Numerical Solution Of Integral Equations Research Articles

The Estimation of the Error at Richardson’s Extrapolation and the Numerical Solution of Integral Equations of the Second Kind

The mode of definition of the error at polynomial Richardson’s extrapolation is described. Along with the table of extrapolations the new magnitudes reflecting expediency and efficiency of extrapolation are entered. On concrete examples it is shown that application of Richardson’s extrapolation to a solution of integral equations has appeared rather effective and gives a solution with a high exactitude. Application of formulas of interpolation leads to a solution in the analytical aspect.

Numerical solution of integral equations for a scalar diffraction problem

Numerical solution of integral equations for a scalar diffraction problem

On the Convergence of the ACA Algorithm for Radiation and Scattering Problems

The adaptive cross approximation (ACA) algorithm, when used to accelerate the numerical solution of integral equations for radiation and scattering problems, sometimes suffers from inaccuracies. These inaccuracies occur when the ACA convergence criterion, which is based on an approximation of the residual relative error, is prematurely satisfied. This paper identifies the two sources of this problem and proposes adaptations of the algorithm that remedy them.

A meshless discrete Galerkin (MDG) method for the numerical solution of integral equations with logarithmic kernels

This paper describes a computational method for solving Fredholm integral equations of the second kind with logarithmic kernels. The method is based on the discrete Galerkin method with the shape functions of the moving least squares (MLS) approximation constructed on scattered points as basis. The MLS methodology is an effective technique for the approximation of an unknown function that involves a locally weighted least square polynomial fitting. The numerical scheme developed in the current paper utilizes the non-uniform Gauss–Legendre quadrature rule for approximating logarithm-like singular integrals and so reduces the solution of the logarithmic integral equation to the solution of a linear system of algebraic equations. The proposed method is meshless, since it does not require any background mesh or domain elements. The error analysis of the method is provided. The scheme is also applied to a boundary integral equation which is a reformulation of a boundary value problem of Laplace’s equation with linear Robin boundary conditions. Finally, numerical examples are included to show the validity and efficiency of the new technique.

An improved method based on Haar wavelets for numerical solution of nonlinear integral and integro-differential equations of first and higher orders

In this paper, a novel technique is being formulated for the numerical solution of integral equations (IEs) as well as integro-differential equations (IDEs) of first and higher orders. The present approach is an improved form of the Haar wavelet methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013). The proposed modifications resulted in computational efficiency and simple applicability of the earlier methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013). In addition to this, the new approach is being extended from IDEs of first order to IDEs of higher orders with initial- and boundary-conditions. Unlike the methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013) (where the kernel function is being approximated by two-dimensional Haar wavelet), the kernel function in the present case is being approximated by one-dimensional Haar wavelet. The modified approach is easily extendable to higher order IDEs. Numerical examples are being included to show the accuracy and efficiency of the new method.

On correct application of one-step inversion of gravity data

One of the main problems on the numerical solution of integral equations is the resolution of input data. Among the integral equations used in geodesy we have the “onestep inversion” based on the first derivative of the Poisson integral, which transforms gravity values on the Earth’s surface to the gravity potential on the reference ellipsoid. In this study, it is shown that the required spatial resolution of the input gravity data on the Earth’s surface for correct one-step inversion depends on the height of the computational region, the fact that if overlooked can cause totally wrong results. Consequently the following two major questions are posed: (i) How could one know whether the spatial resolution of the input gravity data for correct one-step inversion is sufficient? (ii) What should be done if the spatial resolution is not sufficient? By studying the behaviour of the integral kernel, an algorithm is presented which enables an appropriate answer to the former question. In order to address the latter question, a method is proposed to modify the integral kernel which overcomes the adverse effect of insufficient spatial resolution of the input gravity data. Our answers, which possess the novelty of the study, are numerically verified by means of real and simulated gravity data. The numerical results approve the efficiency of the proposed method in solving the problem of insufficient spatial resolution of the input gravity data for correct one-step inversion.

Sinc approximation for numerical solution of integral equation arising in conductor-like screening model for real solvent

In this study, we present the existence of solutions for some nonlinear integral equations known as Hammerstein type which forms the basis for the conductor-like screening model for real solvents (COSMO-RS). By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo’s theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a powerful numerical approach based on the Sinc approximation to solve the equation which forms the basis for the conductor-like screening model for real solvents (COSMO-RS). The approach is based on the recognition of the Hammerstein integral equation for the determination of the chemical potential of a surface segment as a function of screening charge density. Then convergence of this technique is discussed by preparing a theorem which shows an exponential type convergence rate and guarantees the applicability of that.

Two-dimensional wavelets for numerical solution of integral equations

Purpose: In this paper, we shall investigate the numerical solution of two-dimensional Fredholm integral equations (2D-FIEs). Methods: In this work, we apply two-dimensional Haar wavelets, to solve linear two dimensional Fredholm integral equations (2D-FIEs). Using 2D Haar wavelets and their properties, 2D-FIEs of the second kind reduce to a system of algebraic equations. Results: The numerical examples illustrate the efficiency and accuracy of the method. Conclusions: In comparison with other bases (for example, polynomial bases), one of the advantages of this method is, although the involved matrices have a large dimension, they contain a large percentage of zero entries, which keeps computational effort within reasonable limits.

