Showing papers on "Integro-differential equation published in 1994"
TL;DR: In this paper, it was shown that a third-order recursive ordinary differential equation for the total correlation function of the hard-sphere fluid can be derived from the Percus-Yevick integral equation using Baxter's factorization method.
Abstract: We present an analytic expression containing no imaginary terms for the radial distribution function of hard spheres in the Percus-Yevick approximation up to the distance of 4σ (though extendable beyond that), where σ is the hard-sphere diameter. It is shown that a third-order recursive ordinary differential equation for the total correlation function of the hard-sphere fluid can be derived from the Percus-Yevick integral equation using Baxter's factorization method. We have solved this differential equation with its boundary conditions and obtained a result which is equivalent to that obtained by Smith and Henderson, but which contains only real functions. This result is useful in perturbation theories where the evaluation of integrals involving the radial distribution function is required since the expressions presented here are now integrable.
103 citations
TL;DR: In this article, the multidimensional Burgers equation with a viscosity term and a random force modelled by a functional of time-space white noise, Wick products, is considered.
Abstract: We consider the multidimensional Burgers equation with a viscosity term and a random force modelled by a functional of time-space white noise, Wick products. Then we show that the nonlinear equation (B) can be transformed into a linear, stochastic heat equation with a noisy potential. This heat equation is solved explicitly in the following two cases.
78 citations
TL;DR: In this paper, a three-dimensional curvilinear hybrid finite-element-integral equation approach is developed to model arbitrarily curved geometries without approximations, where the advantages of the finite element and integral equation methods are used to eliminate the disadvantages of both methods.
Abstract: A three-dimensional curvilinear hybrid finite-element–integral equation approach is developed. The hybrid finite-element–integral equation method is formulated in general curvilinear coordinates so that arbitrarily curved geometries can be modeled without approximations. The advantages of the finite-element and the integral equation methods are used to eliminate the disadvantages of both methods. For modeling regions of inhomogeneous or anisotropic materials, in which the integral equation method is difficult to implement, the finite-element method is used. The problem with the finite-element method of accurately terminating the computational mesh is eliminated by application of the exact boundary integral equation. The hybrid finite-element–integral equation code is validated on a range of composite curvilinear geometries against measured data or analytical solutions.
77 citations
TL;DR: In this paper, a new general formula for the differentiation of such integrals is proposed, and a gradient of the integral is represented as the sum of integrals taken over a volume and over a surface.
48 citations
TL;DR: In this paper, the authors studied three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation delta u/ delta x0+ delta u=aun+ epsilon f(u), where a and n are arbitrary real constants, f is a smooth function, and 0 < 1.
Abstract: We study three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation delta u/ delta x0+ Delta u=aun+ epsilon f(u), where a and n are arbitrary real constants, f is a smooth function, and 0< epsilon <<1.
46 citations
TL;DR: In this article, a nonlinear superposition formula of the Novikov-Veselov equation is proved under certain conditions, and some particular solutions of the NVE are given as an illustrative application of the obtained result.
Abstract: A nonlinear superposition formula of the Novikov-Veselov equation is proved under certain conditions. Some particular solutions of the Novikov-Veselov equation are given as an illustrative application of the obtained result.
44 citations
TL;DR: In this article, nonlinear superposition formulae of the differential-difference analogue of the KdV equation and two-dimensional Toda equation are proved rigorously.
Abstract: In this paper, nonlinear superposition formulae of the differential-difference analogue of the KdV equation and two-dimensional Toda equation are proved rigorously. Some particular solutions of the differential-difference analogue of the KdV equation are given as an illustrative application of the obtained result.
40 citations
TL;DR: In this paper, a nonlinear Hamilton-Jacobiobian equation for the velocity potential was derived for the relativistic flow of collisionless matter and the velocity of the field may always be derived from a potential which however may be a multi-valued function of the space-time coordinates.
Abstract: Beginning with a relativistic action principle for the irrotational flow of collisionless matter, we compute higher order corrections to the Zel'dovich approximation by deriving a nonlinear Hamilton-Jacobi equation for the velocity potential. It is shown that the velocity of the field may always be derived from a potential which however may be a multi-valued function of the space-time coordinates. In the Newtonian limit, the results are nonlocal because one must solve the Newton-Poisson equation. By considering the Hamilton-Jacobi equation for general relativity, we set up gauge-invariant equations which respect causality. A spatial gradient expansion leads to simple and useful results which are local --- they require only derivatives of the initial gravitational potential.
39 citations
TL;DR: In this article, the Pearson equation is used to convert the hypergeometric equation into a self-adjoint equation and a generalization of this idea to difference equations on linear lattices enables a proof of Barnes's second lemma by using his first lemma and gives extensions of some of Ramanujan's formulas.
