Abstract
In this paper, several results and theorems about the high-order strongly generalized Hukuhara differentiability of function defined via the fuzzy Riemann improper integral (in the sense of Wu) have been established. Then, some properties dealing with the partial derivatives of fuzzy Laplace transform for a fuzzy function of two real variables have been proved. Afterwards, an algorithm of fuzzy Laplace transform for solving second-order fuzzy partial differential equations has been proposed. Finally, two numerical examples, including the heat equation under fuzzy initial conditions, have been studied to justify the efficiency of the algorithm.
1. Introduction
Partial differential equations (PDEs) are extremely useful for the modeling of a variety of natural, physical, and biological phenomena. They have several engineering applications and intervene in many domains of science. Many researchers have investigated fuzzy differential equations (FDEs) and fuzzy partial differential equations (FPDEs) using the rigorous tool of the fuzzy Laplace transform. For example, Allahviranloo et al., Eljaoui et al., and Salahshour et al. have extended this method to solve different kinds of fuzzy differential problems: FDEs of first or second order, FPDEs, and fuzzy integral differential equations [1–5].
In this vein, we have studied the improper fuzzy Riemann integrals by establishing some important results about the continuity and differentiability of a fuzzy improper integral depending on a given crisp parameter in [6]. These results have been then employed to prove some fuzzy Laplace transform properties, which we have used to solve linear FPDEs of first order, under generalized Hukuhara differentiability.
For recent works about partial differential equations, their theory, and necessary materials, one can see [7–9] and the references therein.
The main purpose of this article is to present a fuzzy Laplace transform method for solving FPDEs of second order. To achieve this goal, we begin by developing some results about high-order Hukuhara differentiability of a function defined by a fuzzy improper integral.
The remainder of this work is organized as follows. Section 2 is reserved for preliminaries and recalls some important results about the continuity and differentiability of fuzzy improper integrals that we will need in the sequel. In Section 3, the main results about the high-order differentiability of fuzzy improper integral are studied and new properties of fuzzy Laplace transform are proved. Then, in Section 4, the procedure for solving FPDEs of second order by the fuzzy Laplace transform is proposed. Section 5 deals with numerical examples. Section 6 is reserved for the discussion of the obtained results. In the last section, we present conclusion and further research topic.
2. Preliminaries
Now we recall some basic results which are useful through the rest of this paper.
2.1. Fuzzy Numbers and Functions
A fuzzy number is a function verifying the following four assumptions:(1) is normal, i.e., for which (2) is fuzzy convex(3) is upper semicontinuous(4)The closure of its support is compact (see [10])
Denote as the space of all fuzzy numbers.
For , let be the -level set of . Then, is a nonempty compact interval of . For all , we have
Define the Hausdorff distance on by .
Then, is a complete metric space (for more details, see [11]).
Definition 1 (see [1]). We define a fuzzy number in parametric form as a couple of mappings , , verifying the following properties:(1) is bounded increasing, left continuous in , and right continuous at 0(2) is bounded decreasing, left continuous in , and right continuous at 0(3), for all The length of is level-wise given by .
Theorem 1 (see [12]). Let be a fuzzy function defined on Suppose that for all , the maps are integrable on , and and , for every . Then, is fuzzy Riemann integrable on and we have
For , if there exists an element in such that , then is called the Hukuhara difference of and , which we denote by .
Definition 2 (see [1]). A mapping is said to be strongly generalized differentiable at , if there exists such that for all very small, there exist the H-differences(i); and the limits (ii); and the limits (iii); and the limits (iv) and the limits
Theorem 2 (see [10]). If is a strongly generalized differentiable function on in the sense of Definition 2, (iii) or (iv), then , for each .
So, we can consider only Case (i) or (ii) of Definition 2 almost everywhere in .
Theorem 3 (see [13]). Let be a fuzzy strongly generalized differentiable function on ; then, and are differentiable. Moreover,(1)If is (i)-differentiable, then .(2)If is (ii)-differentiable, then .
