Fuzzy Analytical Solution to Vertical Infiltration

In this article, we examine the solution of the fuzzy linear vertical infiltration equation, which represents the water movement in porous media in that part which is called the vadose zone. This zone is very important for semi-arid areas, due to complex phenomena related to the moisture content in it. These phenomena concern the interchange of moisture content between the vadose zone and the atmosphere, groundwater and vegetation, transfer of moisture and vapor and retention of moisture. The equation describing the problem is a partial differential parabolic equation of second order. The calculation of water flow in the unsaturated zone requires the knowledge of the initial and boundary conditions as well as the various soil parameters. But these parameters are subject to different kinds of uncertainty due to human and machine imprecision. For that reason, fuzzy set theory was used here for facing imprecision or vagueness. As the problem concerns fuzzy differential equations, the generalized Hukuhara (gH) derivative was used for total derivatives, as well as the extension of this theory for partial derivatives. The results are the fuzzy moisture content, the fuzzy cumulative infiltration and the fuzzy infiltration rate versus time. These results allow researchers and engineers involved in Irrigation and Drainage Engineering to take into account the uncertainties involved in infiltration.

By introducing the diffusivity D (cm 2 /s): properties of cumulative infiltration I and infiltration rate v 0 . In general, exact nonlinear solutions are derived for specific forms of the soil-water relationship [10] [11] [12] [13] [14]. Richards's equation is also linearized by considering exponential form of the hydraulic conductivity and the moisture content vs. the pressure head [14] [15]. Approximate solutions are also derived using the form of Brooks and Corey of hydraulic conductivity and the moisture content vs. the pressure head and considering a rectangular profile of moisture content [16] [17] [18]. Su et al. [19], solved the equation of Richards, using a new method based on the Principle of Least Action and the Variational Principle.
The numerical methods-finite difference and finite element-presented in [20]- [33] overcome most of the limitations to result to analytical solutions.
The calculation of water flow in the unsaturated zone requires the knowledge of the initial and boundary conditions as well as of the various soil parameters.
Until today, these conditions and parameters were assumed well-defined, and this assumption is based principally in measurements. But they are subject to different kinds of uncertainty, due to human and machine imprecision. In many cases the uncertainties were considered in statistical terms as random variables Today the fuzzy set theory provides methods for introducing imprecise information in a possibilistic sense. Zadeh [34] initially introduced the fuzzy set theory for facing imprecision or vagueness and since then this theory has been applied in various fields of science. In the present work, the solution of a linear one-dimensional vertical infiltration equation with fuzzy initial and boundary conditions is presented. This equation is a parabolic partial differential equation, Journal of Software Engineering and Applications describing the vertical water movement in a porous medium. The problem of fuzzy differential equations is related to mathematical modelling and engineering applications. Initially differentiable fuzzy functions were studied by Puri and Ralescu [35], who generalized and extended the concept of Hukuhara differentiability of set valued mappings to the class of fuzzy mappings (H-derivative, Hukuhara, [36]). Also, Kaleva [37] and Seikkala [38] developed a theory for fuzzy differential equations. Many related works have been carried out in theoretical and applied topics for fuzzy differential equations with H-derivative [37] [38] [39] [40]. But in some cases, this method suffers certain disadvantages that lead to solutions with increasing support as time t increases [41] [42]. This proves that in some cases the H-derivative solution is not a good generalization of the corresponding crisp case. Bede and Gal [43] mention: "This approach has the disadvantage that it leads to solutions with increasing support, fact which is solved by interpreting a fuzzy differential equation as a system of differential inclusions. But this last-mentioned approach has at its turn some shortcomings.
The main shortcoming is that one cannot talk about the derivative of a fuzzy-number-valued function, since a fuzzy differential equation is directly interpreted with the help of differential inclusions without having a derivative". In order to overcome the above deficiency, the generalized Hukuhara differentiability (gH-differentiability) was introduced by Bede and Gal [44] and Stefanini and Bede [45]. In that case, the solution exists under much less restrictive conditions, but it does not always exist. Recently the general differentiability (g-differentiability) concept is proposed, which further extends the gH-differentiability (Bede and Stefanini [46], Stefanini and Bede [47]). This new derivative is defined for a larger class of fuzzy functions than the Hukuhara derivative. Allahviranloo et al. [48] introduced the (gH-p) differentiability for partial derivatives as an extension of the above theory.
In this paper, as is stated above, the case of linear vertical infiltration is studied, with imprecise boundaries conditions. The diffusivity is considered constant and the crisp problem is solved using the Laplace transform. For the fuzzy solution, the crisp solution is introduced first and then the problem is fuzzified.
Then the problem is solved according to the theories presented in [49] [50], and a fuzzy solution is presented. Consequently, the first derivatives with respect to t and z, as well as the second derivative with respect to z of the problem are examined. The article is organized as follows: in the Materials and Methods section, the physical problem is presented, and the Fuzzy model is applied to it. The mathematical model is formulated, using certain characteristics with fuzzy derivatives, and it is analyzed in its fuzzy form. In the Results and Discussion section, the model is applied in sample soil data, resulting to the fuzzy moisture, the fuzzy cumulative infiltration as well as the fuzzy infiltration rate versus time. The significance and the main advantage of this study, is the introduction of fuzzy logic to solve the problem of vertical infiltration, which is a problem involving partial differential equations and it presents uncertainties in its input variables.

