Approximate solution of a mixed problem for a parabolic equation by means of a special basis of functions

Using the eigenfunctions of two Sturm–Liouville problems (with the same operator of the most general form but two different sets of boundary conditions), we propose a method for construction of specific basis functions such that the corresponding expansions of smooth and piecewise smooth functions lead to fast converging series. The last circumstance can be successfully employed for approximate solution of mixed problems for a parabolic equation when the sought function is approximated in the spatial variables by few basis functions. First, the case of one spatial coordinate is elaborated and the two-dimensional case is briefly discussed in Section 5. The method aims primarily at the case when the sought function is piecewise smooth in the spatial variable and the implementation of the method bases on the concept of generalized solution.