Showing papers on "Integro-differential equation published in 1998"
TL;DR: Local fractional differential equations (LDFDE) as mentioned in this paper is a new class of differential equations, which involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time.
Abstract: We propose a new class of differential equations, which we call local fractional differential equations. They involve local fractional derivatives and appear to be suitable to deal with phenomena taking place in fractal space and time. A local fractional analog of the Fokker-Planck equation has been derived starting from the Chapman-Kolmogorov condition. We solve the equation with a specific choice of the transition probability and show how subdiffusive behavior can arise.
275 citations
Book•
01 Jan 1998
TL;DR: In this paper, Singular integrals in BEM formulations regularization with analytical integration of potentially singular integrals regularization by infinitesimal deformation of the boundary is discussed.
Abstract: CHAPTER 1 INTRODUCTORY NOTES ON SINGULAR INTEGRALS Introduction: Singular integrals Definitions Singular integrals in BEM formulations Regularization Direct limit approach with analytical integration of potentially singular integrals Regularization by infinitesimal deformation of the boundary Analytical regularization Boundary element approximations Collocation approach Galerkin approach Conclusions. CHAPTER 2 EVALUATION OF SINGULAR AND HYPERSINGULAR GALERKIN INTEGRALS: DIRECT LIMITS AND SYMBOLIC COMPUTATION Introduction: Symmetric Galerkin Condition Singular integrals: linear element Coincident integration G First derivative of G Second derivative of G Adjacent integration G and its first derivative Second derivative of G Accuracy Orthotropic elasticity Orthotropic boundary integral equations Singular integrals Fracture calculations Singular integrals: Curved elements Surface derivatives Example calculations Application: electromigration Conclusions. CHAPTER 3 FORMULATION AND NUMERICAL TREATMENT OF BOUNDARY INTEGRAL EQUATIONS WITH HYPERSINGULAR KERNELS Introduction:Some classical theorems General form of boundary integral identities Boundary integral equations Standard boundary integral equations (SBIE) Hypersingular boundary integral equations (HBIE) Evaluation of the free-term coefficients Direct evaluation of singular integrals Three-dimensional problems Limiting process and discretization of the geometry Semi-analytical treatment Final formula Less singular integrals Two-dimensional problems Limiting process and discretization of the geometry Semi-analytical treatment Final formula Numerical examples Strongly singular integrals (CPV) Hypersingular integrals Related works. CHAPTER 4 REGULARIZATION OF BOUNDARY ELEMENT FORMULATIONS BY THE DERIVATIVE TRANSFER METHOD Introduction: Notation Static elasticity Displacement equation 2D problems 3D problems Traction equation Collocation approach Variational approach Elastodynamics Displacement equation Laplace domain Time domain Traction equation Laplace domain Time domain Time domain: variational approach Kirchhoff plates Displacement and gradient equations Moment-shear equations Collocation approach Variation approach Numerical implementation and examples Brazilian test Retangular plate with a slanted crack Square plate with two opposite sides simply-supported and the other ones free Concluding remarks. CHAPTER 5 SINGULAR INTEGRALS AND THEIR TREATMENT IN CRACK PROBLEMS Introduction: The fundamental solution Continuity and discontinuity of the potentials Traction BIE's for fracture modeling Behavior of free term integrals on open surfaces BEM implementation Conclusion. CHAPTER 6 ACCURATE HYPERSINGULAR INTEGRAL COMPUTATIONS IN THE DEVELOPMENT OF NUMERICAL GREEN'S FUNCTIONS FOR FRACTURE MECHANICS Introduction: The boundary element method review Integral equations for displacements and tractions Boundary integral equations Boundary integral equations at crack surfaces Numerical Green's function Complementary solution Fundamental crack opening displacements Final numerical Green's function Stresses at internal points Implementation of NGF Geometric shape function Numerical treatment of the fundamental crack opening integral equation Interpolation to crack opening and its derivatives Examples Conclusions. CHAPTER 7 REGULARIZATION AND EVALUATION OF SINGULAR DOMAIN INTEGRALS IN BOUNDARY ELEMENT METHODS