Definition 3. We say that a mapping is strongly generalized differentiable of the -th order at if there exists , for all , such that for all very small, there exist the H-differences(i) and the limits (ii) and the limits (iii) and the limits (iv) and the limits
Definition 4 (see [1]). If is a continuous mapping such that is fuzzy Riemann integrable on , then is called the fuzzy Laplace transform of . Notice that , where is the Laplace transform of a crisp function .
Theorem 4 (see [1, 2]). Let be a fuzzy-valued function and and be its derivatives on If is (i)-differentiable, then , or if is (ii)-differentiable, then .
Moreover, if and are (i)-differentiable, then . If is (i)-differentiable and is (ii)-differentiable, then
If is (ii)-differentiable and is (i)-differentiable, then
If and are (ii)-differentiable, then .
2.2. Continuity and Differentiability of Fuzzy Improper Integral
In the sequel, denotes one of the intervals or or , where , denotes another interval, and is a nonempty subset of .
Let us recall the properties of continuity and differentiability of a function defined by a fuzzy improper integral that we had established and proved in [6].
Theorem 5 (see [6]). Let satisfying the following conditions:(i) For all is continuous on (ii) For each is continuous on (iii) For all , there exist a couple of nonnegative, continuous crisp functions and , which are integrable on verifying, for all :
Therefore, the fuzzy mapping is continuous on .
Theorem 6 (see [6]). Let verifying the following assumptions:(i) For all is continuous and fuzzy Riemann integrable on (ii) For all is (i)-differentiable on the interval (iii) For all is continuous on (iv) For all is continuous on (v) For all , there exist a couple of continuous crisp functions and , which are integrable on verifying, for all :
Therefore, the fuzzy mapping is (i)-differentiable on and
Moreover, if we replace the assumption by the alternative condition(i) For all is (ii)-differentiable on then the fuzzy function is (ii)-differentiable on and equation (7) remains true.
Theorem 7 (see [6]). Let be a fuzzy function such that satisfies the assumptions above, for all .
Let or (for short) denote the fuzzy Laplace transform of with respect to the time variable . Then,
Theorem 8 (see [6]). Let be a fuzzy-valued function on into . Suppose that the mappings and are fuzzy Riemann integrable on for all for some .(a)If is (i)-differentiable with respect to , then(b)If is (ii)-differentiable with respect to , then
3. High-Order Differentiability of Fuzzy Improper Integral
Theorem 9. We consider a fuzzy-valued function , verifying the following assumptions:(i) For all is continuous and fuzzy Riemann integrable on (ii) For all is strongly generalized differentiable of the second order on the interval (iii) For all and are continuous on (iv) For all and are continuous on (v) For all , there exist four continuous crisp functions , and , which are integrable on verifying, for all :
Therefore, the fuzzy mapping is strongly generalized differentiable of the second order on and
Proof. Using Theorem 6 and since verifies assumptions , then the fuzzy mapping is strongly generalized differentiable on andThen, from the assumptions , the function satisfies the conditions . Hence, and by Theorem 6, is strongly differentiable on and for all , we have
Theorem 10. Let be a fuzzy function. Suppose that the mapping satisfies the assumptions above, for all for some . Let or (for short) denote the fuzzy Laplace transform of with respect to the time variable . Then,
Proof. For fixed , then using Theorem 9, we have
Theorem 11. Let be a fuzzy-valued function on into . Suppose that the mappings , and are fuzzy Riemann integrable on for all for some .(a)If and are (i)-differentiable with respect to , then (b)If is (i)-differentiable and is (ii)-differentiable with respect to , then (c)If is (ii)-differentiable and is (i)-differentiable with respect to , then (d)If and are (ii)-differentiable with respect to , then
Proof. This obviously results from Theorem 4, by fixing and taking the Laplace transforms and derivations with respect to the variable .
