Showing papers on "Integro-differential equation published in 1988"
TL;DR: A method is considered for the integration in time of a partial integro-differential equation using the discretization technique patterned after the idea of Ch.
Abstract: A method is considered for the integration in time of a partial integro-differential equation. The discretization technique employed is patterned after an idea of Ch. Lubich. Error bounds are derived for both smooth and nonsmooth initial data.
149 citations
Book•
01 Jan 1988
TL;DR: In this article, the authors consider the problem of half-plane problems and compare Jones's method with the Wiener-Hopf procedure, and show that the latter is more efficient than the former.
Abstract: Complex Variable and Fourier Transforms: 1.1 Introduction 1.2 Complex variable theory 1.3 Analytic functions defined by integrals 1.4 The Fourier integral 1.5 The wave equation 1.6 Contour integrals of a certain type 1.7 The Wiener-Hopf procedure Miscellaneous examples and results I Basic Procedures: Half-Plane Problems: 2.1 Introduction 2.2 Jones's method 2.3 A dual integral equation method 2.4 Integral equation formulations 2.5 Solution of the integral equations 2.6 Discussion of the solution 2.7 Comparison of methods 2.8 Boundary conditions specified by general functions 2.9 Radiation-type boundary conditions Miscellaneous examples and results II Further Wave Problems: 3.1 Introduction 3.2 A plane wave incident on two semi-infinite parallel planes 3.3 Radiation from two parallel semi-infinite plates 3.4 Radiation from a cylindrical pipe 3.5 Semi-infinite strips parallel to the walls of a duct 3.6 A strip across a duct Miscellaneous examples and results III Extensions and Limitations of the Method: 4.1 Introduction 4.2 The Hilbert problem 4.3 General considerations 4.4 Simultaneous Wiener-Hopf equations 4.5 Approximate factorization 4.6 Laplace's equation in polar co-ordinates Miscellaneous examples and results IV Some Approximate Methods: 5.1 Introduction 5.2 Some problems which cannot be solved exactly 5.3 General theory of a special equation 5.4 Diffraction by a thick semi-infinite strip 5.5 General theory of another special equation 5.6 Diffraction by strips and slits of finite width Miscellaneous examples and results V The General Solution of the Basic Wiener-Hopf Problem: 6.1 Introduction 6.2 The exact solution of certain dual integral equations Miscellaneous examples and results VI Bibliography Index.
131 citations
TL;DR: In this article, general recurrence formulas for various types of one-and two-electron molecular integrals over Cartesian Gaussian functions are derived by introducing basic integrals, which are capable of dealing with any spatial operators in the nonrelativistic forms of the relativistic wave equations, those with the kernel of the Fourier transform, and any order of their derivatives with respect to the function centers in the above integrals.
Abstract: General recurrence formulas for various types of one‐ and two‐electron molecular integrals over Cartesian Gaussian functions are derived by introducing basic integrals. These formulas are capable of dealing with (1) molecular integrals with any spatial operators in the nonrelativistic forms of the relativistic wave equations, (2) those with the kernel of the Fourier transform, (3) those with arbitrarily defined spatial operators so far as the integrals can be expressed in terms of the basic integrals, and (4) any order of their derivatives with respect to the function centers in the above integrals. Thus, the present formulation can cover a large class of molecular integrals necessary for theoretical studies of molecular systems by ab initio calculations, and furthermore provides us with an efficient scheme of computing them by virtue of its recursive nature.
127 citations
TL;DR: In this article, the authors presented a method for the computation of one-and two-electron spin-orbit integrals over Gaussian-type basis functions, using existing nuclear-attraction and electron-repulsion integral codes.
Abstract: Methods for the computation of one- and two-electron spin-orbit integrals over Gaussian-type basis functions are presented. We show that existing nuclear-attraction and electron-repulsion integral codes can be readily adapted for the efficient evaluation of spin-orbit integrals; in particular, one can take advantage of recent advances in the computation of derivative integrals. Recurrence relations for the nuclear attraction integrals are also developed.
64 citations
TL;DR: In this paper, a countable set of integrable dynamical systems is constructed which in the continuous limit turn into the Korteweg-de Vries equation, where the integrals are represented as Lax matrix equations.
