Summary: Many physical phenomena are modelled by nonclassical parabolic boundary value problems with nonlocal boundary conditions. Many different papers studied the second-order parabolic equation, particularly the heat equation subject to the specifications of mass. In this paper, we provide a whole family of new algorithms that improve the CPU time and accuracy of Crandall’s formula shown in [the authors, Appl. Numer. Math. 59, No. 6, 1258–1264 (2009; Zbl 1167.65422
)] (and this algorithm improved the results obtained with BTCS, FTCS or Dufort-Frankel three-level techniques previously used in other works, see [M. Dehghan
, Appl. Numer. Math. 52, No. 1, 39–62 (2005; Zbl 1063.65079
)]) with this kind of problems. Other methods got second or fourth order only when
, while the new codes got
th order for
; therefore, the new schemes require a smaller storage and CPU time employed than other algorithms. We study the convergence of the new algorithms and finally compare the efficiency of the new methods with some well-known numerical examples.