B-polynomial multiwavelets approach for the solution of Abel's integral equation

A numerical method for solving Abel's integral equation as singular Volterra integral equations is presented. The method is based upon Bernstein polynomial (B-polynomial) multiwavelet basis approximations. The properties of B-polynomial multiwavelets are first presented. These properties are then utilized to reduce the singular Volterra integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.     

Modified integral equation solution of viscous flows near sharp corners

Solutions of the biharmonic equation governing steady two dimensional viscous flow of an incompressible newtonian fluid are obtained by employing a direct biharmonic boundary integral equation (BBIE) method in which Green's Theorem is used to reformulate the differential equation as a pair of coupled integral equations. The classical BBIE gives poor convergence in the presence of singularities arising in the solution domain. The rate of convergence is improved dramatically by including the analytic behaviour of the flow in the neighbourhood of the singularities. The modified BBIE (MBBIE) effectively ‘subtracts out’ this analytic behaviour in terms of a series representation whose coefficients are initially unknown. In this way the modified flow variables are regular throughout the entire solution domain. Also presented is a method for including the asymptotic nature of the flow when the solution domain is unbounded.     

Numerical solution of an integral equation for perpetual Bermudan options

We consider perpetual Bermudan options, which have no expiration and can be exercised every T time units. We use the Green's function approach to write down an integral equation for the value of a perpetual Bermudan call option on an expiration date; this integral equation leads to a Wiener–Hopf problem. We discretize the integral in the integral equation to convert the problem to a linear algebra problem, which is straightforward to solve, and this enables us to find the location of the free boundary and the value of the perpetual Bermudan call. We compare our results to earlier studies which used other numerical methods.     

Interval arithmetic error estimation for the solution of Fredholm integral equation

Finite element method with a posteriori estimation using interval arithmetic is discussed for a Fredholm integral equation of the second kind. This approach is general. It leads to the guaranteed L ^∞ asymptotically exact estimate without the usual overestimation when interval arithmetic is used. An algorithm is provided for determination of an approximate solution such that the computed error bound between the exact solution and its approximation in L ^∞ is less than the given tolerance ?. Numerical solution for the equation with only C ¹ kernel illustrates the approach.     

Numerical solution of Laplace's equation as an integral equation of the first kind

A Fourier approximation method is developed for the simple layer potential reformulation of Laplace's equation. The efficacy of the method is demonstrated in computational examples, and also analyzed theoretically.     

Practical applications of an integral equation method for the solution of Laplace's equation

An integral equation method for the solution of Laplace's equation, originally proposed for boundary value problems in a single medium, is here extended to problems involving multiple media. The extended method has been used to compute the internal thermal resistance of electric cables and some numerical results are presented.     

Explicit solution of DMZ equation in nonlinear filtering via solution of ODEs

In this note, we develop a real-time and accurate solution for nonlinear filtering problems based on the Gaussian distribution. Specifically, we present an explicit solution of the Duncan-Mortensen-Zakai (DMZ) equation of the Yau filtering system, which includes the linear filtering system and exact filtering system. The solution is given in terms of a solution of a system of ordinary differential equations. In particular, our method can be implemented in hardware. The complexity of our algorithms is the same as those of Kalman-Bucy filters in the case of linear filtering systems.     

A solution of Fredholm integral equation by means of the spline fit approximation

This paper includes a calculation for two-dimensional and axisymmetric potential flow about an arbitrary body shape by means of the singularity. Vortices are distributed continuously over the body surface using a spline fit function, and the distribution is computed as a solution of Fredholm integral equation. The accuracy of the solution can be found easily by examining whether velocity interior to the boundary surface is regarded as vanishing or not, even if no exact solution exists. The method was applied to potential flows about a circular cylinder, turbine cascades and axisymmetric entrances, and compare with exact analyses or experiments.     

