Abstract
This paper presents a boundary integral equation method for finding the solution of Robin problems in bounded and unbounded multiply connected regions. The Robin problems are formulated as Riemann-Hilbert problems which lead to systems of integral equations and the related differential equations are also constructed that give rise to unique solutions, which are shown. Numerical results on several test regions are presented to illustrate that the approximate solution when using this method for the Robin problems when the boundaries are sufficiently smooth are accurate.
Highlights
A boundary value problem is a problem that involves finding the solution of a differential equation or system of differential equation which meets certain specified requirements or boundary conditions at the end points or along a boundary, usually connected with the physical condition for certain values of the independent variable
This paper considers Laplace’s equation u = in both bounded and unbounded multiply connected regions with a linear combination of Dirichlet and Neumann boundary conditions on the boundary = ∂, generally known as a mixed boundary value problem and commonly called the Robin problem
It has been shown that the problem of conformal mapping, the Dirichlet problem, the Neumann problem, and the mixed Dirichlet-Neumann problem can all be treated as Riemann Hilbert problems as discussed in [ – ]
Summary
A boundary value problem is a problem that involves finding the solution of a differential equation or system of differential equation which meets certain specified requirements or boundary conditions at the end points or along a boundary, usually connected with the physical condition for certain values of the independent variable. This paper considers Laplace’s equation u = in both bounded and unbounded multiply connected regions with a linear combination of Dirichlet and Neumann boundary conditions on the boundary = ∂ , generally known as a mixed boundary value problem and commonly called the Robin problem. The well-known integral equations for RH problem have been employed for solving the Dirichlet problem and the Neumann problem [ ] and the mixed DirichletNeumann problem [ ]. They are uniquely solvable Fredholm integral equations of the second kind. In Section , we show how to treat the integral equations and differential equations numerically.
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