Theorem 12. We consider a fuzzy-valued function , verifying the following assumptions:(i) For all is continuous and fuzzy Riemann integrable on (ii) For all is strongly generalized differentiable of the -th order on (iii) For all and for each is continuous on (iv) For all and for each is continuous on (v) For all and for all , there exist a couple of continuous and integrable crisp functions on verifying, for all :
Therefore, the fuzzy mapping is strongly generalized differentiable of the -th order on and we have
Proof. According to Theorem 6, the result holds true for . Also, by induction, assume that the result is true to the -th order. In addition, let a function , satisfying the conditions . Then, is strongly generalized differentiable of the -th order on andFrom the assumptions and using Theorem 6 and since verifies assumptions , then the fuzzy mapping is strongly generalized differentiable on , that is, is strongly generalized differentiable to the -th order on and we have
4. Fuzzy Laplace Transform for Second-Order Fuzzy Linear Partial Differential Equations
Our aim now is to solve the following second-order linear FPDE using the fuzzy Laplace transform method:where is a fuzzy strongly differentiable function of second order, with continuous partial derivatives. , and are real continuous functions, and , and are continuous fuzzy functions. Without loss of generality, assume that the mappings are all positive.
4.1. Resolution of Equation (21) by Fuzzy Laplace Transform Method
By using fuzzy Laplace transform with respect to , we get
Therefore, we have to distinguish the following 32 cases for solving this last equation.(a)If is (i)-differentiable with respect to and , is (i)-differentiable with respect to and , and is (i)-differentiable with respect to , then using Theorems 10 and 11, we get Denote and . Then, satisfying the following initial conditions: Assume that this leads to where is the solution of system (24) under (25). By the inverse Laplace transform, we obtain(b)If is (i)-differentiable with respect to and , is (i)-differentiable with respect to and (ii)-differentiable with respect to , and is (i)-differentiable with respect to , then we obtain Thus, Assume that this leads to where is the solution of system (29) under (25). By the inverse Laplace transform, we get(c)If is (i)-differentiable with respect to and (ii)-differentiable with respect to , is (i)-differentiable with respect to and , and is (i)-differentiable with respect to , then Assume that this leads to where is the solution of system (32) under (25). By the inverse Laplace transform, we obtain(d)For the -th case from the 29 remaining cases, with , we get a differential system similar to one of the previous systems (24) and (29).
Assume that this leads towhere is the solution of the latter system under initial condition (25).
By the inverse Laplace transform, we have
4.2. Algorithm of Fuzzy Laplace Transform
The steps of the proposed algorithm are as follows:(i)Choose a case from the 32 possible ones according to the differentiability’s type of each from the functions , and with respect to and , respectively.(ii)Replace these functions by their parametric forms to transform equation (21) into an equivalent classical differential system of two linear equations with unknown , for .(iii)Solve this differential system under the given initial and boundary conditions.(iv)Calculate the length of the mappings , and .(v)Deduce the domain of definition for the solution using the nonnegativity of calculated lengths.