Abstract: New constructions of integrable dynamical systems are found that admit representation as Lax matrix equations. A countable set of integrable systems is constructed which in the continuous limit turn into the Korteweg-de Vries equation. For an arbitrary space with finite measure and measure-preserving mapping differential equations are constructed on the space of measurable functions on . Here differentiation is with respect to time and the equations have a countable set of first integrals. Constructions are also given for first integrals of dynamical systems preserving certain differential forms, and new constructions of matrix differential equations having large families of first integrals. Bibliography: 18 titles.
63 citations
TL;DR: Explicit solutions of a model equation describing the motion of melts in the Earth are obtained by an independent variable transformation as discussed by the authors, including periodic wave solutions, solitary wave solutions and weak solutions with compact support.
Abstract: Explicit solutions of a model equation describing the motion of melts in the Earth are obtained by an independent variable transformation. Included are periodic wave solutions, solitary wave solutions and weak solutions with compact support. It is also discussed that if the model equation is modified, the resultant equation can be reduced to the exactly solvable Korteweg-de Vries equation.
34 citations
TL;DR: In this article, the singular integral equation governing the opening of a mode I embedded three-dimensional fracture in an infinite solid was solved by applying the finite element method, and the strategy is to formulate the equation into weak form, and to transfer the differentiation from the singular term, 1/r, in the equation to the test function.
Abstract: The singular integral equation governing the opening of a mode I embedded three-dimensional fracture in an infinite solid was solved by applying the finite element method The strategy is to formulate the equation into weak form, and to transfer the differentiation from the singular term, 1/r, in the equation to the test function A numerical algorithm was thus developed The numerical solutions for circular and elliptical fractures under the action of polynomial pressure distributions were compared with the analytical solutions by Green and Sneddon,12 Irwin,13 Shah and Kobayashi14 and Nishioka and Atluri16 The results have demonstrated that the numerical method reported is accurate and efficient
33 citations
01 Aug 1988
TL;DR: In this paper, the crack tip flux integral, denoted J, is defined over a vanishingly small contour in a normal plane to the crack front, providing the pointwise value of the mechanical energy release rate.
Abstract: The crack tip flux integral, denoted J, for local translations (Mode 1) of a planar, curved crack front has potential application as a characterizing parameter in threedimensional, nonlinear fracture mechanics. The integral, defined over a vanishingly small contour in a normal plane to the crack front, provides the pointwise value of the mechanical energy release rate. Arbitrary loading and material behavior are accommodated in the limit of a vanishing contour shrunk onto the crack front. Numerical evaluation of the integral in the context of finite element analysis necessarily leads to the development of computational procedures for non-vanishing, finite,-sized contours. Such developments have proceeded along two lines with the corresponding introduction of Domain Integrals (integrals over element volumes) and Contour Integrals (line integrals and area integrals) methods. Each approach leads to identical numerical values; selection of a procedure thus becomes a matter of convenience in element meshing and problem specification. This report describes the theoretical basis of each computational procedure, associated numerical algorithms, and details of the implementation in the general purpose analysis system POLO-FINITE. Example problems addressing 2-D and 3-D cracked structures for linear and nonlinear response under thermomechanical loading are presented to illustrate the general applicability of the procedures.
26 citations
TL;DR: In this paper, a new method is established for the numerical solution of the general type of the finite-part integrals and integral equations of the first and second kind, which are frequently encountered in problems of applied mechanics and especially in plane and antiplane elasticity.
26 citations
TL;DR: In this paper, a local error analysis is given of the approximation of the integrals used in boundary integral equation methods for solving Laplace's equation using Taylor series expansions and the asymptotic behaviour of the local truncation error is given in terms of the panel grid size.
Abstract: In this paper a local error analysis is given of the approximation of the integrals used in boundary integral equation methods for solving Laplace's equation. Using Taylor series expansions appriximate expressions are derived for the velocity and velocity-potential integrals. The asymptotic behaviour of the local truncation error is given in terms of the panel grid size.
22 citations
TL;DR: In this paper, the singular integral equation is applied to a number of appropriately selected collocation points on the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation.