Introduction: 2D/3D - FBEM for plasticity at small strains Governing equations Field boundary integral equations for displacements Field boundary integral equations for displacement gradients Regularization for interior source points Regularization for source points on the boundary Discretization and numerical solution Numerical treatment of domain integrals Regular and nearly singular integrals Weakly singular integrals Example Extension to axisymmetric problems Field boundary integral equations for displacements Field boundary integral equations for displacement gradients Numerical treatment of axisymmetric problems FBEM for finite deformation problems Governing equations Field boundary integral equations for displacements Two- and three-dimensional problems Axisymmetric problems Field boundary integral equation for displacement gradients Two- and three-dimensional problems Axisymmetric problems Discretization and numerical solution Example FBEM for damage mechanics FBEM for non-linear fracture mechanics Dual field boundary element method (Dual - FBEM) Example Conclusions. CHAPTER 8 REGULARIZED BOUNDARY INTEGRAL FORMULATION FOR THIN ELASTIC PLATE BENDING ANALYSIS Introduction: Direct boundary integral formulation for thin elastic plate bending problem Governing equation Direct boundary integral formulation Singular boundary integral representations Regularized boundary integral representations Numerical treatment Discretization Applicability of C0 interpolation scheme for the deflection in the regularized boundary integral equation C1-continuous interpolation for the deflection Numerical examples Concluding remarks. CHAPTER 9 COMPLEX HYPERSINGULAR BEM IN PLANE ELASTICITY PROBLEMS Introduction: Advantages of functions of a complex variable Advantages of hypersingular integrals Combined advantages of complex variables and hypersingular equations Brief historical review Complex integral equations Real hypersingular integral equations Complex hypersingular integral equations (CHSIE) Scope of the paper Prerequisities Singular solutions Singular solutions in real variables Singular solutions in complex variables Particular case of Kelvin solution Employing K.M. functions to obtain singular solutions Complex potentials and their properties Complex potentials Particular case of Kelvin solution Limit values of complex potentials Physical meaning of densities Equations of the indirect approach General case Equations for Kelvin's solution Comment Equations of the direct approach General equations Equations for Kelvin's solution Equations for blocky systems and cracks Employing of K.M. functions Comment Complex hypersingular integrals Definitions of hypersingular integrals Connection of direct values of hypersingular integrals with limit values Method of solution: CVH-BEM approach BEM discretization and approximation Choice of approximating functions importance of conjugated polynomials and tip elements Evaluation of crucial integrals Tip elements Integrals from conjugated functions Evaluation of remaining (proper) integrals Straight element Circular arc element Comment Formulae for control Numerical examples Examples regarding approximations Importance of conjugated polynomials Importance of tip elements Influence of element sizes Approximation of boundaries Examples illustrating the range of applications Problems for cracks. CHAPTER 10 SOME COMPUTATIONAL ASPECTS ASSOCIATED WITH SINGULAR KERNELS Introduction: Approximation of boundary densities and geometry in regularized formulations Regularized boundary integral equations Ordinary boundary integral equations - OBIE Derivative boundary integral equations - DBIE Approximations by using standard elements Standard Lagrange-type elements (Slag) Standard Overhauser elements (Sov) Modified Overhauser elements Smooth contour at z Corner at z Numerical examples Conclusions Optimal transformations of the integration variable in numerical computation of nearly-singular integrals Polynomial transformations Optimal transformations Numerical experiments Conclusions Numerical integration of logarithmic and nearly-logarithmic singularity Numerical integrations (wt) - approach (pt) - approach (Tt) - approach (ln) - approach (an) - approach Numerical experiments Conclusions Weak - singularity in 3-d BEM formulations Weakly singular integral Nearly weakly singular integral Numerical experiments Conclusions.
