Application of fuzzy finite difference scheme for the non-homogeneous fuzzy heat equation

A numerical framework based on fuzzy finite difference is presented for approximating fuzzy triangular solutions of fuzzy partial differential equations by considering the type of [gH-p]-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[gH-p]-$$\end{document}differentiability. The fuzzy triangle functions are expanded using full fuzzy Taylor expansion to develop a new fuzzy finite difference method. By considering the type of gH-differentiability, we approximate the fuzzy derivatives with a new fuzzy finite difference. In particular, we propose using this method to solve non-homogeneous fuzzy heat equation with triangular initial-boundary conditions. We examine the truncation error and the convergence conditions of the proposed method. Several numerical examples are presented to demonstrate the performance of the methods. The final results demonstrate the efficiency and the ability of the new fuzzy finite difference method to produce triangular fuzzy numerical results which are more consistent with existing reality.


Introduction
In recent decades, fuzzy set theory has been proven to be a useful tool for modeling systems with uncertainties, giving the models a more realistic look at reality and enabling them to express themselves with a more comprehensive outlook.
The fuzzy derivative concept first appeared in Chang and Zadeh's (1972). Hukuhara's paper Hukuhara (1967) is the starting point for the set-valued and fuzzy differential equations. Puri and Ralescu (1986) suggested the fuzzy differential equations modeling with uncertainty under the concept of H-differentiability. Further studies developed fuzzy differential equations based on the Hukuhara derivative, such as those presented by Kaleva in Kaleva (1987). There are some fuzzy differential equations in this framework, however, for which the diameter of the solution increases as the time t increases (Diamond 2002 To overcome this shortcoming, Bede and Gal introduced the weakly generalized differentiability and the strongly generalized differentiability for the fuzzy functions (Bede and Gal 2005). Moreover, they presented a more general definition of derivatives for the fuzzy functions and their applications for solving fuzzy differential equations Gal 2005, 2006). Stefanini and Bede introduced generalized Hukuhara differentiability (gH-differentiability) Stefanini and Bede (2009) for interval-valued functions by using the concept of generalization of the Hukuhara difference of compact convex set. They showed that this concept of differentiability has relationships with weakly generalized differentiability and strongly generalized differentiability. The disadvantage of the strongly generalized differentiability of a function compared to H-differentiability is that in this case, the fuzzy differential equation has no unique solution (Bede and Gal 2005). Also, in Chalco-Cano et al. (2011), the authors studied relationships between the strongly generalized differentiability and the gH-differentiability , showing the equivalence between these two concepts when the set of switching points of the interval-valued function is finite. Recently, Chalco-Cano et al. (2020) provided a new characterization of the switching points for gH-differentiability and shown that the set of all switching points is at most countable. Partial differential equations explain the majority of phenomena in the fields of mathematics, physics, and engineering. However, mathematical modeling of these phenomena requires a wide variety of data and information. Unfortunately, the measurement of these variables is inherently uncertain. Therefore, the fuzzy partial differential equation is a useful tool for modeling systems with uncertainties (Buckley and Feuring 1999;Allahviranloo 2002;Allahviranloo and Taheri 2009;Bertone et al. 2013;Moghaddam and Allahviranloo 2018;Allahviranloo et al. 2015b;Gouyandeh et al. 2017).
For many fuzzy partial differential equations, analytical solutions are challenging to obtain. Consequently, it is crucial to create some reliable and efficient methods for solving fuzzy partial differential equations. Numerous researchers are presently focusing on the numerical solution of fuzzy partial differential equations, such as difference method (Allahviranloo 2002;Allahviranloo and Afshar 2010), Adomian method (Allahviranloo and Taheri 2009;Pirzada and Vakaskar 2015), finite volume method (Mahmoud and Iman 2011).
In recent years, there has been an increase in interest in the use of the finite difference method to solve fuzzy partial differential equations. According to our knowledge, all papers that have used this method have rewritten the fuzzy partial differential equation as two crisp partial differential equations and solved them using the usual finite difference method. In comparison, this paper is devoted to developing a new fuzzy finite difference method through fuzzy arithmetic and fuzzy Taylor expansion. We approximate the fuzzy derivatives with a fuzzy finite difference by considering the type of gH-differentiability. The fuzzy numerical solution of the fuzzy partial differential equations can be obtained without implicitly embedding them into crisp equations through our method. Even though this paper deals with the fuzzy heat equation, our method can be used to find the numerical solution for a wide variety of fuzzy partial differential equations. Now, let us take a quick look at the contents. Section 2 presents some concepts related to fuzzy numbers and generalized Hukuhara differentiability, as well as some theorems and lemmas used in the central part of the paper. The fuzzy finite difference method for one variable fuzzy functions is discussed in Section 3, and we obtain different formulas for forward, backward, and central difference depending on the type of g H-differentiability. Taking into account the type of [g H − p]-differentiability, we show corresponding formulas for the fuzzy finite difference method of the non-homogeneous heat equation in Section 4. Further, we describe and analyze in detail the convergence condition of the method, as well as truncation error. A full description is given for one of the three examples in Section 5. The last section of the paper discusses conclusions, applications, and future possibilities.

