TY - JOUR TI - Uniform asymptotic approximation of solutions to the conductivity problem with thin open filaments DO - https://doi.org/doi:10.7282/t3-444h-3v53 PY - 2019 AB - The asymptotic approximation of solutions to the conductivity problem with thin filaments is analyzed. While filaments with a closed mid-curve have been looked at previously, this thesis pays particular attention to the case of an open mid-curve, focusing on the extra singularities that can develop around the endpoints of this curve. The argument relies on a primal and dual energy argument as well as an explicit representation for the most singular part of the solution around each of the endpoints of the mid-curve. After proving the energy closeness of the reduced problem to the full problem for a constant conductivity in the inhomogeneity, related problems including those with variable conductivities, anisotropic conductivities, and curved inhomogeneities are all discussed briefly. Finally, numerical simulations are shown for many of the situations in this thesis, illustrating the convergence that has been proven here, as well as convergence that it may be possible to prove in the future. KW - Differential equations, Partial KW - Mathematics LA - English ER -