Fuzzy numerical solution to the uncon ﬁ ned aquifer problem under the Boussinesq equation

In this article, the fuzzy numerical solution of the linearized one-dimensional Boussinesq equation of unsteady ﬂ ow in a semi-in ﬁ nite uncon ﬁ ned 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 ﬂ ow, in order to prove the reliability and ef ﬁ ciency 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.

Colmenares & Neilan ) solutions to the equation but all of the aforementioned problems convey fuzziness (Cox ). In general, until recently, in practice, an attempt has been made to solve a classical mechanical problem introducing many uncertainties in the solution process. Nieto & Rodríguez-López ; Aminikhah ) but also to engineering and hydraulic problems (Guo et al. a, b). When we are studying in fields of physics and engineering, we often meet problems of fuzzy partial differential equations which have to be solved by numerical methods because the exact solution of these problems, as in classical logic, can be found only in some special cases (Boussinesq ; Polubarinova-Kochina ). This case of numerical methods for solving fuzzy differential equations has been rapidly growing in recent years.
During recent years, some analytical and numerical methods have been proposed in order to solve fuzzy differential equations. Initially, the concept of fuzzy derivative was introduced by Chang and Zadeh (Chang & Zadeh ) Allahviranloo et al. () introduced (gH-p) differentiability for partial derivatives as an extension of the above theory. Tzimopoulos et al. in Tzimopoulos et al. (a, b) used the above method and gave a fuzzy analytical solution to a parabolic differential equation and also Tzimopoulos et al. in Tzimopoulos et al. (a, b, ) gave a fuzzy analytical solution to an unconfined aquifer problem described by the Boussinesq equation.
Analytical methods approach real problems with changing parameters in space and time only in limiting cases and this disadvantage leads to the use of numerical methods such as finite difference methods transformed in the fuzzy environment, which approach any physical problem. The numerical method for solving fuzzy differential equations was introduced by Ma et al. (Ma et al. ). Subsequently, numerical solutions of fuzzy differential equations were examined by Friedman (Friedman et al. ), Bede (Bede ) and Abbasbandy (Abbasbandy & Allahviranloo ). The existence of solutions for fuzzy partial differential equations was investigated also by Buckley and Feuring (Buckley & Feuring ); their proposed method works only for elementary partial differential equations. Based on the Seikkala derivative, Allahviranloo (Allahviranloo ), and Kermani and Saburi (Kermani & Saburi ) use a numerical method which is an explicit difference method to solve partial differential equations. Farajzadeh (Farajzadeh ) gives an explicit method for solving fuzzy partial differential equations. Uthirasamy (Uthirasamy ) gives studies on numerical solutions of fuzzy boundary value problems and fuzzy partial differential ing a lake are examined. In the first case there is a sudden rise of the lake's water level, thus the aquifer recharging from the lake, and for the second case a sudden drop of the lake's water level, thus the aquifer discharging to the lake. In both cases special emphasis is given to the aquifer boundary conditions, which are considered to be uncertain, as their uncertainties are the ones that affect to a greater extent the solution of the problem. Moreover, the hydraulic parameters of this problem are considered crisp as well as the geometric parameters. In general the hydraulic communication between the lake and the aquifer has an important effect on the control of the riparian ecosystem and it can also alter the water chemistry. Engineers must have knowledge of the above effects on ecological and hydrological processes for water resource management purposes.

MATERIALS AND METHODS
In order to present in a comprehensive way the methodology used in this work, initially for the understanding of the physical problem and the difficulties it presents but also for the development of its solution, it is considered appropriate to present the flowchart shown in Figure 1. In this flowchart, the methodology of solving the physical problem is developed, determining that the ambiguities of the problem require the use of fuzzy partial differential equations for accurate results. Furthermore, emphasise is given to the two different approaches of solving fuzzy partial differential equations, highlighting the advantages and disadvantages of each method in order to solve the problem.

Physical problem
In the cases of the aquifer illustrated in Figure 2(a) and 2(b), a rise and drop in the lake's water level is observed and the aquifer flow is described by the following Boussinesq equation: or by its linear representation (De Ridder & Zijlstra ): where K ¼ the hydraulic conductivity of the aquifer, S ¼ the specific yield of the aquifer or drainable pore space, D ¼ the depth of the lake, h(x, t) ¼ the depth of the aquifer and x, t ¼ the coordinates (spatial and temporal).
The boundary conditions of the problem are:  while the initial condition is: To facilitate calculations, non-dimensional variables are introduced: The new resulting equation is as follows: with the new boundary and initial conditions: Fuzzy sets and Hukuhara generalized partial derivative Definition 1. We denote by R F the class of fuzzy subsets u:R F ! [0, 1], satisfying the following properties (Puri & Ralescu ; Kaleva ): 3.ũ is upper semi-continuous on R; Then R F is called the space of fuzzy numbers.
Definition 2. Letũ be a fuzzy number. Fundamental concepts in fuzzy theory are the support, the level-sets (or levelcuts) and the core of a fuzzy number. The α-level set of Definition 3. The necessary and sufficient conditions 1. u À α is a bounded monotonic non-decreasing left-continuous function for all α ∈ [0, 1] and right-continuous for α ¼ 0; 2. u þ α is a bounded monotonic non-increasing left-continuous function for all α ∈ [0, 1] and right-continuous for α ¼ 0; Definition 4. The metric structure is given by the transformed Haussdorff distance, in fuzzy sets, Then it is easy to see that D is a metric in R F and has the following properties: Definition 5. The generalized Hukuhara difference of two fuzzy numbersũ,ṽ ∈ R F is defined as follows (Bede & Stefanini ): In terms of α-levels we have: and if the H-difference exists, thenũ⊖ Hṽ ¼ũ⊖ gHṽ ; the conditions of existence ofũ⊖ gHṽ ¼w are w À α non-decreasing and w þ α non-increasing with w À α < w þ α . Definition 6.