172 citations
TL;DR: The quantum dynamical Yang-Baxter (QDYB) equation was introduced by Gervais, Neveu, and Felder as mentioned in this paper, which is a generalization of the classical DYB equation.
Abstract: The quantum dynamical Yang–Baxter (QDYB) equation is a useful generalization of the quantum Yang–Baxter (QYB) equation. This generalization was introduced by Gervais, Neveu, and Felder. Unlike the QYB equation, the QDYB equation is not an algebraic but a difference equation, with respect to a matrix function rather than a matrix. The QDYB equation and its quasiclassical analogue (the classical dynamical Yang–Baxter equation) arise in several areas of mathematics and mathematical physics (conformal field theory, integrable systems, representation theory). The most interesting solution of the QDYB equation is the elliptic solution, discovered by Felder.
In this paper, we prove the first classification results for solutions of the QDYB equation. These results are parallel to the classification of solutions of the classical dynamical Yang–Baxter equation, obtained in our previous paper. All solutions we found can be obtained from Felder's elliptic solution by a limiting process and gauge transformations.
Fifteen years ago the quantum Yang–Baxter equation gave rise to the theory of quantum groups. Namely, it turned out that the language of quantum groups (Hopf algebras) is the adequate algebraic language to talk about solutions of the quantum Yang–Baxter equation.
In this paper we propose a similar language, originating from Felder's ideas, which we found to be adequate for the dynamical Yang–Baxter equation. This is the language of dynamical quantum groups (or ?-Hopf algebroids), which is the quantum counterpart of the language of dynamical Poisson groupoids, introduced in our previous paper.
160 citations
TL;DR: In this article, a new transformation between two integrable hierarchies of the Camassa-Holm equation and the Hunter-Saxton equation is presented, which is the high-frequency limit of the CAMASHA-HOLM equation.
Abstract: In this article we present a new transformation between two integrable hierarchies of the Camassa-Holm equation and the Hunter-Saxton equation. For instance we present a transformation between the Harry-Dym equation and the extended Harry-Dym equation. Moreover, we describe a relationship between the Hunter-Saxton equation, which is the high-frequency limit of the Camassa-Holm equation, and the Sinh-Gordon equation by a reciprocal transformation.
87 citations
TL;DR: In this paper, the electromagnetic coupling in the Kemmer-Duffin-Petiau (KDP) equation was analyzed and it was shown that certain interaction terms in the Hamilton form of the KDP equation do not have a physical meaning and will not affect the calculation of physical observables.
83 citations
Journal Article•
63 citations
TL;DR: A collection of methods for solving the incompressible Navier--Stokes equations in the plane that are based on a pure stream function formulation using fast algorithms for the evaluation of volume integrals, to avoid the ill-conditioning which hampers finite difference and finite element methods in this environment.
Abstract: We present a collection of methods for solving the incompressible Navier--Stokes equations in the plane that are based on a pure stream function formulation. The advantages of this approach are twofold: first, the velocity is automatically divergence free, and second, complicated (nonlocal) boundary conditions for the vorticity are avoided. The disadvantage is that the solution of a nonlinear fourth-order partial differential equation is required. By recasting this partial differential equation as an integral equation, we avoid the ill-conditioning which hampers finite difference and finite element methods in this environment. By using fast algorithms for the evaluation of volume integrals, we are able to solve the equations using O(M) or O(M log M) operations, where M is the number of points in the discretization of the domain.
45 citations
TL;DR: In this article, the authors investigated the asymptotic behavior of the solutions as the distance from the finite end of a semi-infinite cylinder tends to infinity and thus established spatial decay results of the Saint-Venant type.
Abstract: One method of regularizing the initial value problem for the backward heat equation involves replacing the equation by a singularly perturbed hyperbolic equation which is equivalent to a damped wave equation with negative damping. Another regularization of this problem is obtained by perturbing the initial condition rather than the differential equation. For both of these problems, we investigate the asymptotic behavior of the solutions as the distance from the finite end of a semi-infinite cylinder tends to infinity and thus establish spatial decay results of the Saint-Venant type.