5. Numerical Examples
Example 1. We consider the heat equation with fuzzy initial and boundary conditions:where is a positive real number. Here, we have to distinguish only 8 cases.(a)If is (i)-differentiable with respect to and and is (i)-differentiable with respect to , then Using the conditions leads to Solving (39), we get Since , then . Thus, By the inverse Laplace transform, we deduce The lengths of , and are, respectively, given by Hence, this solution is invalid because .(b)If is (i)-differentiable with respect to and and is (ii)-differentiable with respect to , then Solving (44), we get Since , then . Hence, By the inverse Laplace transform, we obtain The lengths of , and are, respectively, given by So, this solution (called solution 1) is valid over , where (see Figures 1–4).(c)If and are (i)-differentiable with respect to and is (ii)-differentiable with respect to , then Solving (49), we obtain Similarly and as in Case 2, we obtain . So, By the inverse Laplace transform, we deduce Hence, this solution is invalid because .(d)If is (i)-differentiable and is (ii)-differentiable with respect to and is (ii)-differentiable with respect to , then Solving (53), we get As in Case 1, we get . Also, by the inverse Laplace transform, Thus, this solution (called solution 2) is valid over , where (see Figures 5–8).(e)If is (ii)-differentiable and is (i)-differentiable with respect to and is (i)-differentiable with respect to , then Solving (56), we get As in Case 2, we get . By the inverse Laplace transform, Thus, this solution (called solution 2) is valid over , where (see Figures 5–8).(f)If and are (ii)-differentiable with respect to and is (i)-differentiable with respect to , then Solving (59), we get As in Case 1, we have . Also, by the inverse Laplace transform, Therefore, this solution is invalid because .(g)If is (ii)-differentiable and is (i)-differentiable with respect to and is (ii)-differentiable with respect to , then Solving (62), we get As in Case 1, we get . Also, by the inverse Laplace transform, Hence, this solution (called solution 2) is valid over , where (see Figures 5–8).(h)If and are (ii)-differentiable with respect to and is (ii)-differentiable with respect to , then Solving (65), we get As in Case 2, we get . Also, by the inverse Laplace transform,Therefore, this solution is invalid because .
Remark 1. Notice that in all cases, if we take , we find the crisp solution of the corresponding classical heat equationFor the graph of the crisp solution, see Figures9 and 10.
Example 2. We consider the following FPDE:Analogously, we obtain the solution’s expression by distinguishing the following cases:(a)If (i)-differentiable with respect to and is (i)-differentiable with respect to , then Therefore, this solution (called solution 3) is valid all over (see Figures 11–14).(b)If (i)-differentiable with respect to and is (ii)-differentiable with respect to , then Thus, this solution is invalid because .(c)If (ii)-differentiable with respect to and is (i)-differentiable with respect to , then Also, this solution is invalid because .(d)If (ii)-differentiable with respect to and is (ii)-differentiable with respect to , thenSo, this solution is invalid because is not (ii)-differentiable with respect to .
Remark 2. Notice that in all cases, if we take , we find the crisp solution of the corresponding classical problem
6. Discussion
The Laplace transform method was used to compute the analytic solution of two linear FPDEs of second order. First, we decomposed each FPDE into a system of two crisp PDEs with the unknown , for , for which we calculated the Laplace transforms using the properties proved in Section 3. Then, using the inverse Laplace transform, we obtained the lower and upper solution’s parts , respectively. Finally, we determined the definition’s domain of these fuzzy solutions by utilizing the positivity of the length for and its partial derivatives , and .
In both numerical examples studied, the fuzzy solution of the second-order FPDE can be expressed as follows:where is the crisp solution of the corresponding classical second-order PDE, obtained by letting , and is an undesirable term, which represents the fuzzy pure part of the fuzzy solution .
This fuzzy pure component results from the modeling choices and steps using fuzzy tools and theory. It also measures the uncertainty and vagueness in the adopted model due to the imprecisions in the initial and boundary conditions or in the fuzzy (respectively, real) second member of the FPDE (respectively, PDE).
On the one hand, we get for Example 1:that is,for solution 1 of (37), given in its parametric form. Hence, we have
On the other hand, we obtainthat is, , for solution 2 of (37), written in its parametric form. So, we have
The presence of the pure fuzzy parts in these two solutions of the fuzzy heat equation can be explained by some uncontrollable parameters or omitted inputs in the modeling of the crisp problem. So, the fuzzy approach is better and more efficient than the ordinary classical way.
Furthermore, we have for Example 2:that is, , for solution 3 of (69), given in its parametric form. Then, we have
In (37), the length of the fuzzy pure part goes to infinity as for solution 2, while the length converges to 0 as for the fuzzy pure part of solution 1. Thus, the first solution is stable, whereas the second solution is unstable. In other words, solution 2 is most uncertain or vague than solution 1 for Example 1.
