A numerical solution for integral equations

A method for the numerical approximation for the solution of Fredholm integral equations of the second kind is presented. The approximation is obtained using cubic B-splines as co-ordinate functions in the variational formulation of the problem.     

The tanh transformation for the solution of singular integral equations

Using a tanh transformation a quadrature formula for the evaluation of singular integrals is obtained. The formula has the same step length h as the formula for regular integrals derived by F. Stenger. These quadrature formulae are valid for end point singularities of any order and their error exhibits an exponential decay when the number of integrations tends to infinity. Using these formulae the solution of singular integral equations does not depend on the order of the end point singularities. Furthermore the collocation points are given by a very simple equation and, in the case of constant coefficients, by a closed-form formula.     

Numerical solution of the integral equations for optimal sequential filtering

Inhomogeneous Fredholm integral equations occur frequently in communication theory where it is desired to determine optimal receivers and filters for signal detection and estimation. In this paper an initial value method is utilized to determine the Fredholm resolvent and the solution of the integral equation. Numerical results are given for a simple example. The method is of particular interest where sequential solutions are desired.     

The numerical solution of certain dual integral equations

Summary Dual integral equations with Hankel kernels are reduced to simple integral equations whose unknowns are physically meaningful quantities. Numerical methods for solving these integral equations are established and applied to a problem in electrostatics.     

A method for the numerical solution of singular integral equations with a principal value integral

A method for the numerical solution of singular integral equations with kernels having a singularity of the Cauchy type is presented. The singular behavior of the unknown function is explicitly built into the solution using the index theorem. The integral equation is replaced by integral relations at a discrete set of points. The integrand is then approximated by piecewise linear functions involving the value of the unknown function at a finite set of points. This permits integration in a closed form analytically. Thus the problem is reduced to a system of linear algebraic equations. The results obtained in this way are compared with the more sophisticated procedures based on Gauss-Chebyshev and Lobatto-Chebyshev quadrature formulae. An integral equation arising in a crack problem of the classical theory of elasticity is used for this purpose.     

On shape differentiation of discretized electric field integral equation

This work presents shape derivatives of the system matrix representing electric field integral equation discretized with Raviart–Thomas basis functions. The arising integrals are easy to compute with similar methods as the entries of the original system matrix. The results are compared to derivatives computed with automatic differentiation technique and finite differences, and are found to be in an excellent agreement. Furthermore, the derived formulas are employed to analyze shape sensitivity of the input impedance of a planar inverted F-antenna, and the results are compared to those obtained using a finite difference approximation.     

Numerical solution of Fredholm integral equations with a logarithmic singularity

Generalized quadrature is used for the numerical solution of two Fredholm integral equations which occur in electrostatics and aerodynamics.     

Isoparametric,finite element,variational solution of integral equations for three-dimensional fields

An improved numerical method, based on a variational approach with isoparametric finite elements, is presented for the solution of the boundary integral equation formulation of three-dimensional fields. The technique provides higher-order approximation of the unknown function over a bounding surface described by two-parameter, non-planar elements. The integral equation is discretized through the Rayleigh–Ritz procedure. Convergence to the solution for operators having a positive-definite component is guaranteed. Kernel singularities are treated by removing them from the relevant integrals and dealing with them analytically. A successive element iterative process, which produces the solution of the large dense matrix of the complete structure, is described. The discretization and equation solution take place one element at a time resulting in storage and computational savings. Results obtained for classical test models, involving scalar electrostatic potential and vector elastostatic displacement fields, demonstrate the technique for the solution of the Fredholm integral equation of the first kind. Solution of the Fredholm equation of the second kind is to be reported subsequently.     

Mutually constrained partial differential and integral equations for an exterior field problem

A new implementation of the mutually constrained partial differential and integral equation method for the exterior 2-dimensional field problem is described. It is shown, that the method is applicable to exterior problems in an inhomogeneous medium. The inhomogeneity is considered in the finite element procedure and in boundary element method, where an adequate Green's function is applied. The temperature distribution around a three-cable system is then computed as an illustration. The eddy-current losses in the cable sheaths are calculated using the Fredholm integral equation of the second kind.     

The use of null field integral equations in magnetic-field problems

The objective of this paper is to discuss the use of the null field integral equations in the solution of eddy current magnetic field problems. Their usage is approached through a development of the "classical" boundary integral equation technique as applied to irregularly shaped two-dimensional objects. Two immediate advantages are discussed, the first being the circumvention of singularities in the integrands. The second is the ability to improve the conditioning of the determination matrix by appropriate choice of null field points. The technique is tested on an irregularly shaped ellipsoid and results compared to those obtained from perturbation theory.     

