include the effects of external forces, and the Boltzmann equation was derived as an approximation to this equation with the assumptions of short-range interactions and low density.
TL;DR: In this article, the authors introduced path integrals using the properties of integral equations of the Fredholm type, a framework adapted for studying the equivalence between the path-integral formalism and other formulations of quantum mechanics.
01 Jan 1972
TL;DR: In this paper, the existence and boundedness of a solution to an integral equation is assumed and conditions are found which ensure the solution has a limit at infinity, where the limit is defined by a set of conditions.
Abstract: Assume the existence and boundedness of a solution to an integral equation. Conditions are found which ensure the solution has a limit at infinity.
01 Jan 1972
TL;DR: The Hopf-Cole solution to Burgers' equation is derived by use of stochastic integrals in this article, where the authors make the change of variables u=cf>x; the motivation for considering
Abstract: In this paper the Hopf-Cole solution to Burgers' equation is derived by use of stochastic integrals. First the equation is written in Hamilton-Jacobi form, and then, following an idea of Freidlin, the solution is differentiated along a Brownian motion. Let « be a solution to the (backwards) Cauchy problem for Burgers' equation ut + uux +| uxx = 0, t < T,xe R, x; the motivation for considering
t + Hl + Uxx = Ü, t along the diffusion governed by the linear part of the operator, i.e., the Brownian motion yT = z + br — bs, s ^r ^ T, where b is a standard Brownian motion starting at zero and zeR and s< T are fixed. Then by Ito's lemma [1, p. 32] Presented to the Society, April 10, 1971 under the title A probabilistic derivation of the Hopf-Cole solution to Burgers' equation; received by the editors April 19, 1971. AMS 1970 subject classifications. Primary 35Q99, 35K55; Secondary 60HO5.
TL;DR: A Fredholm integral equation of the second type is developed for the biopotentials of single cells and methods for handling two singularities arise in the numerical solution.
TL;DR: In this paper, the authors studied the solutions of a wave equation which describes a spin-zero particle in the Coulomb field of a nucleus and analyzed its asymptotic properties.
Abstract: We have studied the solutions of a wave equation which describes a spin‐zero particle in the Coulomb field of a nucleus. An interesting feature of this equation is that the kernel is not of the Fredholm type. The behavior of the momentum space wavefunction for large momentum is not determined solely by the angular momentum state but, as in the cases of the Dirac and Klein‐Gordon equations, it depends on the electric charge as well. Our analysis of the asymptotic properties is based on a Mellin transformation of the momentum space equation. This leads to a singular integral equation with a Cauchy‐type kernel which may be treated by standard methods. The equation is shown to have unique solutions.
TL;DR: In this paper, a solution of an integral equation of the theory of viscoelasticity is proposed based on a creep test on a polymer specimen with an included elastic element, which improves the convergence of the method of approximations.
Abstract: A solution of an integral equation of the theory of viscoelasticity is proposed. This solution, based on a creep test on a polymer specimen with an included elastic element, improves the convergence of the method of approximations.
TL;DR: In this paper, an integro-differential equation approach for solving problems of acoustic scattering and radiation from fluid-immersed elastic bodies is described, where a consistent set of integral and differential equations relating the incident pressure field to the surface pressure and displacements is developed for a submerged elastic spherical shell and results obtained using a numerical solution technique are compared with the classical modal expansion solution.
TL;DR: In this paper, the standing-wave integral equation is considered in the complex momentum plane, and the solution of this equation is used to inquire about some properties of the reactance matrix.
Abstract: The standing-wave integral equation is considered in the complex momentum plane. This equation is solved, and the solution is used to inquire about some properties of the reactance matrix. As an illustrative application, the pole expansion of the reactance matrix and the two-potential problem are then investigated.
01 Jan 1972
TL;DR: In this paper, it was shown that there exist generalizations of Burgers' equation which admit closed-form solutions and the analog of the terms added to the original Burgers equation is the inclusion of forcing terms in the Navier-Stokes equation.
Abstract: We have seen that there exist generalizations of Burgers' equation which admit closed form solutions The analog of the terms added to the original Burgers equation is the inclusion of forcing terms in the Navier-Stokes equation
TL;DR: In this paper, an approximative method for solving the integral equation of the three-dimensional contact problem of the theory of elasticity in the case when the contact domain is represented by two paral lel s t rips of different widths was proposed.
Abstract: We suggest an approximative method for solving the integral equation of the three-dimensional contact problem of the theory of elasticity in the case when the contact domain is represented by two paral lel s t r ips of different widths~ The method of the solution is based on the reduction of the integral equation to an infinite sys tem of l inear algebraic equations for which the application of the reduction method is jus t i fied.
01 Jun 1972
TL;DR: In this paper, a semi-infinite strip held rigidly on its short end is considered, and the effect of material properties on the stress intensity factor is presented, where an exact formulation of the problem in terms of a singular integral equation is provided.
Abstract: A semi-infinite strip held rigidly on its short end is considered. Loads in the strip at infinity (far away from the fixed end) are prescribed. Integral transform technique is used to provide an exact formulation of the problem in terms of a singular integral equation. Stress singularity at the strip corner is obtained from the singular integral equation which is then solved numerically. Stresses along the rigid end are determined and the effect of the material properties on the stress intensity factor is presented. The method can also be applied to the problem of a laminate composite with a flat inclusion normal to the interfaces.
TL;DR: In this paper, coupled integral equations for dynamical scattering were developed from the general integral equation, and the results were given in the forward scattering approximation and in the column approximation, respectively.
Abstract: Abstract The coupled integral equations for dynamical scattering are developed from the general integral equation. The results are given in the forward scattering approximation. Extension to bade scattering is briefly mentioned. Expressions for distorted crystals are derived both in the column approximation and beyond. The formulation is suggested to be very useful as a basis for perturbation methods.
TL;DR: In this paper, a Cauchy system was proposed to reduce the Fredholm integral equation to a time-like variable in which λ plays the role of the time variable.
Abstract: Let a circular flat punch penetrate a finitely thick slab resting on a rigid foundation. Lebedev and Ufliand showed that the determination of the stresses and displacements can be reduced to solving the Fredholm integral equation {ie181-1} We show how to reduce this integral equation to a Cauchy system in which λ plays the role of the time-like variable. Numerical experiments show the computational efficacy.