45 citations
TL;DR: In this paper, the authors give explicitly N -soliton solutions of a (2C 1)-dimensional equation, which is obtained by unifying two directional generalizations of the potential KdV equation: the closed ring with the potential KP equation, and the Calogero-Bogoyavlenskij-Schiff equation.
Abstract: We give explicitly N -soliton solutions of a (2C 1)-dimensional equation, xtC xxxz=4CxxzCxxz=2C@ 1 x zzz=4 D 0. This equation is obtained by unifying two directional generalizations of the potential KdV equation: the closed ring with the potential KP equation, and the Calogero-Bogoyavlenskij-Schiff equation. This equation is also a reduction of the KP hierarchy. We also find the Miura transformation which yields the same ring of the corresponding modified equations. The study of higher-dimensional integrable systems is one of the central themes in integrable systems. A typical example of a higher-dimensional integrable system is obtained by modifying the Lax operators of a basic equation, the potential KdV (p-KdV) equation in this paper. The Lax pair of the p-KdV equation have the form L.x;t/D@ 2 xCx.x;t/ (1) T.x;t/D.L.x;t/ 3 2/
43 citations
TL;DR: In this paper, the authors analyse a model for radiative heat transfer in materials that are conductive, grey and semitransparent, and prove the existence of weak solution in the cases where coercivity can be obtained.
Abstract: In this work we analyse a model for radiative heat transfer in materials that are conductive, grey and semitransparent. Such materials are for example glass, silicon, water and several gases. The most important feature of the model is the non-local interaction due to exchange of radiation. This, together with non-linearity arising from the well-known Stefan-Boltzmann law, makes the resulting heat equation non-monotone. By analysing the terms related to heat radiation we prove that the operator defining the problem is pseudomonotone. Hence, we can prove the existence of weak solution in the cases where coercivity can be obtained. In the general case, we prove the solvability of the system using the technique of sub and supersolutions.
37 citations
TL;DR: In this paper, a method for reconstructing the complex permittivity of a bounded inhomogeneous object from measured scattered-field data is presented, which extends the method previously developed for the TM case to the more complicated TE case.
Abstract: A method for reconstructing the complex permittivity of a bounded inhomogeneous object from measured scattered-field data is presented. This paper extends the method previously developed for the TM case to the more complicated TE case. In the TM case, the electric-field integral equation involves an integral operator whose integrand was simply a product of the background Green's function, contrast, and field. In the TE case, the magnetic field is polarized along the axis of an inhomogeneous cylinder of arbitrary cross section and the corresponding integral equation contains derivatives of both the background Green's function and the field. The nonlinear inversion based upon the modified-gradient method as presented in the literature is applied to the magnetic-field equation. However, the integral equation can also be formulated as an electric-field integral equation for the two transversal components of the electric field. Again, the integrand is a product of the background Green's function, contrast, and electric-field vector. The derivatives are operative outside the integral. In this paper, the latter formulation will be taken as a point of departure to develop a nonlinear inversion scheme using the modified-gradient method.
TL;DR: In this article, a new technique is presented for transferring the domain integrals in the boundary integral equation method into equivalent boundary integrals, which has certain similarities to the dual reciprocity method (DRM) in the way radial basis functions are used to approximate the body force term.
Abstract: In this paper a new technique is presented for transferring the domain integrals in the boundary integral equation method into equivalent boundary integrals. The technique has certain similarities to the dual reciprocity method (DRM) in the way radial basis functions are used to approximate the body force term. However, the resulting integrals are evaluated in a much simpler way. Several examples are presented to demonstrate the validity and accuracy of the proposed paper.
Journal Article•
TL;DR: In this article, the local dynamics of a second-order nonlinear difference-differential equation with large delay was studied and it was shown that the structure of solutions is determined by the dynamics of an equation of the Ginzburg-Landau type, which plays the role of a normal form.