Equation (6) mentioned in the introduction, is called infiltration equation by
Philip [51], because it describes the vertical water flow in a porous medium. For 1 0 θ θ > , the above equation with initial and boundary conditions provided by Equation (7), describes the vertical infiltration of water by applying a constant moisture content at z = 0.
In the above equation, Philip [9] has estimated the diffusivity as follows: he has considered that k is constant by matching linear and non-linear values of infiltration rate ( ) and has obtained the value: In Equation (12), K s = the hydraulic conductivity at saturation, K r = the residual hydraulic conductivity, θ s = the moisture content at saturation, and θ r = the residual moisture content. He now poses in Equation (9): and Equation (9) becomes: with initial and boundaries conditions: The solution of this equation is ( [50]): while the infiltration rate is:

Generalities of the Fuzzy Model
Note: In order to facilitate the readers non-familiar with the fuzzy theory, we describe here some definitions concerning preliminaries of fuzzy theory and some definitions about the differentiability.

Definition 1. Membership Function
A fuzzy set U  on a universe set X is a mapping

Definition 2. Closure
Let X be a Banach space and U  be a fuzzy set on X. We define the a-cuts of U  as ( )

Definition 3. Space of All Compact and Convex Sets
Let Ҡ(X) the family of all nonempty compact convex subsets of a Banach space.
The space of all compact and convex fuzzy sets on X is denoted as Ƒ(X).

Definition 4. α-Cut Forms
. According to representation theorem of Negoita and Ralescu [39] and the theorem of Goetschel and Voxman [52], the membership function and the α-cut form of a fuzzy number U  , are equivalent and in particular the α-cuts ,

Definition 5. gH-Differentiability (Bede and Stefanini, [46]) Let
if and only if one of the following two cases holds: is decreasing as functions of α, and derivative is a fuzzy number.

Definition 8. Implication of g-Differentiability
The gH-differentiability implies g-differentiability, but the inverse is not true.

Definition 9. [gH-p] Differentiability
A fuzzy-valued function U  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 by are real valued functions and partial differentiable with respect to x. We say that (Khastan et al. [49], Allahviranloo et al. [48], Mondal and Roy [53]): with respect to x. We say that [48] [49]:

Formulation
We write Equation (14), in its fuzzy form as follows: with the new initial and boundary conditions: We can find solutions to the fuzzy problem (Equation (19)) and the initial and boundary conditions (Equation (20) We will hereby restrict ourselves to the solution of the (1,1) system, which is described in detail.

Solution of the (1,1) System
Boundary conditions Initial condition ( ) By setting F − = Θ in Equation (22), we take the following Laplace transformation: with boundary conditions: The solution of Equation (26) The first derivative w.r.t. z is: The variable B(s) should be equal to 0, in order to satisfy the boundary condition (Equation (27) For the first condition for z = 0, we have: Applying now the inverse Laplace transform [56] to Equation (32) we obtain the following equation: In Equation (34) we set G + = Θ and we take the following Laplace transformation: with boundary conditions: Applying the same process as in case 1, we have: In Equation (39) the fuzzy number A  is as follows: The second boundary condition is: We apply now the "L'Hospital Rule"

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Thus, it is proven that the initial and boundary conditions of Equations ( (23) and (34) Second Derivative of C versus z: ( ) In the right part of Equation (41) we apply the theorem 1 of Bede and Gal [43]: For any , a b R ∈ with , 0 a b ≥ , or , 0 a b ≤ , and any fuzzy number By substituting in Equation (41) the above expressions of ( ) ( ) ( )   1  2  3 , , , , , F z t F z t F z t , we obtain: As proven above, Equation (39) satisfies Equations ( (23) and (34)), or their equivalent fuzzy Equation (19), provided that functions ( ) ( ) where:  F z t is positive for every value of η, ξ and subsequently for all values of x and t. In order to examine the positivity of ( ) 3 , F z t , for a large spectrum of real soils, we examined 9 soils, whose properties are shown in Table 1. Soils 1 to 6 were taken from Nie et al. [57], while soils 7 to 9 were taken respectively from Evangelides [58], Sakellariou-Makrantonaki [59] and Sismanis [60]. The diffusion coefficients D were calculated from the Van

Fuzzy Infiltration rate and fuzzy cumulative Infiltration
The fuzzy infiltration rate is: and the fuzzy cumulative infiltration is:

First Case, Sample Number 7, Loamy Sand
For the first case the following are valid: In Figure 4, the soil water profiles

Second Case, Sample Number 8, Sandy Loam
For the second case the following are valid: 3 3 0. 35 In Figure 8, the soil water profiles at distances z = 300, 400, 700, and 1100 cm respectively from the origin. In Figure 9, the membership function of the ( ) 0 , z t Θ Θ  is    illustrated in real times t = 5, 10, 30 and 60 min at z = 70cm. In Figure 10, the fuzzy cumulative infiltration I α      vs t is illustrated, while in Figure 11, the fuzzy infiltration rate [ ] 0 v α  versus time is shown for soil sample number 8.

Third Case, Sample Number 9, Sandy Loam
For the third case the following are valid: 3 3 0.42 cm cm s θ = , 3 3 0.05 cm cm r θ = ,  In Figure 12, the soil water profiles at distances z = 400, 500, 1000, and 1400 cm respectively from the origin. In Figure 13, the membership function of the Journal of Software Engineering and Applications

Conclusions
The Bede and Stefanini [46] theory with the generalized Hukuhara (g-H) derivative, as well as its extension on differential equations [48], allows researchers to solve practical problems, useful in engineering. It is now possible for engineers to take the fuzziness of various parameters involved into consideration, when calculating and deciding on their work. As is pointed in the remark, the linearized solution estimates well the phenomena of cumulative Infiltration and Infiltration rate for real soils.