Fuzzy numerical solution to the unconfined aquifer problem under the Boussinesq equation

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Abstract
In this article, the fuzzy numerical solution of the linearized one-dimensional Boussinesq equation of unsteady flow in a semi-infinite unconfined aquifer bordering a lake is examined. The equation describing the problem is a partial differential parabolic equation of second order. This equation requires knowledge of the initial and boundary conditions as well as the various soil parameters. The above auxiliary conditions are subject to different kinds of uncertainty due to human and machine imprecision and create ambiguities for the solution of the problem, and a fuzzy method is introduced. Since the physical problem refers to a partial differential equation, the generalized Hukuhara (gH) derivative is used, as well as the extension of this theory regarding partial derivatives. The objective of this paper is to compare the fuzzy numerical and analytical results, for two different cases of the physical problem of an aquifer's unsteady flow, in order to prove the reliability and efficiency of the proposed fuzzy numerical scheme (fuzzy Crank–Nicolson scheme). The comparison of the methods is based on the transformed Haussdorff metric, which shows that the distances between the analytical and numerical results tend to zero.

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