In Example 2, the crisp part of the solution 3 is and its fuzzy pure part is , for which the length diverges to the infinity as (respectively, ). Hence, the uncertainty is increasing as the value of or increases, and this solution is unstable.
In general, as approaches 1, the fuzziness and uncertainty become smaller and completely disappear, for , yielding the crisp solution of the classic real problem.
Moreover, note that the existence, form, and asymptotic behavior of the solution depend on the choice of the kind of each used fuzzy partial derivative. Indeed, solution 2 of Example 1 is unstable, and it is obtained if we assume that is (i)-differentiable with respect to and it is (ii)-differentiable with respect to and is (ii)-differentiable with respect to (see the fourth case (d)), while solution 1 is stable and is obtained provided that is (i)-differentiable with respect to and and is (ii)-differentiable with respect to (see the second case (b)). But, in the other cases, there is no valid fuzzy solution.
In Example 2, we note that solution 3 exists and is unstable, and it is valid only if we assume that is (i)-differentiable with respect to and is (ii)-differentiable with respect to .
Furthermore, the uniqueness of the solution is lost in Example 1 (we have two (solutions 1 and 2)), although this unicity is preserved in Example 2.
Consequently, the study of these examples demonstrates that the type of fuzzy partial derivatives used influences the existence, uniqueness, and stability of the fuzzy solution(s) for a second-order FPDE under the strong generalized differentiability assumption.
Now, we proceed to the graphic interpretation of Figures 5–8. Firstly, in the solution 2 of (37), the lower and upper parts of this solution are proportional to the crisp solution of the corresponding classic equation, for all . Indeed, we have
So, all the obtained graphs are in fact images of the crisp solution’s graph (see Figures 9 and 10) by the dilation of ratio equal to for the graph of and equal to for the graph of .
Secondly, for each valid fuzzy solution (1, 2, and 3), in both examples, the maximum value of the upper solution is greater than the maximum value of the lower solution . For instance, we get the following results for solution 2, , and :and for solution 1 and , we havewhile we get for solution 3 and :
Finally, the graphs of the solutions and are almost equal and coincide with the crisp solution’s graph for close to 1, as it is shown in the graph of solution 3 for (see Figures 13 and 14).
7. Conclusions
Theorems of high differentiability for a fuzzy function defined via a fuzzy improper integral have been investigated and proved, which have been employed to prove some results related to the partial derivatives of the fuzzy Laplace transform. Then, using the Laplace transform method, the solutions for linear FPDEs of second order have been given. For future research, one can apply this method to solve nonlinear FPDEs of first or high order. The influence of the choice of the kind of the fuzzy partial derivatives on the fuzzy solutions and their existence, uniqueness, and asymptotic behavior have been discussed.
Data Availability
The graph data used to support the findings of this study are included within the supplementary information file. This file contains programs developed by using the free Python software to plot each of the fourteen figures in the manuscript. For more information and documentation about Python, the readers can consult the website https://www.python.org.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to extend their gratitude to the anonymous referees for their in-depth reading, criticism of, and insightful comments on earlier versions of this paper.
Supplementary Materials
Here we give a concise description of the supplementary material used in this article. All the fourteen Python programs that we developed are similar and can be summarized as follows: Step 1: call the necessary Python libraries in the heading—numpy, matplotlib.pyplot, mpltoolkits.mplot3d, and sklearn.datasets. Step 2: set up the axes using “plt.axes” and define the mapping to plot using “def.” Step 3: define the intervals for the values of each of the variables , and using “np.linspace.” Step 4: plot the function using “ax.plotsurface.” Step 5: set up the figure’s title using “ax.settitle” and the axis labels using “ax.setxlabel” and “ax.setylabel.” (Supplementary Materials)