Abstract: An alternate proof of the classical beta integral is given by using the Pearson equation whose solution converts the hypergeometric equation into a self-adjoint equation. A generalization of this idea to difference equations on linear lattices enables a proof of Barnes’s second lemma by using his first lemma and gives extensions of some of Ramanujan’s formulas. Similar analysis on q-quadratic lattices gives an extension of Askey’s integral on the real line and the corresponding basic bilateral sum of Gosper. Another q-quadratic lattice gives an extension of the Askey–Wilson integral. A quadratic lattice is used to evaluate a principal value integral on the real line, whose q-analogue is shown to be equivalent to the Askey integral extension. A list of various beta integrals is also presented.
36 citations
TL;DR: In this paper, a boundary element formulation for the Navier-Stokes equations was presented, in which the well-known closed-form fundamental solution for the steady Stokes equations is employed.
Abstract: This paper presents a boundary element formulation for the permanent Navier–Stokes equations in which the well-known closed-form fundamental solution for the steady Stokes equations is employed. In this way, from the integral representation formulae for the Stokes' equations, an integral equation is found in which the original non-linear convective terms of the Navier–Stokes equations appear as a domain integral. Additionally, the method of dual reciprocity is used to transform the domain integral to boundary integrals (this method is closely related to the method of particular integrals also used in the literature to transform domain integrals to boundary integrals). Numerical results are presented for the three-dimensional internal flow in a cylindrical container with a rotating cover, in which the accuracy of the method is shown.
31 citations
TL;DR: In this article, the authors studied the quantum Knizhnik-Zamolodchikov equation of level 0 associated with the spin 1/2 representation of Uq(SL2) and found an integral formula for solutions in the case of an arbitrary total spin and mod q mod < 1.
Abstract: We study the quantum Knizhnik-Zamolodchikov equation (1984) of level 0 associated with the spin 1/2 representation of Uq(SL2). We find an integral formula for solutions in the case of an arbitrary total spin and mod q mod <1. In the formula, different solutions can be obtained by taking different integral kernels with the cycle of integration being fixed.
TL;DR: Work on the reduction of independent variables in the Schr\"odinger equation for two-particle systems from six to three is extended to give a closed expression of the variational equation for states of arbitrary total angular momentum.
Abstract: Work on the reduction of independent variables in the Schr\"odinger equation for two-particle systems from six to three is extended to give a closed expression of the variational equation for states of arbitrary total angular momentum. The method consists of writing the Hamiltonian in six coordinates, by a suitable choice of transformation equations, and evaluating integrals involving real angular-momentum functions. An existing variational equation for P states follows as a special case of the present formalism.
TL;DR: In this paper, the spectral relations for the first kind with singular kernel were obtained by using Krein's method for the Fredholm integral equation of the first type with a singular kernel.
Abstract: In this paper, we are going to obtain the spectral relations for the Fredholm integral equation of the first kind with certain singular kernel, by using Krein's method.
TL;DR: In this article, the Walsh functions and the Laplace transform are combined to solve the one-group linear transport equation in a planar geometry considering anisotropic scattering, and numerical simulations are presented.
TL;DR: In this paper, a new extension of the sinh-Gordon equation in two spatial dimensions is obtained from the negative Kadomtsev-Petviashvili equation and the Miura transformation.
TL;DR: In this paper, a new class of fractional integrals connected with balls in ℝn is introduced and investigated, where a Riesz potential is represented by a composition of such integrals.
Abstract: The purpose of the paper is to introduce and to investigate a new class of fractional integrals connected with balls in ℝn. A Riesz potentialI
Ω
α
ρ over a ball Ω is represented by a composition of such integrals. Using this representation we obtain necessary and sufficient solvability conditions for the equationI
Ω
α
ρ =f in the space Lp(Ωw) with a power weight w(x) and solve the equation in a closed form. The investigation is based on a special Fourier analysis adopted for operators commuting with rotations and dilations in ℝn.
TL;DR: In this article, a method to solve the time dependent Schrodinger equation subject to time dependent boundary conditions was proposed, and the results obtained by guessing are seen to be a consequence of the invariance of the Schroffinger equation under scaling of the space-time coordinates and/or the existence of time dependent invariants associated with the studied Hamiltonian.
TL;DR: An integral equation method for the Dirichlet problem for the biharmonic equation is proposed in this article, which leads to a $2 \times 2$ matrix integral equation system.
Abstract: An integral equation method for the Dirichlet problem for the biharmonic equation is proposed. It leads to a $2 \times 2$ matrix integral equation system. By taking suitable norms on the spaces of density functions, the Fredholm operator theory can be used to prove the solvability. The kernels in this system are relatively complicated. Therefore, especially when a high-order polynomial approximation is used for a numerical purpose, it is costly to evaluate the integrals that appear in the numerical system. A discrete Galerkin method that has shown superb convergence is proposed here, as in [K. Atkinson, J. Integral Equations Appl., 1 (1988), pp. 343–363] and elsewhere. When the boundary functions are smooth, exponential convergence is observed.
TL;DR: In this paper, the authors used the Laplace transform on the I -interval explored by the optical depth variable to obtain the resolvent kernel of the Schwarzschild-Milne integral equation.
