This work involve a boundary integral equation method to find the non-uniquely solvable numerical solution of the Interior Riemann-Hilbert problem on a region with corners. The integral equation was derived based on the Fredholm integral equation of the second kind with continuous kernel and the solvability of the integral equation and its equivalence to the problem is reviewed the derived integral equation in this research for the non-uniquely solvable interior Riemann-Hilbert problem on a region with corners will be computed in achieving this aim, this study developed two numerical formulas where the Nystrom method with the Gaussian quadrature rule are implemented. So that, the singularities are eliminated during numerical integration. Numerical examples on four test regions with 0ff-corners are presented to demonstrate the effectiveness of this formulation.
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