A note on efficient techniques for the second-order parabolic equation subject to non-local conditions

Many physical phenomena are modelled by non-classical parabolic boundary value problems with non-local boundary conditions. In [M. Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl. Numer. Math. 52 (2005) 39–62], several methods were compared to approach the numerical solution of the one-dimensional heat equation subject to the specifications of mass. One of them was the (3,3) Crandall formula. The scheme showed in Eq. (64) in that paper is of order O(h2), not of order O(h4) as proposed by that author. However, it is possible with several changes to derive a Crandall algorithm of order O(h4). Here, we compare the efficiency of the new method with the previous results in the same tests, and we reach errors 103 to 105 times smaller with the new scheme.