Showing papers on "Summation equation published in 2004"
TL;DR: In this paper, the authors studied more general integral equations and established regularity, radial symmetry, and monotonicity of the solutions. And they also considered subcritical cases, super critical cases, and singular solutions in all cases; and obtained qualitative======¯¯properties for these solutions.
Abstract: Let $n$ be a positive integer and let $ 0 < \alpha < n.$
In this paper, we study more general integral equation
$ u(x) = \int_{R^n} \frac{1}{|x-y|^{n-\alpha}} K(y) u(y)^p dy.
We establish regularity, radial symmetry, and monotonicity of the
solutions. We also consider subcritical cases, super critical
cases, and singular solutions in all cases; and obtain qualitative
properties for these solutions.
162 citations
TL;DR: In this paper, the physical and analytical properties of a stabilized form of the electric field integral equation are discussed for closed and open perfectly conducting geometries, and it is demonstrated that the modified equation provides a well-conditioned formulation for smooth geometry in both the high and low-frequency limits.
Abstract: The physical and analytical properties of a stabilized form of the electric field integral equation are discussed for closed and open perfectly conducting geometries. It is demonstrated that the modified equation provides a well-conditioned formulation for smooth geometries in both the high- and low-frequency limits; an instability remains near the edges of open geometries, requiring future consideration. The surface Helmholtz decomposition is used to illustrate the mechanism of the stabilization procedure, and the relevance of this mechanism to the numerical discretization of the equation is outlined.
152 citations
TL;DR: A combination of Taylor and Block-Pulse functions on the interval [0,1], that is called Hybrid functions, is used to estimate the solution of a linear Fredholm integral equation of the second kind.
124 citations
TL;DR: In this article, a nonlinear quadratic integral equation of Volterra type in the Banach space of real functions defined and continuous on a bounded and closed interval is studied.
Abstract: We study a nonlinear quadratic integral equation of Volterra type in the Banach space of real functions defined and continuous on a bounded and closed interval. With the help of a suitable measure of noncompactness, we show that the mentioned integral equation has monotonic solutions.
90 citations
TL;DR: In this paper, it was shown that if a linear difference equation is of order ≥ 1/2, then the Newton polygon can be obtained from the equation, and the solution of the equation can be computed with a constant ρ > 0.
Abstract: Let $f(z)$ be an entire function of order less than $1/2.$ We consider an
analogue of the Wiman-Valiron theory rewriting power series of $f(z)$ into
binomial series. As an application, it is shown that if a transcendental entire
solution $f(z)$ of a linear difference equation is of order $\chi < 1/2,$
then we have %$\chi$ is obtained from the Newton polygon for the equation, and
$\log M(r,f) = Lr^{\chi}(1 + o(1))$ with a constant $L > 0.$
69 citations
TL;DR: In this article, a Galerkin discretization scheme for a modified form of the electric field integral equation for smooth, three-dimensional, perfectly conducting scatterers is presented.
Abstract: The details of a Galerkin discretization scheme for a modified form of the electric field integral equation are outlined for smooth, three-dimensional, perfectly conducting scatterers. Limitations of the divergence conforming finite-element bases in preserving the self-stabilizing properties of the electric field integral equation operator are indicated. A numerically efficient alternative is outlined which relies on an operator-based Helmholtz decomposition. The condition number of the resulting matrix equation is demonstrated to be frequency independent for scattering from a perfectly conducting sphere at various frequencies.
68 citations
TL;DR: In this paper, two integral equation methods are considered which can be inverted to provide the surface velocity and/or pressure given a measurement of the pressure on an imaginary surface in the near field of a vibrating or scattering body.
Abstract: Two integral equation methods are considered which can be inverted to provide the surface velocity and/or pressure given a measurement of the pressure on an imaginary surface in the nearfield of a vibrating or scattering body. This problem is central to nearfield acoustical holography (NAH). The integral equations are discretized using boundary element methods (BEMs). The integral equation methods considered are (1) an indirect formulation method based on the single layer integral equation and (2) a direct formulation method based on a system of equations derived from the Helmholtz–Kirchhoff integral equation. The formation of integral equations from the mentioned methods will not involve the explicit inversion of matrices, but instead will require this inversion to be done implicitly. Since these methods back-track the sound field from the measurement surface to the surface of the source/vibrator they are ill-posed in nature and Tikhonov regularization are used to stabilize the reconstruction. Problems a...
