Accurate computation of the field in Pippard's nonlocal superconductivity model (Q1904064)

The authors apply the Galerkin finite element method to the following boundary value problem of a Fredholm type integro-differential equation \[ -\bigl( a_2 (x) u'(x) \bigr)' + a_0 (x)u(x) + \int^{+1}_{-1} K(x,y) u(y)dy = f(x), \quad x \in (-1,1), \] \[ a_2 (-1) u'(-1) = b_0, \quad a_2 (1)u'(1) = b_1, \] for given functions \(a_0 (x)\) and \(a_2 (x)\). The motivation of this work is to numerically solve a nonlocal superconductivity model developed by Pippard. The presented theory is illustrated by several numerical examples.