Preliminaries
The purpose of this section is to introduce the general terms and definitions used to describe fuzzy operations and the necessary notations.
The triangular fuzzy number a ∈ R T is defined as an ordered triple a = (a 1 , a 2 , a 3 ) with a 1 ≤ a 2 ≤ a 3 . Some properties of the triangular fuzzy number are discussed in Kaufmann and Gupta (1985), but we will describe some of the properties of this class of numbers here which are used in this paper.
provided that c is a triangular fuzzy number.

Remark 2.2
In the rest of this paper, all fuzzy numbers and fuzzy functions will be considered triangular. Additionally, all the lemmas and theorems will be proved on the assumption that the generalized Hukuhara difference exists. Proposition 2.3 Consider a, b and c are triangular fuzzy numbers and Hukuhara difference exists, then

Proof
Case 1. We have Case 2. According to assumption a = c (−1)b and Hukuhara difference exists, so Thus, the proof is complete.
(1). The other case (λ 1 and λ 2 are negative constants) can be proved in a similar way and we omit the details.
Definition 2.5 (See Bede 2013) Let y : (a, b) → R T is a fuzzy-valued function such that y(t) = y 1 (t), y 2 (t), y 3 (t) , where y 1 (t), y 2 (t) and y 3 (t) are real-valued differentiable functions on (a, b is a triangular fuzzy number. In general, if , then y is generalized Hukuhara differentiable function on (a, b).

Remark 2.6
We assume that the notations C k gH ([a, b], R T ) stand for all triangular fuzzy function f and it's first k, gH-derivatives which are defined on [a, b] and fuzzy continuous (Allahviranloo et al. 2015a). Throughout the rest of this paper, y(t) ∈ C j gH ([a, b], R T ) for j = 1, ..., n − 1 and t ∈ [a, b] with no switching point on [a, b]. Moreover, for simplicity -When y ( j) 3 (t) , we will use the notation denote y 1 (t) , we will use the notation denote y ( j) ii.gH (t) to show y   3 (t) in these triangular fuzzy functions.) In particular, we have the following cases to show the all kind of gH-differentiability for y ( j) gH (t) of order j, when j = 0, 1, 2.
Case 1. If y(t), y gH (t) and y gH (t) are [i−g H]−differentiable, we have Theorem 2.8 Let y : [a, b] → R T be a triangular fuzzy function such that y ∈ C n gH ([a, b], R T ) with no switching points in [a, b]. Then, for j = 1, 2, ..., n, there are the following different scenarios i.
.., n are integrable. We will prove parts (i) and (iii); the other parts are similar, and we omit the details. By using Remark 2.6 and Definition 2.7 , we get which proves this case.
Next, we are going to prove a crucial theorem to all the different cases in Remark 2.6, which will be used in the following sections. Actually, we will obtain four terms of the fuzzy Taylor's expansion about the point t k for t k ≤ t and t ≤ t k by considering different type of gH-differentiability for y(t), y gH (t) and y gH (t). where and ♦ can be one of the ⊕, ⊕(−1) or (−1).

