In this paper, we propose a comprehensive methodological framework for solving the fuzzy groundwater flow problem in a simpler and faster way based on numerical analysis. In particular, a novel simplified matrix explicit inverse formula is proposed as an efficient method to solve the fuzzy Finite Difference numerical scheme, called Crank–Nicolson implicit scheme. The main advantage of the proposed method is that it offers an efficient and simpler solution to the algebraic tridiagonal system of equations that appeared in the fuzzy Crank–Nicolson scheme, without affecting the accuracy of the results. Without doubt, the simulation of an unconfined aquifer flow, using the Finite Difference Method, is often time-consuming due to the costly calculations required to solve the implicit Finite Difference schemes. This process becomes even more difficult when the physical problem is solved in its fuzzy form where the calculations become more complex and greatly increase. However, problem uncertainties are not considered negligible and must be included in the final calculations for a more rational management of the water body. This creates the need to find a new and faster way to solve the fuzzy implicit schemes, as proposed in this work. The simplified matrix explicit inverse formula was applied to the fuzzy Crank–Nicolson scheme to solve the fuzzy partial differential equation of Boussinesq and, hence, the simulation of the physical problem. Its results were compared with the corresponding results of the Thomas algorithm, which until today, is the most common method of solving the tridiagonal system of equations. The case under consideration refers to a sudden rise in the lake's water level, thus resulting in the aquifer recharging from the lake. The performance of the proposed method was more than satisfactory in terms of calculation accuracy and reliability where the numerical results are matched with the differences appearing from the 12 decimal point onward between the two examined methods. The proposed method tested also considered the running times, achieving much better times compared with the Thomas algorithm, where the differences ranged from a few seconds to several hours depending on the examined time and space steps.

在本文中，我们提出了一个全面的方法框架，用于基于数值分析以更简单、更快速的方式解决模糊地下水流动问题。特别是，提出了一种新的简化矩阵显式逆公式作为求解模糊有限差分数值格式的有效方法，称为 Crank-Nicolson 隐式格式。该方法的主要优点是它为出现在模糊 Crank-Nicolson 格式中的代数三对角方程组提供了一种有效且更简单的解，而不会影响结果的准确性。毫无疑问，由于求解隐式有限差分方案所需的计算成本高昂，使用有限差分法模拟无侧限含水层流通常非常耗时。当物理问题以模糊形式解决时，这个过程变得更加困难，计算变得更加复杂并大大增加。然而，问题的不确定性被认为是不可忽略的，必须将其包括在最终计算中，以便对水体进行更合理的管理。这就需要找到一种新的更快的方法来解决模糊隐式方案，正如这项工作中提出的那样。将简化的矩阵显式逆公式应用于模糊 Crank-Nicolson 格式以求解 Boussinesq 的模糊偏微分方程，从而模拟物理问题。将其结果与 Thomas 算法的相应结果进行了比较，直到今天，Thomas 算法仍是求解三对角方程组的最常用方法。正在考虑的案例是指湖水位突然上升，从而导致含水层从湖中补给。所提出方法的性能在计算精度和可靠性方面非常令人满意，其中数值结果与两种检查方法之间从小数点后 12 点开始出现的差异相匹配。所提出的测试方法还考虑了运行时间，与 Thomas 算法相比获得了更好的时间，其中差异范围从几秒到几个小时，具体取决于检查的时间和空间步长。所提出方法的性能在计算精度和可靠性方面非常令人满意，其中数值结果与两种检查方法之间从小数点后 12 点开始出现的差异相匹配。所提出的测试方法还考虑了运行时间，与 Thomas 算法相比获得了更好的时间，其中差异范围从几秒到几个小时，具体取决于检查的时间和空间步长。所提出方法的性能在计算精度和可靠性方面非常令人满意，其中数值结果与两种检查方法之间从小数点后 12 点开始出现的差异相匹配。所提出的测试方法还考虑了运行时间，与 Thomas 算法相比获得了更好的时间，其中差异范围从几秒到几个小时，具体取决于检查的时间和空间步长。