Abstract: This paper deals with the local dynamics of a second-order nonlinear difference-differential equation with large delay. It is shown that, in the vicinity of critical cases, the structure of solutions is determined by the dynamics of an equation of the Ginzburg-Landau type, which plays the role of a normal form.
TL;DR: In this paper, a pointwise solution for the time dependent Schrodinger equation on Rd with potentials and initial conditions which can grow exponentially at infinity and belong to the class of smooth Laplace transforms of complex measures on Rd is presented.
Abstract: We construct a pointwise solution for the time dependent Schrodinger equation on Rd with potentials and initial conditions which can grow exponentially at infinity and belong to the class of smooth Laplace transforms of complex measures on Rd. The methods used are both analytic and probabilistic and the result can be looked upon as an extension of rigorously defined Feynman path integrals to the case of potentials which can strongly grow at infinity. An appendix with the calculation of some Wiener integrals is also presented.
TL;DR: It is proved unconditional stability and the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution are obtained.
Abstract: We propose and analyze the spectral collocation approximation for the partial integrodifferential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.
TL;DR: In this article, an integral equation of the first kind with Riesz kernel is discussed and a local conditional pointwise estimate holds at a point if the solution has some additional smoothness properties in a neighbourhood of this point.
Abstract: In this paper, an integral equation of the first kind with Riesz kernel is discussed. Since the kernel of this integral equation is analytic, this problem is severe ill-posed. We prove that, for solutions of the integral equation, a local conditional pointwise estimate holds at a point if the solution has some additional smoothness properties in a neighbourhood of this point.
01 Jan 1998
TL;DR: In this paper, the authors considered a stochastic Burgers equation and showed that the gradient of the corresponding transition semigroup Pt φ does exist for any bounded φ; and can be estimated by a suitable exponential weight.
Abstract: — We consider a stochastic Burgers equation. We show that the gradient of the corresponding transition semigroup Ptφ does exist for any bounded φ; and can be estimated by a suitable exponential weight. An application to some Hamilton-Jacobi equation arising in Stochastic Control is given.
TL;DR: In this paper, the derivation of a very simple equation of state for a hard-disc fluid is discussed, in the form of a simple rational function that fulfils the requirements of being exact to first order in density and containing a (single pole) singularity at the close-packed density.
Abstract: The `derivation' in a standard course in statistical thermodynamics of a very simple equation of state for a hard-disc fluid is discussed. This equation has the form of a simple rational function that fulfils the requirements of being exact to first order in density and containing a (single pole) singularity at the close-packed density. This approach is in the same spirit as that used by Boltzmann in 1898 to propose an equation of state for hard spheres.
TL;DR: In this paper, the spinless Salpeter equation is transformed into integral and integro-differential equations, and the action of the square-root operator is analyzed in the form of a square root operator.
Abstract: The spinless Salpeter equation presents a rather particular differential operator. In this paper we rewrite this equation into integral and integro-differential equations. These kinds of equations are well known and can be more easily handled. We also present some analytical results concerning the spinless Salpeter equation and the action of the square-root operator.
TL;DR: In this paper, the non-linear Poisson-Boltzmann (PB) equation for a uniformly char ged platelet, confined together with co- and counter-ions to a cylindrical cell, is solved semi-analytically by transforming it into an integral equation and solving the latter iteratively.
Abstract: The non-linear Poisson-Boltzmann (PB) equation for a circular, uniformly char ged platelet, confined together with co- and counter-ions to a cylindrical cell, is solved semi-analytically by transforming it into an integral equation and solving the latter iteratively. This method proves efficient and robust, and can be readily generalized to other problems based on cell models, treated within non-linear Poisson-like theory. The solution to the PB equation is computed over a wide range of physical conditions, and the resulting osmotic equation of state is shown to be in semi-quantitative agreement with recent experimental data for Laponite clay suspensions, in the concentrated gel phase.
TL;DR: An improved volume integral equation method for computing electromagnetic scattering by particles in the free space or imbedded in layered media is introduced in this article, where the singularity of Green's function is reduced by moving derivatives to test and expansion functions.