A. First order
A fuzzy-valued function H of two variables is a rule that assigns to each ordered pair of real numbers (x, t), in a set D a unique fuzzy number denoted byH(x, t). Let The same is valid for @H α (x 0 , t 0 ) @t Definition 7.

B. Second order
LetH(x, t):D ! R F , and @H α (x 0 , t 0 )=@x i,gH be [gH-p]differentiable at (x 0 , t 0 ) ∈ D with respect to x. We say that Transform of the fuzzy problem

System of crisp problems
The problem is transformed now with the help of the generalized Hukuhara difference and Hukurara partial derivative to the following fuzzy partial differential equation: with the new boundary and initial conditions: We can find solutions to the fuzzy Equation (3)  System (1,1): System (2,1): System (1,2): System (2,2): where for all the above systems defined (Figure 3):

Fuzzy finite difference
The main idea in the finite difference method is to replace the derivative in a partial differential equation with difference quotients.
Assume that u is a function of the independent crisp variables x and t. Subdivide the x-t plane into sets of equal rectangles of sides h, k, by equally spaced grid lines parallel to O x , defined by x i ¼ ih, i ¼ 0, 1, 2, . . . , m, and equally spaced grid lines parallel to O t , defined by t j ¼ jk, j ¼ 0, 1, 2, . . . , n, where m and n are positive integers with h ¼ L=m and k ¼ T=n (Figure 4).
For the development of the fuzzy finite difference scheme we will recall the fuzzy linear partial differential Equation (3): with the non-dimensional boundary and initial conditions: If System 1 satisfies Equation (5) then the fuzzy Crank-Nicolson scheme develops as follows.

RESULTS
For the aquifer of Figure  System 1 is considered here, since it gives a physical solution to the problem in both the recharging and discharging cases.
In both cases fuzziness is introduced on H(0,τ)(c ¼ 0.15), that is ( Figure 5): Then the analytical solution becomes (Tzimopoulos et al. a, b): Also, the following obtained: System 1 satisfies the recharging case and then: [H(0, t)] ¼ [0:85 þ 0:15α, 1:15 À 0:15α] where ffiffiffiffi ffi qt p After using the property of subtraction between fuzzy numbers and the property of a fuzzy singleton (Hanss ) for the parameter H 0 then System 1 also satisfies the discharging case with the following form regarding the analytical solution: Thus, the fuzzy Crank-Nicolson scheme for both cases is: where r ¼ dt=(dx) 2 .
Generally, when numerical methods are applied, comparison with the analytical solution of the problemwhen it existsis a way of verifying the results. In this physical problem the analytical solution exists and for this reason, the fuzzy Crank-Nicolson scheme is applied below and compared with the results of the analytical solution. Also, we should mention that the fuzzy Crank-Nicolson scheme is solved with the Thomas method (algorithm) which is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations (Smith ).
Another very important point to note when using differential equations is that it is a well-posed problem, as according to Jacques Hadamard (Hadamard ).
We used the fuzzy Crank-Nicolson scheme (11) to approximate the fuzzy analytical solutions (10a) and (10b) for different times and points. The optimal spatial and temporal steps that emerged were dt ¼ 0.0000018, dx ¼ 0.01 thus r ¼ 0.018 for the recharging case, and dt ¼ 0.00000108, dx ¼ 0.01 thus r ¼ 0.0108 for the discharging case. Figures 6 and 7 illustrate the dimensional depth profiles as a function of x for the recharging and discharging cases respectively. As a first point of view according to the above graphs, we can observe that the fuzzy numerical results tend to approach the fuzzy analytical solution satisfactorily. However, a more reliable comparison method, which comes to confirm the visual results, is based on the transformed Haussdorff distance. This metric (Equation (2)) was applied  to compare the final results and Figure 11 presents the transformed Haussdorff distances for both cases in three different times.
Based on the results, it is observed that the maximum distances between the fuzzy numbers of the two methods appeared in the results for level α-cut ¼ 0, which is absolutely reasonable since in this α-cut level the results are extracted taking into account the full percentage of fuzziness. In addition, it is observed that the distances of the fuzzy numbers decrease as the observation times increase, Figure 9 | Dimensional depth profile for the discharging case, for t ¼ 120d in 3D mode.