Effects of polydispersity and anisotropy in colloidal and protein solutions: An integral equation approach

Application of integral equation theory to complex fluids is reviewed, with particular emphasis to the effects of polydispersity and anisotropy on their structural and thermodynamic properties. Both analytical and numerical solutions of integral equations are discussed within the context of a set of minimal potential models that have been widely used in the literature. While other popular theoretical tools, such as numerical simulations and density functional theory, are superior for quantitative and accurate predictions, we argue that integral equation theory still provides, as in simple fluids, an invaluable technique that is able to capture the main essential features of a complex system, at a much lower computational cost. In addition, it can provide a detailed description of the angular dependence in arbitrary frame, unlike numerical simulations where this information is frequently hampered by insufficient statistics. Applications to colloidal mixtures, globular proteins and patchy colloids are discussed, within a unified framework.

Automatic simulation of electrochemical transients at cylindrical wire electrodes, by the adaptive Huber method for Volterra integral equations

Numerical solutions of integral equations describing controlled-potential transient experiments at cylindrical wire or fiber electrodes were rarely attempted in the past, seemingly due to the lack of adequate approximations to the kernel functions arising in such cases. In order to overcome this difficulty, in the former work of the present author a highly accurate approximation to the kernel function representing cylindrical diffusion in the absence of homogeneous reactions has been designed. In the present work the approximation is combined with the recently developed adaptive Huber method for solving electrochemical integral equations of Volterra type. The resulting method is tested on examples of kinetic models. The performance of the method is found similar to that previously reported for integral equations involving kernels specific for planar electrodes. The method provides automatic solutions with an accuracy that can be effectively set up by choosing an appropriate value of the error tolerance parameter. Errors corresponding to the range from about 10 −2 (relative to the maximum solution value) down to about 10 −6 or even less, can be easily achieved at a modest computational cost.

The Goos-Hänchen effect for surface plasmon polaritons

By means of an impedance boundary condition and numerical solution of integral equations for the scattering amplitudes to which its use gives rise, we study as a function of its angle of incidence the reflection of a surface plasmon polariton beam propagating on a metal surface whose dielectric function is ɛ1(ω) when it is incident on a planar interface with a coplanar metal surface whose dielectric function is ɛ2(ω). When the surface of incidence is optically more dense than the surface of scattering, i.e. when |ɛ2(ω)|≫|ɛ1(ω)|, the reflected beam undergoes a lateral displacement whose magnitude is several times the wavelength of the incident beam. This displacement is the surface plasmon polariton analogue of the Goos-Hänchen effect. Since this displacement is sensitive to the dielectric properties of the surface, this effect can be exploited to sense modifications of the dielectric environment of a metal surface, e.g. due to adsorption of atomic or molecular layers on it.

Modified integral current methods in electrodynamics of nonhomogeneous media

The main difficulty in numerical solution of integral equations of electrodynamics is associated with the need to solve a high-order system of linear equations with a dense matrix. It is therefore relevant to develop numerical methods that lead to linear equation systems of lower order at the cost of more complex evaluation of the coefficients. In this article we propose a method for solving linear equations of electrodynamics which is a modification of the integral current method. The main distinctive feature of the proposed method is double integration of the electric Green's tensor in the process of algebraization of the original integral equation. The solutions of the system of linear equations are thus integral means of the electric field inside the anomaly constructed by the proposed transformation formula. We prove convergence and derive error bounds for both the solution of the integral equation and the electromagnetic field components evaluated from approximate transformation formulas.

Theory of coupled open resonators

A theory of coupled open resonators with an inclined beam splitter is constructed on the basis of numerical solution of integral equations with allowance for ohmic and diffraction losses caused by finiteness of mirror sizes. High selectivity of longitudinal modes is shown to be possible with the effective extraction in the form of a wave beam and excess of the useful radiation power over the internal loss.

Product integration methods based on discrete spline quasi-interpolants and application to weakly singular integral equations

Quadrature formulae are established for product integration rules based on discrete spline quasi-interpolants on a bounded interval. The integrand considered may have algebraic or logarithmic singularities. These formulae are then applied to the numerical solution of integral equations with weakly singular kernels.

Method of boundary integral equations as applied to the numerical solution of the three-dimensional Dirichlet problem for the laplace equation in a piecewise homogeneous medium

A Dirichlet problem is considered in a three-dimensional domain filled with a piecewise homogeneous medium. The uniqueness of its solution is proved. A system of Fredholm boundary integral equations of the second kind is constructed using the method of surface potentials, and a system of boundary integral equations of the first kind is derived directly from Green’s identity. A technique for the numerical solution of integral equations is proposed, and results of numerical experiments are presented.