Abstract: An improved volume integral equation method for computing electromagnetic scattering by particles in the free space or imbedded in layered media is introduced. We apply method of moment with rooftop expansion and test function for the solution of volume integral equation. The singularity of Green’s function is reduced by moving derivatives to test and expansion functions. Comparison with Mie theory and other numerical codes are made.
Patent•
22 Jul 1998TL;DR: A programmable very large scale integration (VLSI) chip and method for the analog solution of a family of partial differential equations commonly encountered in engineering and scientific computing are presented in this article.
Abstract: A programmable Very Large Scale Integration (VLSI) chip and method for the analog solution of a family of partial differential equations commonly encountered in engineering and scientific computing: The Laplace equation, the diffusion or conduction equation, the wave equation, the Poission equation, the modified diffusion equation, the modified wave equation, and the wave equation with damping.
01 Jan 1998
TL;DR: In this article, certain integral equations arise in the problem of computing the function w = f (z) that maps a simply connected region D, with boundary Γ and containing the origin, conformally onto the interior or exterior of the unit circle 1w 1 = 1.
Abstract: We shall discuss certain integral equations that arise in the problem of computing the function w = f (z) that maps a simply connected region D, with boundary Γ and containing the origin, conformally onto the interior or exterior of the unit circle 1w 1 = 1. In the case when Γ is a Jordan contour, we obtain Fredholm integral equations of the second kind \(\phi (s) = \pm \int_{\Gamma } {N(s,t)\phi (t)dt + g(s)}\) where φ(s) known as the boundary correspondence function, is to be determined and N(s, t) is the Neumann kernel. We shall discuss an iterative method for numerical computation of the Lichtenstein—Gershgorin equation and present the case of a degenerate kernel and also of the Szego kernel. The case when Γ has a corner yields Stieltjes integral equations and is presented in Chapter 12.
TL;DR: In this article, a reversible system of ODEs possessing some first integrals is considered and a simple description of the first integral space is given, under the auxiliary assumption that the integrals are symmetric (i.e., they do not change sign under the action of the reversing involution).
TL;DR: In this article, the eigenvalues and eigenfunctions for a family of singular integral equations are explicitly found for a class of singular functions, and it is shown how their discrete spectrum becomes continuous as the equation degenerates.
Abstract: Eigenvalues and eigenfunctions are explicitly found for a family of singular integral equations. It is shown how their discrete spectrum becomes continuous as the equation degenerates.
TL;DR: In this article, the boundary exact controllability of dynamical elasticity equations for incompressible materials with memory has been investigated by applying the HUM (Hilbert Uniqueness Method) due to J.L.Lions.
Abstract: This paper is concerned with the boundary exact controllability of the equation u 00 u Z t 0 g(t ) u( )d = r p where Q is a nite cylinder )0; T ( , is a bounded domain of R n ; u = (u1(x; t); ; u2(x; t)), x = (x1; ; xn) are n dimensional vectors and p denotes a pressure term. The result is obtained by applying HUM (Hilbert Uniqueness Method) due to J.L.Lions. The above equation is a simple model of dynamical elasticity equations for incompressible materials with memory.
TL;DR: A new, high order accurate method for the rapid, parallel evaluation of certain integrals in potential theory on general three-dimensional regions, which avoids the problems associated with using quadrature methods to evaluate an integral with a singular kernel.
TL;DR: In this article, Boyadjiev et al. proposed a fractional generalization of the Free Electron Laser (FEL) equation, and the solution is obtained by a method that combines the variation of parameters and successive approximations.
TL;DR: In this paper, the authors studied an integral equation of first kind that arises in the study of inverse problems for nonlinear differential equations and obtained conditions for this integral equation to have a unique solution.
Abstract: We study an integral equation of first kind that arises in the study of inverse problems for nonlinear differential equations. The peculiarity of the equation is that the argument of the unknown function is a given function of two variables. We obtain conditions for this integral equation to have a unique solution.