Welcome to boundary integral equation methods

Integral Equation Methods for Free Boundary Problems∗

We outline a unified approach for treating free boundary problems arising in Finance using integral equation methods. Starting with the PDE formulations of the free boundary problems, we show how to derive nonlinear integral equations for the free boundaries in a variety of Finance applications. Methods to treat theoretical (existence, uniqueness) questions and analytical and numerical approxim...

Integral Equation Methods for Solving Laplace's Equation with Nonlinear Boundary Conditions: the Smooth Boundary Case

A nonlinear boundary value problem for Laplace's equation is solved numerically by using a reformulation as a nonlinear boundary integral equation. Two numerical methods are proposed and analyzed for discretizing the integral equation, both using product integration to approximate the singular integrals in the equation. The first method uses the product Simpson's rule, and the second is based o...

Multistep and multistage boundary integral methods for the wave equation

We describe how time-discretized wave equation in a homogeneous medium can be solved by boundary integral methods. The time discretization can be a multistep, Runge-Kutta, or a more general multistep-multistage method. The resulting convolutional system of boundary integral equations belongs to the family of convolution quadratures of Ch. Lubich. The aim of this work is two-fold. It describes a...

The Boundary Integral Equation Method

Fifth-order Numerical Methods for Heat Equation Subject to a Boundary Integral Specification

In this paper a fifth-order numerical scheme is developed and implemented for the solution of the homogeneous heat equation ut = αuxx with a nonlocal boundary condition as well as for the inhomogeneous heat equation ut = uxx+s(x, t) with a nonlocal boundary condition. The results obtained show that the numerical method based on the proposed technique is fifth-order accurate as well as L-accepta...