The solution of the variational equation in (26) goes as follows. One introduces a fixed time increment and interpolate the unknown with piecewise linear shape functions both spatially and temporally in order to solve (26). The nodes of the time shape functions are located at , where k is an integer. Now, assume that has been computed at , (). One then sets in (26) and discretise the unknown at spatially. Subsequent use of Galerkin's method then reduces (26) to a system of linear equations for the discretised at , thus yielding an approximate solution for (26). In a similar manner one obtains a numerical solution of (31).
In the special case of planar cracks, the inner integrals required for the discretisation of (26) and (31) (i.e. the integrals with respect to and in (26)) allow analytical evaluation. The remaining outer integrals(i.e. those with respect to in (26)) are then evaluated numerically.