Abstract: A new method is proposed, by using some special quadrature rules, for the numerical evaluation of the general type of finite-part singular integrals and integral equations with logarithmic singularities. In this way the system of such equations can be numerically solved by reduction to a system of linear equations. For this reduction, the singular integral equation is applied to a number of appropriately selected collocation points on the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation. An application is given, to the determination of the intensity of the logarithmic singularity in a simple crack inside an infinite, isotropic solid.
TL;DR: The Mangler-type principal value integrals play an important role in the formulation and solution of the related hypersingular integral equations for crack problems in plane elasticity.
TL;DR: In this paper, an integral equation method is described for solving the mise-a-la-masse potential problem in a linear, homogeneous, and anisotropic half-space in which there are embedded equipotential conductors.
TL;DR: In this article, the existence of nonconstant periodic traveling wave solutions was shown by converting the generalized Korteweg-deVries equation into a nonlinear integral equation using the Green's function method and employing the Schauder's fixed point theorem to establish the result.
Abstract: The two-dimensional generalized Korteweg-deVries equation is considered. We show the existence of nonconstant periodic traveling wave solutions. The equation is converted into a nonlinear integral equation using the Green's function method. We then employ the Schauder's fixed point theorem to establish the result. An example is given for the case of f(u) = 2u 3,i.e., the modified KdV equation
TL;DR: In this paper, the boundary integral equation method is converted to a pair of integral equations by using Green's third identity, and additional constraint equations are derived to enforce uniqueness in such situations.
Abstract: In the boundary integral equation method, the biharmonic equation is converted to a pair of integral equations by using Green's third identity. In multiply connected domains, for a particular exceptional geometry the integral equations do not have a unique solution. Additional constraint equations are derived to enforce uniqueness in such situations. Two example problems are solved to demonstrate the effectiveness of the constraints.
TL;DR: The Green's differential equation as mentioned in this paper expresses the relationship between the field parameters and the differential geometry of equipotential surfaces, and it is suggested that the equation be referred to as Green's equation.
Abstract: Some important aspects of a little-known equation in electrostatics are discussed. The equation in question expresses the relationship between the field parameters and the differential geometry of equipotential surfaces. It is suggested that the equation be referred to as Green's differential equation.
TL;DR: In this article, the authors analyzed the symmetries of the derivative nonlinear Schrodinger (DNLS) equation and derived a new constant of motion, which may be formally considered as a charge and which is related to the helicity of the physical system.
Abstract: From the analysis of the symmetries of the derivative nonlinear Schrodinger (DNLS) equation, we obtain a new constant of motion, which may be formally considered as a charge and which is related to the helicity of the physical System. From comparison of these symmetries and those of the soliton solutions, we draw conclusions about the number of constraints that must be imposed and the way a Liapunov functional must be constructed in order to study the solitons' stability. We also examine the relationship between the stability with respect to form and the symmetries that are broken by the soliton solutions. We complete the analysis with some numerical simulations: we solve the DNLS equation taking a slightly perturbed soliton as an initial condition and study its temporal evolution, finding that, as expected, they are stable with respect to form.
TL;DR: In this paper, the time-dependent equation of radiative transfer for isotropic scattering has been solved by integral equation technique in terms of X- and Y-functions appropriate for the problem.
Abstract: The time-dependent equation of radiative transfer for isotropic scattering has been solved by integral equation technique in terms ofX- andY-functions appropriate for the problem. It is seen thatX- andY-functions are reducible to the corresponding function for steady-state problems by simply changing the Laplace transform parameters-i.e., byS→0.
TL;DR: In this article, the binding energy of three-quark systems has been approximated by an integro-differential equation which is equivalent to the Faddeev equation in an adiabatic approximation.
Abstract: Employing a recently derived integro-differential equation which is equivalent to the Faddeev equation, in an adiabatic approximation, we obtain accurate lower bounds to the binding energy of three-quark systems close to the upper bounds provided by the hypercentral approximation to the hyperspherical expansion method. Calculations have been performed for the Martin and the spin-dependent Ono-Schoberl potential.
TL;DR: In this article, it was shown that the second order differential equation where L is rational, has a non-rational meromorphic solution w = win the plane only if w(z)is also a solution of some Riccati equation with rational coefficients or of an equation = n=2, 3.4 or 6.