Numerical algorithm for the solution of linear two-dimensional integral equations of the first kind

A numerical algorithm is proposed for the solution of two-dimensional integral equations of the first kind, to which some inverse problems of heat conduction are reduced.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 6, pp. 1103–1108, December, 1977.     

Fast multipole method as an efficient solver for 2D elastic wave surface integral equations

The fast multipole method (FMM) is very efficient in solving integral equations. This paper applies the method to solve large solid-solid boundary integral equations for elastic waves in two dimensions. The scattering problem is first formulated with the boundary element method. FMM is then introduced to expedite the solution process. By using the FMM technique, the number of floating-point operations of the matrix-vector multiplication in a standard conjugate gradient algorithm is reduced from O(N ²) to O(N ^1.5), where N is the number of unknowns. The matrix-filling time and the memory requirement are also of the order N ^1.5. The computational complexity of the algorithm is further reduced to O(N ^4/3) by using a ray propagation technique. Numerical results are given to show the accuracy and efficiency of FMM compared to the boundary element method with dense matrix.     

The numerical solution of Cauchy singular integral equations with application to fracture

The present study investigates a numerical algorithm for solving systems of Cauchy singular integral equations of the second kind such as those which often occur in the analysis of interface crack problems. The algorithm takes advantage of many standard subroutines for performing numerical integrations and can be easily applied to equations which are defined over different intervals of the dependent variable. The solution technique is illustrated by analyzing two homogeneous center cracked panels: one loaded in tension and the other loaded in shear and bending. In the second example problem, the presence of crack face friction strongly couples the underlying singular integral equations. The numerical results are compared to closed form elasticity solutions and are shown to be extremely accurate. In addition, the study also illustrates the feasibility of using various assumed forms of the undetermined functions. By assuming these slightly altered forms, many rather complex problems are either solved directly or reduced in complexity.     

A numerical solution of singular integral equations without using special collocation points

A new technique for the solution of singular integral equations is proposed, where the unknown function may have a particular singular behaviour, different from the one defined by the dominant part of the singular integral equation. In this case the integral equation may be discretized by two different quadratures defined in such a way that the collocation points of the one correspond to the integration points of the other. In this manner the system is reduced to a n × n system of discrete equations and the method preserves, for the same number of equations, the same polynomial accuracy. The main advantage of the method is that it can proceed without using special collocation points. This new technique was tested in a series of typical examples and yielded results which are in good agreement with already existing solutions.     

Computational methods for solving static field and eddy current problems via Fredholm integral equations

Two-dimensional static field problems can be solved by a method based on Fredholm integral equations (equations of the second kind). This has numerical advantages over the mote commonly used integral equation of the first kind. The method is applicable to both magnetostatic and electrostatic problems formulated in terms of either vector or scalar potentials. It has been extended to the solution of eddy current problems with sinusoidal driving functions. The application of the classical Fredholm equation has been extended to problems containing boundary conditions: 1) potential value, 2) normal derivative value, and 3) an interface condition, all in the same problem. The solutions to the Fredholm equations are single or double (dipole) layers of sources on the problem boundaries and interfaces. This method has been developed into computer codes which use piecewise quadratic approximations to the solutions to the integral equations. Exact integrations are used to replace the integral equations by a matrix equation. The solution to this matrix equation can then be used to directly calculate the field anywhere.     

Boundary integral equations for thin bodies

A boundary integral equation formulation for thin bodies which uses CBIE (conventional BIE) only is well known to be degenerate. A mixed formulation for a thin rigid scatterer which combines CBIE and HBIE (hypersingular BIE) is motivated by examining the discretized form of the integral equations, and this formulation is shown to be non-degenerate for thin non-rigid inclusion problems. A near-singular integration procedure, useful for singular integrals as well, is presented. Finally, numerical examples for acoustic wave scattering from rigid and soft scatterers are presented.     

A numerical method for the solution of certain classes of nonlinear Volterra integro-differential and integral equations

A direct numerical method is presented for the solution of classes of non-linear, singular Volterra integral and integro-differential equations of the types where y is the unknown and the Fs and Gs are given but to a large extent arbitrary functions of their arguments. The adaptation of the method to other equations, in particular with different kernels, is immediate.     

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.