Munira Ismail

Boundary integral equations with the generalized Neumann kernel for Laplace's equation in multiply connected regions

This paper presents a new boundary integral method for the solution of Laplace’s equation on both bounded and unbounded multiply connected regions, with either the Dirichlet boundary condition or the Neumann boundary condition. The method is based on two uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. Numerical results are presented to illustrate the efficiency of the proposed method.

Solving Robin problems in bounded doubly connected regions via an integral equation with the generalized Neumann kernel

This paper presents a new boundary integral equation method for the solution of Robin problems in bounded doubly connected regions. We show how to reformulate the Robin problems as a Riemann-Hilbert problem which leads to systems of integral equations. Related differential equations are also constructed that give rise to unique solutions. Numerical results on several test regions are presented to illustrate the solution technique for the Robin problems when the boundaries are sufficiently smooth.

A new numerical approach for uniquely solvable exterior Riemann-Hilbert problem on region with corners

Numerical solution for uniquely solvable exterior Riemann-Hilbert problem on region with corners at offcorner points has been explored by discretizing the related integral equation using Picard iteration method without any modifications to the left-hand side (LHS) and right-hand side (RHS) of the integral equation. Numerical errors for all iterations are converge to the required solution. However, for certain problems, it gives lower accuracy. Hence, this paper presents a new numerical approach for the problem by treating the generalized Neumann kernel at LHS and the function at RHS of the integral equation. Due to the existence of the corner points, Gaussian quadrature is employed which avoids the corner points during numerical integration. Numerical example on a test region is presented to demonstrate the effectiveness of this formulation.

An integral equation approach for the numerical solution of the Riemann problem on a simply connected region with corners

In this paper we introduce a new integral equation for computing the numerical solution of the Riemann problem in a simply connected region bounded by curves having a finite number of corners in the complex plane. The solution to this problem may be characterised as the solution to a singular integral equation on the boundary. By using results of Hille and Muskhelishvili, the theory is extended to include boundaries with corners which are rarely used for numerical computation. Following Swarztraubers derivation of an integral equation for the numerical solution of Dirichlet problem in a region of general shape, we use the Picard iteration method to obtain an iterative formula which removes singularities during numerical integration. The result is a direct method which does not involve conformal mapping. The numerical examples given demonstrate the effectiveness of this new strategy.