Abstract: The Schwarzschild-Milne integral equation−which is equivalent to the transfer equation with appropriate boundary conditions−is solved in an homogeneous plane-parallel medium of any optical extent (i.e. infinite, semi-infinite or finite). Local scattering of light is assumed isotropic. The three proposed methods use the Laplace transform on the I -interval explored by the optical depth variable. The first method is based on the Sobolev classical scheme yielding the resolvent kernel of the Schwarzschild-Milne equation in terms of the resolvent function. In the other two methods, the Laplace transform of the appropriate Green distribution is calculated from an integral equation of the Schwarzschild-Milne type (Ambartsumian method) or of the Cauchy type (a method from Leonard-Mullikin-Yanovitskii). Auxiliary functions defined in the complex plane are needed in the three methods. Their restrictions in the range [-1, +1] have already been introduced in the literature and have a clear physical meaning. The solution is provisionally written as an inverse Laplace transform of these auxiliary functions. Further developments based on the theorem of residues need a systematic study of the auxiliary functions in the complex plane (to be achieved in a forthcoming article).
TL;DR: In this paper, the system of equations for each subregion, which is derived from the boundary integral equation, is transformed to an equation similar to the stiffness equation of the finite element method.
Abstract: The system of equations for each subregion, which is derived from the boundary integral equation, is transformed to an equation similar to the stiffness equation of the finite element method. The global equation is made by the superposition of these matrix equations. The present approach is derived theoretically and is compared with the existing ones in order to indicate its features. Then, it will be applied to analysis of two dimenisonal elastic and potential problems
TL;DR: In this paper, a strongly nonlinear integral equation of the Hammerstein type is considered, which arises from an important engineering application, and a realistic condition for the linear operator included in the equation is found which gives a precise, constructively built, solution.
TL;DR: A Galerkin method for the approximation of a fourth order integro-differential equation is given, which will result in a dense stiffness matrix but is overcome by use of an iteration scheme which is shown to converge to the numerical solution.
TL;DR: In this article, the singular element integrals are computed analytically by a program in the Maple language, after recasting the integrals by coordinate transformations that render all singularities one-dimensional.
Abstract: Skin effect at high frequencies and electrostatics of good conductors can be formulated as an integal equation, whose solution by finite elements requires evaluation of integrals with Green's function kernels. The singular element integrals are computed analytically by a program in the Maple language, after recasting the integrals by coordinate transformations that render all singularities one-dimensional. Tables of singular integrals are given for elements up to order 4. >
TL;DR: In this paper, a forward stochastic transport model is proposed for the transport of a doublet and a higher multiplet densities, which can be converted into a simple functional form which is then used to establish a connection with the backward equation and a compact derivation of the moments equations.
TL;DR: In this paper, the Cauchy data of a Kirchhoff plate are coupled by a system of four integral equations, which contains hypersingular kernels up to the order r -3.
Abstract: The four Cauchy data of a Kirchhoff plate are coupled by a system of four integral equations The first two of these are known In this paper we derive the third integral equation It contains hypersingular kernels up to the order r
-3
TL;DR: In this article, a new type of nonlinear difference-differential equations is proposed and its transformation relation to the discrete KdV equation is found, and based on it an N-soliton solution is derived.
TL;DR: In this article, a boundary integral equation for the notch problem of antiplane elasticity with a weaker singular kernel (logarithmic) was proposed. But the authors took the distributed dislocation density as the unknown function and the resultant force function was taken as the right hand term of the integral equation.
Abstract: In this paper the notch problem of antiplane elasticity is discussed and a new boundary integral equation is formulated. In the problem, the distributed dislocation density is taken to be the unknown function. Unlike the usual choice, the resultant force function is taken as the right hand term of the integral equation; therefore, a new boundary integral equation for the notch problem of antiplane elasticity with a weaker singular kernel (logarithmic) is obtained. After introducing a particular fundamental solution of antiplane elasticity, the notch problem for the half-plane is discussed and the relevant boundary integral equation is formulated. The integral equations derived are compact in form and convenient for computation. Numerical examples demonstrated that high accuracy can be achieved by using the new boundary equation.
TL;DR: In this paper, a class of models of the Enskog-Vlasov equation based on discretization of the velocity space is presented, and a proposal of solution of two problems is given.
Abstract: In order to have, a kinetic equation suited to liquid dynamics and phase transitions, the intermolecular potential is split into a repulsive hard-core and an attractive tail. The hard-core is treated as in the revised Enskog equation, whereas the tail enters the equation only linearly, in a mean-field term. Such an equation is called the Enskog-Vlasov equation. The goal of the paper is to present a construction of a class of models of the Enskog-Vlasov equation, based on the idea of discretization of the velocity space. A proposal of solution of two problems is given: discrete velocity models of the Enskog collisional operator; discrete velocity models of kinetic equations with self-consistent forces. The conservation equations which follow from the proposed models have the structure of the capillarity equations with a vander Waals-like pressure formula, if the Kac limit is imposed on the attractive tail.