Universiti Teknologi Malaysia Institutional Repository

Analytical and numerical methods for the Riemann problem

Murid, Ali H. M. and Nasser, Mohamed M. S. (2003) Analytical and numerical methods for the Riemann problem. The Proceedings of Annual Fundamental Science Seminar 2003 . pp. 41-50.

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Abstract

In this survey we consider the classical and new methods for computing the analytical and numerical solutions to the Riemann problem, a class of boundary value problems for analytic functions, in a simply connected region Ω+ with smooth boundary Γ = ∂Ω+ in the complex plane. The classical methods for solving this problem based on reducing the Riemann problem to the Dirichlet problem or to the Hilbert problem where it is required the availability of a suitable conformal mapping from Ω+ onto the unit disk D. Recently, the authors introduce a new method for solving the Riemann problem by transforming its boundary condition to a Fredholm integral equation of the second kind with the generalized Neumann kernel. This method has several advantages in terms of numerical operations as well as ease in programming. This paper sketches these classical and new methods and shows the advantages of our method for solving the Riemann problem using Fredholm integral equations.

Item Type:Article
Uncontrolled Keywords:Riemann problem, Hilbert problem, Dirichlet problem, Fredholm integral equation, eigenvalue, index of functions.
Subjects:Q Science > QA Mathematics
Divisions:Science
ID Code:3835
Deposited By: Assoc. Prof. Dr. Ali Hassan Mohamed Murid
Deposited On:26 Jun 2007 07:41
Last Modified:01 Jun 2010 03:12

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