Higher order numerical solution of the integral equation for the two-dimensional neumann problem

Interest in the problem of two-dimensional potential flow in arbitrary multiply-connected domains has been stimulated by the need to calculate flow about multiple airfoil configurations consisting of slats and flaps detached from the main airfoil. General methods of solution are based on the use of a singularity distribution over the boundary. The distribution is obtained as the solution of an integral equation over the boundary. In implementing this solution various investigators approximate the boundary by an inscribed polygon, whose faces are small flat surface elements. The singularity on each element is taken as constant by some investigators and linearly varying by others. This paper systematically investigates the effectiveness of higher order approximations of the integral equation, including use of curved surface elements and parabolically-varying singularity. It is found that the approach using flat elements with constant singularity is mathematically consistent as is the next higher-order approach with parabolic elements and linearly varying singularity. The popular approach based on flat elements with linearly varying singularity is shown to be mathematically inconsistent, and examples are presented for which the effect of element curvature is greater than that of the singularity derivative. A number of examples are presented to show that: (1) the higher order solutions give very little increase in accuracy for the important case of exterior flow about a convex body: (2) for bodies with substantial concave regions and for interior flows in ducts, the use of parabolic elements and linearly varying singularity can give a dramatic increase in accuracy; and (3) the use of still higher order solutions leads to a rather small additional gain in accuracy.     

Improved integral equation solution for the first passage time of leaky integrate-and-fire neurons

An accurate calculation of the first passage time probability density (FPTPD) is essential for computing the likelihood of solutions of the stochastic leaky integrate-and-fire model. The previously proposed numerical calculation of the FPTPD based on the integral equation method discretizes the probability current of the voltage crossing the threshold. While the method is accurate for high noise levels, we show that it results in large numerical errors for small noise. The problem is solved by analytically computing, in each time bin, the mean probability current. Efficiency is further improved by identifying and ignoring time bins with negligible mean probability current.     

The use of approximate functions for the numerical solution of Laplace' equation by an integral equation

In this paper an integral equation method will be outlined to solve Laplace' equation numerically in a finite area S. The method uses either a function which is an approximation of the unknown potential of a particular solution which is only a good approximation in a part of S. The method is also valid if the approximate function is not a solution of Laplace' equation.     

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

This paper presents new formulations of the boundary–domain integral equation (BDIE) and the boundary–domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Numerical solution of 3D elastostatic inclusion problems using the volume integral equation method

A volume integral equation technique developed by Lee and Mal (J. Appl. Mech. Trans. ASME 64 (1997) 23) has been extended to investigate three-dimensional stress problems with multiple inclusions of various shapes. Based on the volume integral formulation, displacement continuity and traction equilibrium along the interfaces between the matrix and the inclusions are automatically satisfied. While the embedding matrix is represented by an integral formulation, only the inclusion parts are discretized into finite elements (isoparametric quadratic 10-node tetrahedral or 20-node hexahedral elements are used in the present study). A number of numerical examples are given to show the accuracy and effectiveness of the proposed method.     

Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with a variable coefficient

In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.     

The use of Sherman-Morrison formula in the solution of Fredholm integral equation of second kind

We consider a constructive method for the solution of Fredholm integral equations of second kind. This method is based on a simple generalization of the well-known Sherman-Morrison formula to the infinite dimensional case. In particular, this method constructs a sequence of functions, that converges to the exact solution of the integral equation under consideration. A formal proof of this convergence result is provided for the case of Fredholm integral equations with integral kernel. Finally, a boundary value problem for the Laplace equation is considered as an example of the application of the proposed method.     

An integral equation method for the numerical solution of Laplace's equation without Green's function

The common way to establish an integral equation for the solution of Laplace's equation uses the Green's function of the given equation. It will be shown in this paper that an integral equation can also be constructed by using a particular solution of the Laplace equation as the Kernel of the integral equation.     

Random solution of a stochastic integral equation :almost sure and mean square convergence of successive approximations

A non-linear random integral equation of the Volterra type of the form was considered by Tsokos (1960), where ω?Ω, the underlying set of the probability measure space (Ω A, P). He was concerned with the existence of a unique random solution to this equation, where a random solution is defined to be a second-order stochastic process, x(t; ω), which satisfies the equation almost surely. The present paper shows that a sequence of successive approximations, x_n)tω) converges to the unique random solution at each t≥0 with probability one and in mean square under the conditions of Tsokos' theorem. The rate of convergence of the successive approximations and a bound on the mean square error of approximation are also given.     

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.

     

On the approximate solution of the delay integral equation of the statistical theory of radiation damage

In the statistical theory of radiation damage, the mean number of atoms displaced in the atomic cascade is given by a delay integral equation with specified initial conditions. Numerical procedures the use spline functions in conjuction with appropriate quadrature rules are presented for the construction of continuous approximations to the mean number of displaced atoms, represented by the delay integral. The methods presented are shown to be stable and to be of order (m + 1) for spline functions of degree m. Finally, the method for quadratic splines is used to compute the mean number of displaced atoms for atomic collisions with Firsov potentials, and with truncated Coulomb potentials.