Introduction to nonlinear differential and integral equations.

First published in 1960 by the U. S. Atomic Energy Commission.

Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle

The problem considered is that of determining the shape of a planar acoustically sound-soft obstacle from knowledge of the far-field pattern for one time-harmonic incident field. Two methods, which are based on the solution of a pair of integral equations representing the incoming wave and the far-field pattern, respectively, are proposed and investigated for finding the unknown boundary. Numerical resultsare included which show that the methods give accurate numerical approximations in relatively few iterations.

On the Zeros of the Solutions to Nonlinear Hyperbolic Equations with Delays

2002 Mathematics Subject Classification: Primary 35В05; Secondary 35L15

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.

Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.

Population inversion between the sup(3)Hsub(4) and the sup(3)Fsub(4) excited states of Tmsup(3+) investigated by means of numerical solutions of the rate equations system in Tmsup(3+)-doped and Tmsup(3+), Hosup(3+)-codoped fluoride glasses

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Energy transfer rates and population inversion of sup(4)Isub(11/2) excited state of Ersup(3+) investigated by means of numerical solutions of the rate equations system in Er:LiYFsub(4) crystal

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert's method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.

Solvabilty of generalized Hammerstein integral equations on unbounded domains, with sign-changing kernels

In this work we study an Hammerstein generalized integral equation u(t)=∫_{-∞}^{+∞}k(t,s) f(s,u(s),u′(s),...,u^{(m)}(s))ds, where k:ℝ²→ℝ is a W^{m,∞}(ℝ²), m∈ℕ, kernel function and f:ℝ^{m+2}→ℝ is a L¹-Carathéodory function. To the best of our knowledge, this paper is the first one to consider discontinuous nonlinearities with derivatives dependence, without monotone or asymptotic assumptions, on the whole real line. Our method is applied to a fourth order nonlinear boundary value problem, which models moderately large deflections of infinite nonlinear beams resting on elastic foundations under localized external loads.

Eigenstrain boundary integral equations with local Eshelby matrix for stress analysis of ellipsoidal particles

Aiming at the large scale numerical simulation of particle reinforced materials, the concept of local Eshelby matrix has been introduced into the computational model of the eigenstrain boundary integral equation (BIE) to solve the problem of interactions among particles. The local Eshelby matrix can be considered as an extension of the concepts of Eshelby tensor and the equivalent inclusion in numerical form. Taking the subdomain boundary element method as the control, three-dimensional stress analyses are carried out for some ellipsoidal particles in full space with the proposed computational model. Through the numerical examples, it is verified not only the correctness and feasibility but also the high efficiency of the present model with the corresponding solution procedure, showing the potential of solving the problem of large scale numerical simulation of particle reinforced materials.

Analytical and numerical solutions of inward spherical solidification of a superheated melt with radiative-convective heat transfer and density jump at freezing front

Analytical and numerical solutions of a general problem related to the radially symmetric inward spherical solidification of a superheated melt have been studied in this paper. In the radiation-convection type boundary conditions, the heat transfer coefficient has been taken as time dependent which could be infinite, at time,t=0. This is necessary, for the initiation of instantaneous solidification of superheated melt, over its surface. The analytical solution consists of employing suitable fictitious initial temperatures and fictitious extensions of the original region occupied by the melt. The numerical solution consists of finite difference scheme in which the grid points move with the freezing front. The numerical scheme can handle with ease the density changes in the solid and liquid states and the shrinkage or expansions of volumes due to density changes. In the numerical results, obtained for the moving boundary and temperatures, the effects of several parameters such as latent heat, Boltzmann constant, density ratios, heat transfer coefficients, etc. have been shown. The correctness of numerical results has also been checked by satisfying the integral heat balance at every timestep.

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.

On the nonlinear integral transform of an ocean wave spectrum into an along-track interferometric synthetic aperture radar image spectrum

We present a new nonlinear integral transform relating the ocean wave spectrum to the along-track interferometric synthetic aperture radar (AT-INSAR) image spectrum. The AT-INSAR, which is a synthetic aperture radar (SAR) employing two antennas displaced along the platform's flight direction, is considered to be a better instrument for imaging ocean waves than the SAR. This is because the AT-INSAR yields the phase spectrum and not only the amplitude spectrum as with the conventional SAR. While the SAR and AT-INSAR amplitude spectra depend strongly on the modulation of the normalized radar cross section (NRCS) by the long ocean waves, which is poorly known, the phase spectrum depends only weakly on this modulation. By measuring the phase difference between the signals received by both antennas, AT-INSAR measures the radial component of the orbital velocity associated with the ocean waves, which is related to the ocean wave height field by a well-known transfer function. The nonlinear integral transform derived in this paper differs from the one previously derived by Bao et al. [1999] by an additional term containing the derivative of the radial component of the orbital velocity associated with the long ocean waves. By carrying out numerical simulations, we show that, in general, this additional term cannot be neglected. Furthermore, we present two new quasi-linear approximations to the nonlinear integral transform relating the ocean wave spectrum to the AT-INSAR phase spectrum.