  This paper aims at obtaining, by means of integral transforms, analytical
approximations in short times of solutions to boundary value problems for the
one-dimensional reaction-diffusion equation with constant coefficients. The
general form of the equation is considered on a bounded generic interval and
the three classical types of boundary conditions, i.e., Dirichlet as well as
Neumann and mixed boundary conditions are considered in a unified way. The
Fourier and Laplace integral transforms are successively applied and an exact
solution is obtained in the Laplace domain. This operational solution is proven
to be the accurate Laplace transform of the infinite series obtained by the
Fourier decomposition method and presented in the literature as solutions to
this type of problem. On the basis of this unified operational solution, four
cases are distinguished where innovative formulas expressing consistent
analytical approximations in short time limits are derived with respect to the
behavior of the solution at the boundaries. Compared to the infinite series
solutions, the analytical approximations may open new perspectives and
applications, among which can be noted the improvement of numerical efficiency
in simulations of one-dimensional moving boundary problems, such as in Stefan
models.