Proof
Since y ∈ C 4 gH ([a, b], R T ) with no switching points, so y (i) gH , i = 0, 1, 2, 3, 4 are integrable on [a, b]. We want to prove Case 1, therefore y(t), y gH (t) and y gH (t) are [i − g H]−differentiable. According to Theorem 2.8, we can write and By integration from each side of equation (3), we conclude that By continuing this process Applying the integral operator to y i.gH (ξ 2 ) gives With the similar manner, Therefore, by integration of (4), we get that t t−Δt Therefore, But we have With repeated integrals, we have t ξ 1 In this case, we can conclude In the same way, the other cases outlined in the theorem are also proven using Theorem 2.8 .
with respect to x without any switching point on D and -if the type of [g H − p]−differentiability of both u(x, t) and u x gH (x, t) are the same, then u

Finite difference methods
Our goal here is to describe the fundamentals of the fuzzy finite difference method. To accomplish this, we will first show you how to obtain the finite difference formula for the first and second derivatives of a triangular fuzzy function y(t). Now, we describe the essential details of finite difference methods. First, we select an integer N > 0 and divide the interval [a, b] into (N + 1) equal sub-intervals whose endpoints are the mesh points t i = a + iΔt, for ([a, b], R T ), so based on the different types of differentiability that mentioned in Remark 2.6, the first and second gH-derivative of this fuzzy function can be approximated by fuzzy finite difference method as follows Case 3.1 Consider y(t) and y gH (t) -The first fuzzy forward difference.
Hence, by using the Case 1 in Theorem 2.9, we have solve for y i.gH (t) yields By using the fuzzy mean value theorem in Allahviranloo et al. (2015b), for all i = 0, 1, ..., N , there are ξ + ∈ (t, t + Δt) such that where the term Δt 2 y (ξ + ) is called truncation error of the forward fuzzy finite difference approximation. Moreover, the properties of the Hausdorff distance (Lakshmikantham et al. 2006) are implied that as Δt → 0. Therefore, Δt should be sufficiently small to get a good approximation. Finally, for sufficiently small Δt, the first forward fuzzy finite difference approximation of y i.gH (t) is Δt .
-The first fuzzy backward difference.
To obtain the backward fuzzy finite difference formula, using Theorem 2.9 (case 1), we can write Rearranging equation (7) gives For having a more useful approximation value for y i.gH (t), by using the fuzzy mean value theorem in Allahviranloo et al. (2015b), there are ξ − ∈ (t − Δt, t) such that So by considering Δt is small enough, the approximation value obtained for the first-order gH-derivative is equal to Δt .
-The first fuzzy central difference.
We use Hukuhara to subtract Eq. (5) from Eq. (7) and divide by 2Δt, then we obtain On the other hand, given the Hausdorff distance properties, it can be seen that When Δt → 0, we have the following equation is the first fuzzy central difference approximation of y i.gH (t).
-The second-order fuzzy central difference.
To obtain an appropriate approximation for the second-order derivative of the fuzzy function y(t), (5) is added to (7), then the equations are rearranged and divided by Δt 2 Hence, for Δt sufficiently small, the appropriate approximation obtained for the second-order derivative y i.gH (t i ) is equal to we obtain the similar approximation value for y i.gH (t) and y i.gH (t) and there is no need to repeat the process of obtaining these approximate values. Accordingly, the type of gH-differentiability of the second-order derivative has no effect on the obtained values, and these approximation values all depend on the fuzzy function y(t) and its first-order derivative y [ g H].
Here, for the first and second gH-derivatives, we present the relevant fuzzy finite difference formulas by considering the type of gH-differentiability. We will not elaborate on the proof in these cases since it is the same as Case 3.1.
-The first fuzzy forward difference.
-The first fuzzy backward difference.
-The first fuzzy central difference.
-The second-order fuzzy central difference.
-The first fuzzy forward difference.
-The first fuzzy central difference.
-The second-order fuzzy central difference. • The first fuzzy forward difference.
• The first fuzzy central difference.
• The second-order fuzzy central difference.

