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.

翻译：本文旨在通过积分变换，获得常系数一维反应扩散方程边值问题在短时间内的解析近似解。方程的一般形式定义在有界泛型区间上，并统一考虑了三种经典边界条件，即狄利克雷、诺伊曼及混合边界条件。依次应用傅里叶和拉普拉斯积分变换，在拉普拉斯域中得到精确解。该算子解被证明是傅里叶分解法得到的无穷级数的精确拉普拉斯变换，而该无穷级数在文献中作为此类问题的解给出。基于这一统一的算子解，区分了四种情况，根据解在边界处的行为导出了在短时间极限下表示一致解析近似的新公式。与无穷级数解相比，这些解析近似可能开辟新的视角和应用，其中包括在斯蒂芬模型等一维移动边界问题模拟中提高数值效率。