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 ( h 2 ) , not of order O ( h 4 ) as proposed by that author. However, it is possible with several changes to derive a Crandall algorithm of order O ( h 4 ) . 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.