The non-homogeneous fuzzy heat equation
In mathematical physics, motion or transport of particles, i.e., ions, molecules, etc., from higher concentration to lower concentration is modeled by the diffusion equation with appropriate boundary and initial conditions. Heat conduction in a rod is a prototypical diffusion equation. Consider a uniform rod of length L which is insulated everywhere except at its two ends and the temperature is transmitted non-uniformly from beginning to end. This temperature is denoted by u(x, t), and x is a coordinate in space, t represents time. Measuring the temperature is an uncertain problem, and this vagueness may appear in the initial and boundary conditions. Suppose the temperature at the ends are kept at a fixed fuzzy temperature of u(0, t) and u(L, t), respectively. The problem is to find the future temperature along the rod by considering the given fuzzy initial temperature u(x, 0). In this case, the above problem is formulated as the following fuzzy non-homogeneous initial-boundary-value heat equation where f (x), g(t), h(t) and F(x, t) are triangular fuzzy func- This equation has a unique solution in different states of [gHp]-differentiatiability (Allahviranloo et al. 2015b) and the main purpose of this section is to obtain an approximate fuzzy solution for the fuzzy heat equation using the fuzzy finite difference method. Suppose that u(x, t) is the exact fuzzy solution of equation (8) provided that the types of [g H − p]−differentiability with respect to x and t are the same. The basic idea is to replace all the derivatives in equation (8) by corresponding difference approximation.
, by considering the type of [g H− p]−differentiability, the following different situations will be happen In this case, the heat equation will be as follows -Forward Difference in time: Since u(x, t) is [(i) − p]−differentiable with respect to t, then different cases 1, 2, 3 and 4 in Theorem 2.9 can be used, in which Therefore, according to Section 3, we obtain -Central Differences in Space: Due to the fact that u x gH (x, t) is a [(i) − p]−differentiable function , all cases 1, 2, 7 and 8, which are expressed in Theorem 2.9, can be used. So let us take case 1 Adding and re-arranging: Now, substitute equations (11) and (13) into the main equation (9), accordingly To obtain an approximation solution for equation (9) using the fuzzy finite difference method, we must divide the domain [0, L] × [0, T ] into a set of mesh points. Here, we subdivide the domain [0, L] × [0, T ] into N x + 1 and N t + 1 equally mesh points Now, consider U n k denotes the mesh function that approximates u(x k , t n ) for k = 0, ..., N x and n = 0, ..., N t . By putting mesh point (x k , t n ) into equation (9), the following formula is obtained where μ = Δt Δx 2 .
-Truncation error: Now, we want to investigate the truncation error of the scheme (14). The truncation error, T (x, t), is the difference between two side of equation when the exact solution u(x k , t n ) is replaced with the approximation value U n k , hence In fact we showed that the approximate solution U n k obtained by finite difference method (14) converges to the exact solution u(x k , t n ) provided that μ ≤ 1 2 for sufficiently large value of N t , besides D(u The following algorithm summarizes the proposed fuzzy finite difference.
Algorithm 4.1 1. Choose N t and N x iii. set the boundary value U n+1 . In the following, we will briefly consider the other case of the [g H − p]−differentiability for equation (8). The whole process of proof for the following situations is the same as in case 1, so we will not go into details and we just express the algorithm.
Case 2. Consider the following fuzzy heat equation iii. set the boundary value U n+1

Numerical examples
We will solve a few examples of the fuzzy finite difference method in this section to illustrate its efficiency and accuracy in solving the fuzzy heat equation. All calculations were performed on a PC running Mathematica software. In the following example, the fuzzy finite difference method is explained in detail.
By placing the values (x k , t n ) in the exact solution, it is easy to verify U n k = u(x k , t n ). Then, the exact solution of the fuzzy heat equation (18) is obtained by this method.
According to the procedure outlined in Algorithm 4.2, Δt and Δx should be considered large such that Δt Δx 2 ≤ 1 2 . We consider N x = 2 and N t = 50. So we have many sub-intervals and it is not possible to show the approximate numbers, U n j , and only the approximate and exact solutions are shown in Fig. 2. In addition, Fig. 3 represents the logarithm of the error for various N t .

Conclusion
We presented the new fuzzy finite difference method for approximating the fuzzy triangular solution of the fuzzy nonhomogeneous heat equation with triangular initial-boundary conditions. To do this, the fuzzy Taylor expansion was extended according to the type of g H−differentiability, and the finite difference formulas for the first and second derivatives of a triangular fuzzy function y(t) were obtained. Moreover, the convergence conditions for solving the fuzzy heat problem were also investigated. Several 1 (Left) for Example 5.3 numerical examples were presented to demonstrate the performance of the methods. The final results demonstrated the efficiency and the ability of the new fuzzy finite difference method to produce triangular fuzzy numerical results which are more consistent with existing reality. Even though this paper deals with the fuzzy non-homogeneous heat equation, our method can be used to find the numerical solution for a wide variety of fuzzy partial differential equations. The fuzzy numerical solution of the fuzzy partial differential equations can be obtained without implicitly embedding them into crisp equations through our method.