A NOTE ON THE APPROXIMATION OF PDES WITH UNBOUNDED COEFFICIENTS -- THE SPECIAL ONE-DIMENSIONAL CASE

Abstract: We consider the spatial approximation of the Cauchy problem for a linear uniformly parabolic PDE of second order, with nondivergent operator and unbounded timeand space-dependent coefficients, where equation’s free term and initial data are also allowed to grow. We concentrate on the special case where the PDE has one dimension in space. As in [10], we consider a suitable variational framework and approximate the PDE problem’s generalised solution in the spatial variable, with the use of finite-difference methods, but we obtain, for this case, consistency and convergence results sharper than the corresponding results obtained in [10] for the more general multidimensional case.