In this research we present a method based on using Exponential Basis Functions (EBFs) to
solve a class of time dependent engineering problems. The solution is first approximated by a
summation of EBFs satisfying the differential equation and then completed by satisfying the
time dependent boundary conditions as well as the initial conditions through a collocation
method. This can be performed by considering two approaches. In the first one the solution is
split into three parts, i.e. a homogeneous solution obtained by homogeneous boundary
conditions, a homogeneous solution obtained by non-homogeneous solution and finally a
particular solution induced by source terms. In the second approach the solution is split into
two parts, i.e. a homogeneous solution and a particular solution induced by source terms. The
two approaches are then employed to construct a time marching algorithm for the solution of
problems over a long period of time.

We shall present the details of the application of the two approaches introduced to some
mathematical and engineering problems. The details of the time marching algorithm proposed
are explained. Several problems are solved to show the capabilities of the approaches used.
Some benchmark problems are also devised and solved for further studies. It is shown that
the one of the introduced approaches is capable of solving a class of problems with moving
boundaries.