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A note on efficient techniques for the second-order parabolic equation subject to non-local conditions. (English) Zbl 1167.65422
Summary: Many physical phenomena are modelled by non-classical parabolic boundary value problems with non-local boundary conditions. In [M. Dehghan, Appl. Numer. Math. 52, No. 1, 39–62 (2005; Zbl 1063.65079)], several methods were compared to approach the numerical solution of the one-dimensional heat equation subject to specifications of mass. One of them was the (3,3) Crandall formula. The scheme displayed in Eq. (64) in that paper is of order $O\left({h}^{2}\right)$, not of order $O\left({h}^{4}\right)$ as proposed by that author. However, it is possible with several changes to derive a Crandall algorithm of order $O\left({h}^{4}\right)$. Here, we compare the efficiency of the new method with the previous results in the same tests, and we reach errors ${10}^{3}$ to ${10}^{5}$ times smaller with the new scheme.

##### MSC:
 65M06 Finite difference methods (IVP of PDE)