57 citations
TL;DR: In this paper, a second kind integral equation formulation for the Dirichlet problem for the Laplace equation in two dimensions, with the boundary conditions specified on a collection of open curves, is presented.
56 citations
TL;DR: In this article, the authors prove the existence and regularity of the density of the real-valued solution to a three-dimensional stochastic wave equation using the tools of Malliavin calculus on the appropriate Gaussian space related to the noise.
Abstract: We prove the existence and regularity of the density of the real-valued solution to a three-dimensional stochastic wave equation. The noise is white in time and has a spatially homogeneous correlation whose spectral measure μ satisfies ∫ R 3 μ(dξ)( 1+|ξ| 2) - η<∞ , for some η ∈(0,1 2 ) . Our approach uses the mild formulation of the equation given by means of Dalang's extended version of Walsh's stochastic integration. We apply the tools of Malliavin calculus on the appropriate Gaussian space related to the noise. An extension of Dalang's stochastic integral to the Hilbert-valued setting is needed. Let S3 be the fundamental solution to the three-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral ∫ 0 t ds∫ R 3 μ(dξ)| cal F S 3(s)(ξ)| 2 and upper bounds for ∫ 0 t ds∫ R 3 μ(dξ)|ξ|| cal F S 3(s)(ξ)| 2 in terms of powers of t. These estimates, together with a suitable mollifying procedure for S3, are crucial in the analysis of the inverse of the Malliavin variance.
52 citations
TL;DR: In this paper, the complementary mild-slope equation is derived from Hamilton's principle in terms of stream function, and the numerical results of the present model are compared to the exact linear theory and the existing mild slope equations that have been derived from the velocity-potential formulation.
Abstract: A new depth-integrated equation is derived to model a time-harmonic motion of small-amplitude waves in water of variable depth. The new equation, which is referred to as the complementary mild-slope equation here, is derived from Hamilton's principle in terms of stream function. In the formulation, the continuity equation is satisfied exactly in the fluid domain. Also satisfied exactly are the kinematic boundary conditions at the still water level and the uneven sea bottom. The numerical results of the present model are compared to the exact linear theory and the existing mild-slope equations that have been derived from the velocity-potential formulation. The computed results give better agreement with those of the exact linear theory than the other mild-slope equations. Comparison shows that the new equation provides accurate results for a bottom slope up to 1.
50 citations
07 Jun 2004
TL;DR: In this paper, it was shown that the difference equation xn+1 = f (xn,xn−1,..., xn−k ), n= 0, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 24
Abstract: where the initial conditions x−1, x0 are arbitrary real numbers and a, b, c, d are positive constants. Recently, there has been a lot of interest in studying the global attractivity, boundedness character, and the periodic nature of nonlinear difference equations. For some results in this area, see, for example, [1–13], we recall some notations and results which will be useful in our investigation. Let I be some interval of real numbers and the function f has continuous partial derivatives on Ik+1, where Ik+1 = I × I × ··· × I (k + 1− times). Then, for initial conditions x−k, x−k+1, . . . , x0 ∈ I , it is easy to see that the difference equation xn+1 = f ( xn,xn−1, . . . ,xn−k ) , n= 0,1, . . . , (1.2)
TL;DR: In this article, the Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions (13) that can be considered as a generalization of the sine-Gordon equation.
Abstract: The Singular Manifold Method (SMM) is applied to an equation in 2 + 1 dimensions (13) that can be considered as a generalization of the sine-Gordon equation. SMM is useful to prove that the equation has two Painleve branches and, therefore, it can be considered as the modified version of an equation with just one branch, that is the AKNS equation in 2 + 1 dimensions. The solutions of the former split as linear superposition of two solutions of the second, related by a Backlund-gauge transforma- tion. Solutions of both equations are obtained by means of an algorithmic procedure derived from these transformations.
TL;DR: This paper proposes a new combined field integral equation, converts it into variational form, establishes its coercivity in the natural trace spaces for electromagnetic fields, and concludes existence and uniqueness of solutions for any frequency.