Adaptive Monte Carlo methods for matrix equations with applications

This paper discusses empirical studies with both the adaptive correlated sequential sampling method and the adaptive importance sampling method which can be used in solving matrix and integral equations. Both methods achieve geometric convergence (provided the number of random walks per stage is large enough) in the sense: e ν ≤ c λ ν , where e ν is the error at stage ν , λ ∈ ( 0 , 1 ) is a constant, c > 0 is also a constant. Thus, both methods converge much faster than the conventional Monte Carlo method. Our extensive numerical test results show that the adaptive importance sampling method converges faster than the adaptive correlated sequential sampling method, even with many fewer random walks per stage for the same problem. The methods can be applied to problems involving large scale matrix equations with non-sparse coefficient matrices. We also provide an application of the adaptive importance sampling method to the numerical solution of integral equations, where the integral equations are converted into matrix equations (with order up to 8192×8192) after discretization. By using Niederreiter’s sequence, instead of a pseudo-random sequence when generating the nodal point set used in discretizing the phase space Γ , we find that the average absolute errors or relative errors at nodal points can be reduced by a factor of more than one hundred.

Closed-form error estimates for the numerical solution of Fredholm integral equations of the second kind

Closed-form algebraic formulae are derived for the local error incurred in the numerical solution of integral equations by iterated collocation methods, the analysis being illustrated by application to Fredholm integral equations of the second kind. The novel error analysis uses an asymptotic approach, in the small parameter of the numerical mesh size, applied to a finite-rank degenerate-kernel orthogonalpolynomial approximation of the exact kernel. It is proved that, under suitable conditions, the discrepancy between our theoretically predicted error and the actual numerical error tends to zero at the rate ‖K − KM‖, where M is the rank of the degenerate-kernel approximation. A leading-order error analysis is validated on three increasingly accurate projection methods applied to both smooth and sharply peaked kernels, our error predictions being demonstrated to be exponentially convergent to experimentally obtained global numerical errors. The new method is demonstrated to be cost-effective relative to standard extrapolation.

Sparsification of the Impedance Matrix in the Solution of the Integral Equation by Using the Maximally Orthogonalized Basis Functions

The higher order vector basis functions defined in large patches have been utilized in the numerical solution of integral equations in this paper to sparsify the impedance matrix and relieve the memory pressure. The physical explanation for the sparsification of the impedance matrix is also elucidated. Furthermore, the maximally orthogonalized bases have been applied to improve the condition number of the impedance matrix. The scaling factor was reformed to speed up the iteration convergence in the numerical solution. Finally, the iterative method for sparse matrix equations is applied to improve the solution efficiency. Some numerical results are provided to illustrate the excellent performance both in the sparsification of the impedance matrix and solution efficiency for numerical analysis of the scattering problem.

A variational method using Alpert multiwavelets

The numerical solution of variational problems is usually achieved by numerical solution of the Euler--Lagrange differential equations or by Rayleigh--Ritz direct methods. In 1975, Chen and Hsiao showed how Walsh functions could be used in a direct method to solve several model variational problems. This Rayleigh--Ritz method was generalized by Sloss and Blyth in 1998. In 2004, Hsiao modified the Walsh function method to provide a Haar wavelet direct method and illustrated how this could be used for the solution of a few model problems. We extend this by applying Alpert multiwavelets to the direct solution of variational problems. Alpert multiwavelets were developed for the numerical solution of integral equations and provide a generalization of Haar wavelets. Alpert multiwavelets have the advantage of being expressible as simple polynomials over disjoint subintervals allowing for ease of computation. The method is applied to an example which allows comparisons with the results for Haar wavelets to be made. Some convergence results are given. References B. K. Alpert. Sparse Representation of Smooth Linear Operators. PhD Thesis, Yale University, 1990. B. K. Alpert, G. Beylkin, D. Gines and L. Vozovoi. Adaptive Solution of Partial Differential Equations in Multiwavelet Bases. Journal of Computational Physics , 182, 149--190, 2002. doi:10.1006/jcph.2002.7160 R. Y. Chang and M. L. Wang. Shifted Legendre direct method for variational problems. J. Optimization Theory Appl. , 39, 299--307, 1983. C. F. Chen and C. H. Hsiao. A Walsh series direct method for solving variational problems. J. Franklin Inst. , 300, 265--280, 1975. I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. , 41(7), 909--996, 1988. C. H. Hsiao. Haar wavelet direct method for solving variational problems. Mathematics and Computers in Simulation , 64, 569--585, 2004. doi:10.1016/j.matcom.2003.11.012 C. Hwang and Y. P. Shih. Laguerre series direct method for variational problems. J. Optimization Theory Appl. , 39, 143--149, 1983. R. S. Schechter. The Variational Method in Engineering . McGraw--Hill, NewYork, 1967. B. G. Sloss and W. F. Blyth. A variational method using Walsh functions. Nonlinear Analysis, Theory, Methods and Applications , 32, 549--561, 1998. D. F. Walnut. An introduction to wavelet analysis . Birkhauser, Boston, 2002.

On the numerical solution of integral equations for some main problems in plane elasticity

On the numerical solution of integral equations for some main problems in plane elasticity