Solving Multi-connected BVPs with Uncertainly Defined Complex Shapes

The paper presents the interval fast parametric integral equations system (IFPIES) applied to solve multi-connected boundary value problems (BVPs) with uncertainly defined complex shapes of a boundary. The method is similar to the fast PIES, which uses the fast multipole method to speed up solving BVPs and reduce RAM utilization. However, modelling uncertainty in the IFPIES uses interval numbers and directed interval arithmetic. Segments created the boundary have the form of the interval Bézier curves of the third degree (curvilinear segments) or the first degree (linear segment). The curves also required some modifications connected with applied directed interval arithmetic. In the presented paper, the reliability and efficiency of the IFPIES solutions were verified on multi-connected BVPs with uncertainly defined complex linear and curvilinear domain shapes. The solutions were compared with the ones obtained by the interval PIES only due to the lack of examples of solving uncertainly defined BVPs in the literature. All presented tests confirm the high efficiency of the IFPIES method.