Abstract: Many boundary integral equation methods used in the simulation of direct electromagnetic scattering of a time-harmonic wave at a perfectly conducting obstacle break down when applied at frequencies close to a resonant frequency of the obstacle. A remedy is offered by special indirect boundary element methods based on the so-called combined field integral equation. However, hitherto no theoretical results about the convergence of discretized combined field integral equations have been available. In this paper we propose a new combined field integral equation, convert it into variational form, establish its coercivity in the natural trace spaces for electromagnetic fields, and conclude existence and uniqueness of solutions for any frequency. Moreover, a conforming Galerkin discretization of the variational equations by means of ${\rm div}_\Gamma$-conforming boundary elements can be shown to be asymptotically quasi-optimal. This permits us to derive quantitative convergence rates on sufficiently fine, uniformly shape-regular sequences of surface triangulations.
TL;DR: In this paper, the boundary integral equation approach, displacement domain integral equation (DDE), and strain domain integral equations (SDE) were applied to calculate the elastostatic field of an isotropic elastic half-plane containing possibly isotropics, orthotropic and anisotropic inclusions.
Abstract: The boundary integral equation approach, also known as the boundary element method, the displacement domain integral equation approach and the strain domain integral equation approach are applied to calculate the elastostatic field of an isotropic elastic half-plane containing possibly isotropic, orthotropic and anisotropic inclusions. Numerical examples are given to show the characteristics of these three integral equation approaches.
TL;DR: In this paper, a nonhomogeneous linear delay difference equation with continuous variable is considered and an asymptotic result for the solutions is obtained by the use of a positive root with an appropriate property of the so called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equations.
Abstract: We consider a nonhomogeneous linear delay difference equation with continuous variable and establish an asymptotic result for the solutions. Our result is obtained by the use of a positive root with an appropriate property of the so called characteristic equation of the corresponding homogeneous linear (autonomous) delay difference equation. More precisely, we show that, for any solution, the limit of a specific integral transformation of it, which depends on a suitable positive root of the characteristic equation, exists as a real number and it is given explicitly in terms of the positive root of the characteristic equation and the initial function.
TL;DR: In this paper, the authors used the neutral particle, single-collision kernels to produce benchmark solutions in one-dimensional planar and spherical geometries, where the solution procedure begins with the formulation of an inhomogeneous integral equation for time-dependent transport.
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TL;DR: In this article, the radial symmetry property of positive solutions of an integral equation arising from some higher order semi-linear elliptic equations on the whole space was considered and the main result is Theorem 1 below.
Abstract: In this paper, we consider radial symmetry property of positive solutions of an integral equation arising from some higher order semi-linear elliptic equations on the whole space $\mathbf{R}^n$. We do not use the usual way to get symmetric result by using moving plane method. The nice thing in our argument is that we only need a Hardy-Littlewood-Sobolev type inequality. Our main result is Theorem 1 below.
TL;DR: In this paper, the authors discuss a population equation with diffusion and explain the origin of this equation and study it with the theory developed by G. Fragnelli and G. Nickel.
01 Jan 2004
TL;DR: In this paper, the authors proved a pointwise estimate for the difference between the solution of an abstract heat equation and a wave equation which results from the heat equation by a time singular perturbation.
Abstract: R. Ikehata recently proved some integral estimate for the difference between the solution of an abstract heat equation and the solution of an abstract wave equation which results from the heat equation by a time singular perturbation. The estimate is obtained if the initial values are chosen appropriately. We prove a pointwise estimate which improves the above result for large times into several directions, and we also establish the optimality of this estimate for the wave equation in an exterior domain. Our proofs rely on the spectral theorem for unbounded self-adjoint operators.
TL;DR: In this article, a class of partial integro-differential equations defined on a spatially extended domain that arise in the modeling of neuronal networks was studied. And it was shown that any solution of ODE that satisfies a fairly relaxed growth condition is also a solution of the integral equation.
TL;DR: In this paper, the authors considered a class of Schrodinger equations with a symmetric double-well potential and an external, both repulsive and attractive, nonlinear perturbation and showed that, under certain conditions and in the limit of large barrier between the two wells, the reduction of the time-dependent equation to a two-mode equation gives the dominant term of the solution with precise estimate of the error.
Abstract: We consider a class of Schrodinger equations with a symmetric double-well potential and an external, both repulsive and attractive, nonlinear perturbation. We show that, under certain conditions and in the limit of large barrier between the two wells, the reduction of the time-dependent equation to a two-mode equation gives the dominant term of the solution with a precise estimate of the error.