Abstract: It is shown that the second order differential equation where L is rational, has a non-rational meromorphic solution w=win the plane only if w(z)is also a solution of some Riccati equation with rational coefficients or of an equation = n=2, 3.4 or 6. In this case and respectively.
TL;DR: In this article, the authors presented an analytic methodology for determining the temperature and radiosity distributions in a rectangular, gray fin array through a primitive formulation that explicitly couples the two distributions.
Abstract: This paper presents an analytic methodology for determining the temperature and radiosity distributions in a rectangular, gray fin array through a primitive formulation that explicitly couples the two distributions. The primitive formulation, used in conjunction with the Green's function method, produces a set of coupled nonlinear Fredholm integral equations for temperature and radiosity that must be solved simultaneously. Accurate numerical results are produced rapidly and efficiently using the trapezoidal method and a standard iterative scheme. Initially, an integrodifferential equation is derived from the one-dimensional steady-state energy balance. This mixed mode equation displays the coupling between the temperature and radiosity. In deriving the energy balance, a temperature-dependent thermal conductivity is included in terms of a Taylor series expansion. The Green's function method is used to convert the integro-differential equation describing the conservation of energy into an equivalent integral equation. The second integral equation is developed from the balance of radiant energy for a diffuse-gray surface. The coupled integral equations are solved simultaneously by a simple numerical integration scheme. A parametric study considering the effects of the conduction-radiation number, emissivity, spacings, and thermal conductivity is presented.
TL;DR: In this article, a non-homogeneous Stokes equation instead of a homogeneous one was applied to the Navier-Stokes equation and the estimate of the Hs+1 norm and the order of the L 2 norm of the errors was obtained.
Abstract: Three dimensional initial boundary value problem of the Navier-Stokes equation is considered. The equation is split in an Euler equation and a non-stationary Stokes equation within each time step. Unlike the conventional approach, we apply a non-homogeneous Stokes equation instead of homogeneous one. Under the hypothesis that the original problem possesses a smooth solution, the estimate of theHs+1 norm, 0≦s<3/2, of the approximate solutions and the order of theL2 norm of the errors is obtained.
TL;DR: In this article, an approximate evolution equation for a one-dimensional probability distribution of a single-degree-of-freedom system driven by Ornstein-Uhlenbeck noise is presented.
Abstract: By use of an operator method, we construct a novel approximate evolution equation for a one-dimensional probability distribution of a single-degree-of-freedom system driven by Ornstein-Uhlenbeck noise. This equation is an integro-differential equation of the time-convolutionless type and its steady-state solution is presented. A class of non-linear equations with multiplicative noise is considered.
TL;DR: In this article, the problem of finding upper and lower bounds to the natural frequencies of free vibration of a circular plate with stepped radial density has been investigated, and the integral equation formulation is shown to provide more accurate upper bounds than the differential equation formulation for both the clamped and simply supported plates.
TL;DR: The second in a series of papers on the quantum nonlinear Schrodinger equation is as discussed by the authors, where the authors present the solution of the equation explicitly in terms of the free fields.
TL;DR: Inversion algorithms have been derived by D. V. Widder and T. T. Haimo for the dual Weierstrass-Laguerre transform as discussed by the authors.
TL;DR: In this article, the regions of growth and decay of the solutions are determined and the corresponding Stokes parameters are obtained for a nonlinear ordinary differential equation obtained as the reduction of the three-wave equation by a new technique due to Gurarii and Mateev.
TL;DR: In this article, a particular numerical method is discussed for solving the Hill-Wheeler equation, which is an integral equation that arises in the generator coordinate method analysis of problems in nuclear physics.
Abstract: A particular numerical method is discussed for solving the Hill–Wheeler equation, which is an integral equation that arises in the generator coordinate method analysis of problems in nuclear physics. The method is applicable to scattering problems with finite range potentials and exploits prior knowledge of the asymptotic form of the solution to convert the Hill–Wheeler equation into a Fredholm integral equation of the first kind. The resulting Fredholm equation is ill‐posed, which often results in numerically unstable solutions. The method of regularization is used to produce numerically stable, approximate solutions. Optimizing the choice of regularization parameter is discussed and calculations are presented for an example problem.