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TL;DR: In this article, the modified non-linear Schroedinger equation is constructed using an Ansatz for the wave function, which has time and azimuthal angle dependence introduced previously in the study of spinning boson stars and Q-balls.
Abstract: Fullerenes and nanotubes consist of a large number of carbon atoms sitting on the sites of a regular lattice For pratical reasons it is often useful to approximate the equations on this lattice in terms of the continuous equation At the moment, the best candidate for such an equation is the modified non-linear Schroedinger equation In this paper, we study the modified non-linear Schroedinger equation, which arises as continuous equation in a system describing an excitation on a hexagonal lattice This latter system can eg describe electron-phonon interaction on fullerene related structures such as the buckminster fullerene and nanotubes Solutions of this modified non-linear Schroedinger equation, which have solitonic character, can be constructed using an Ansatz for the wave function, which has time and azimuthal angle dependence introduced previously in the study of spinning boson stars and Q-balls We study these solutions on a sphere, a disc and on a cylinder
Our construction suggests that non-spinning as well as spinning solutions, which have a wave function with an arbitrary number of nodes, exist We find that this property is closely related to the series of well-known mathematical functions, namely the Legendre functions when studying the equation on a sphere, the Bessel functions when studying the equation on a disc and the trigonometric functions, respectively, when studying the equation on a cylinder
TL;DR: Here, the solution in one, two and three dimensional for the Volterra-Fredholm integral equation of the first kind is obtained in the space L"2(@W)xC[0,T], T<~.
TL;DR: In this article, the adaptive integral method (AIM) was used to solve the scattering problem of mixed dielectric/conducting objects, which can reduce the memory requirement for storage and speed up the matrixvector multiplication in the iterative solver.
Abstract: This paper presents the adaptive integral method (AIM) utilized to solve scattering problem of mixed dielectric/conducting objects. The scattering problem is formulated using the Poggio-Miller- Chang-Harrington-Wu-Tsai (PMCHWT) formulation and the electric field integral equation approach for the dielectric and conducting bodies, respectively. The integral equations solved using these approaches can eliminate the interior resonance of dielectric bodies and produce accurate results. The method of moments (MoM) is applied to discretize the integral equations and the resultant matrix system is solved by an iterative solver. The AIM is used then to reduce the memory requirement for storage and to speed up the matrix-vector multiplication in the iterative solver. Numerical results are finally presented to demonstrate the accuracy and efficiency of the technique.
TL;DR: In this article, the use of orthonormal wavelet basis functions for solving singular integral scattering equations is investigated and it is shown that these basis functions lead to sparse matrix equations which can be solved by iterative techniques.
Abstract: The use of orthonormal wavelet basis functions for solving singular integral scattering equations is investigated. It is shown that these basis functions lead to sparse matrix equations which can be solved by iterative techniques. The scaling properties of wavelets are used to derive an efficient method for evaluating the singular integrals. The accuracy and efficiency of the wavelet transforms are demonstrated by solving the two-body T-matrix equation without partial wave projection. The resulting matrix equation which is characteristic of multiparticle integral scattering equations is found to provide an efficient method for obtaining accurate approximate solutions to the integral equation. These results indicate that wavelet transforms may provide a useful tool for studying few-body systems.
TL;DR: Necessary and sufficient conditions are obtained for the existence of positive solutions of a second-order half-linear dynamic equation on time scales.
TL;DR: In this article, a well-posedness result for an initial-boundary value problem with only integral conditions over the spatial domain for a one-dimensional quasilinear wave equation is presented.
Abstract: This paper presents a well-posedness result for an initial-boundary value problem with only integral conditions over the spatial domain for a one-dimensional quasilinear wave equation. The solution and some of its properties are obtained by means of a suitable application of the Rothe time-discretization method.
TL;DR: In this paper, the existence of solutions of an integral equation related to traffic flow is proved using a technique associated with measures of noncompactness, which can be seen as a non-compact approach.
Abstract: Using a technique associated with measures of noncompactness we prove the existence of solutions of an integral equation related to traffic flow.
TL;DR: In this article, an image reconstruction algorithm for the inverse conductivity problem based on reformulating the problem in terms of integral equations is presented, where the values of injected electric currents and of the corresponding induced boundary potentials, as well as the boundary values of the electrical conductivity.