On Fuzzy Improper Integral and Its Application for Fuzzy Partial Differential Equations

Abstract

We establish some important results about improper fuzzy Riemann integrals; we prove some properties of fuzzy Laplace transforms, which we apply for solving some fuzzy linear partial differential equations of first order, under generalized Hukuhara differentiability.

1. Introduction

Wu introduced in [1] the improper fuzzy Riemann integral and presented some of its elementary properties; then he studied numerically this kind of integrals.

This notion was exploited by certain researchers to study fuzzy differential equations (FDEs) of first or second order utilizing fuzzy Laplace transform, namely, by Allahviranloo and Ahmadi in [2], then by Salahshour et al. (see [3, 4]), and by ElJaoui et al. in [5].

The objective of this paper is to study the improper fuzzy Riemann integrals by establishing some important results about the continuity and the differentiability of a fuzzy improper integral depending on a given parameter.

These results are then employed to prove some fuzzy Laplace transform’s properties, which we use to solve fuzzy partial differential equations (FPDEs).

The organization of the remainder of this work is as follows. Section 2 is reserved for preliminaries. In Section 3, the main results are proved and new properties of fuzzy Laplace transform are investigated. Then, in Section 4, the procedure for solving first-order FPDEs by fuzzy Laplace transform is proposed. Section 5 deals with some numerical examples. In Section 6, we present conclusion and a further research topic.

2. Preliminaries

By we meant the set of all nonempty compact convex subsets of , which is endowed with the usual addition and scalar multiplication. Denote (see [6]) where(1)is normal; that is, for which ,(2) is convex in the fuzzy sense,(3) is upper semicontinuous,(4)the closure of its support is compact.For , denotes the -level set of .

Then, it is obvious that for all , , andLet be a function which is defined by the identity where is the Hausdorff distance defined in . Then, it is clear that is a complete metric space (for more details about the metric see [7]).

Definition 1 (see [2]). One defines a fuzzy number in parametric form as a couple of mappings and , , 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 following identity holds true (see [8]):

Theorem 2 (see [1]). One considers a fuzzy valued function defined on . Suppose that, for all fixed , the crisp functions are integrable on , for every , and that there exist two positive constants and such that and for every . Then is fuzzy Riemann integrable (in the sense of Wu) on , its improper fuzzy integral , andFor , if there exists an element such that , then is called the Hukuhara difference of and , which we denote by .

Definition 3 (see [2]). A mapping is said to be strongly generalized differentiable at , if there exists , such that (i)for all   being very small, there exist ; ; and the limits  or(ii)for all   being very small, there exist ; ; and the limits  or(iii)for all   being very small, there exist ; ; and the limits  or(iv)for all   being very small, there exist ; ; and the limits

The next theorem permits us to consider only case (i) or case (ii) of Definition 3 almost everywhere in the domain of the mappings studied.

Theorem 4 (see [9]). If is a strongly generalized differentiable function on in the sense of Definition 3, (iii) or (iv), then for each .

Theorem 5 (see, e.g., [10]). We consider a fuzzy function which is represented by , for all : (1)If is (i)-differentiable, then the crisp functions and are differentiable and .(2)If is (ii)-differentiable, then the crisp functions and are differentiable and .

Definition 6 (see [2]). If is a continuous mapping such that is fuzzy Riemann integrable on then is called the fuzzy Laplace transform of which one denotes by Denote by the classical Laplace transform of a crisp function , and then

Theorem 7 (see [2]). Let be a fuzzy valued function and its derivative on . Then, if is (i)-differentiableor if is (ii)-differentiableprovided that the Laplace transforms of and exist.

3. Continuity and Differentiability of Fuzzy Improper Integral

In this section, denotes one of the intervals, or or , where , denotes another interval, and is a nonempty subset of .

Lemma 8. Let , be two fuzzy valued functions, which are fuzzy Riemann integrable on , in the sense of Wu (see [1]), such that the real function is integrable on , and then

Proof. From identity (4), we have

Theorem 9. Let be a fuzzy function, satisfying the following conditions: () For all , is continuous on .()For each , is continuous on .()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 .

Proof. Let and let be a sequence of elements of , which converges to as . For , , and , we haveThusBy tending and using assumption , we obtainThereforeFrom and , we deduce that the mappings , , and are all integrable on .
On the other hand, we get the following inequality from Lemma 8:That is,By assumption , we have as .
So, by the dominated convergence theorem, as .
From inequality (22), we deduce that as .
Consequently, is continuous on .

Lemma 10. One considers two fuzzy valued functions , , which are fuzzy Riemann integrable on (in the sense of Wu), such that exists for all , then is fuzzy Riemann integrable on , the Hukuhara difference is well defined, and

Proof. Let ; that is, . It is clear that there exist positive constants , , , and such that, for all in , we haveHenceand similarlyThen from Theorem 2, is fuzzy Riemann integrable on .
By “linearity” of the fuzzy integral, we get Thus, exists and .

Theorem 11. One considers a fuzzy valued function , verifying the following assumptions: () For all , is continuous and fuzzy Riemann integrable on .()For all , is (i)-differentiable on the interval .()For all , is continuous on .()For all , is continuous on .()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 andMoreover, if one replaces assumption by the alternative condition ()for all , is (ii)-differentiable on ,then the fuzzy function is (ii)-differentiable on and (29) remains true.

Proof. Assume that ()–() hold true. Let , being very small, and define the auxiliary functions For fixed , we havewhere the existence of the Hukuhara differences is ensured by the (i)-differentiability of and by Lemma 10.
Analogously, we get From assumptions ()–(), we deduce that and satisfy conditions ()-() of Theorem 9.
On the other hand, using the finite increments theorem, we obtainSimilarly, we haveInequalities (33) and (34), which are obviously also true for , ensure that and satisfy condition of Theorem 9.
Applying the latter theorem, we getTherefore, is (i)-differentiable at andThe proof under assumption instead of is similar to the first case.

Theorem 12. One considers a fuzzy function . Suppose that the mapping satisfies 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 11 we have

Theorem 13. Let be a fuzzy valued function on into . Suppose that the mappings and are fuzzy Riemann integrable on , for all for some . Consider the following:(a)If is (i)-differentiable with respect to , then(b)If is (ii)-differentiable with respect to , then

Proof. This is a direct result of Theorem 12, by fixing and taking the Laplace transforms with respect to .

4. Fuzzy Laplace Transform Algorithm for First-Order Fuzzy Partial Differential Equations

Our aim now is to solve the following first-order FPDE using the fuzzy Laplace transform method under strongly generalized differentiability:where is a fuzzy function of , , is a real constant, and , , and are fuzzy valued functions, such that is linear with respect to . For short, assume that (case is similar).

By using fuzzy Laplace transform with respect to , we getTherefore, we have to distinguish the following cases for solving (42):(a)Case : If is (i)-differentiable with respect to and , then by Laplace transform where and . Using Theorems 12 and 13 we get the following differential system: satisfying the following initial conditions: Assume that this leads to where is solution of system (44) under (45). By the inverse Laplace transform we get(b)Case : If is (i)-differentiable with respect to and (ii)-differentiable with respect to , then by Theorems 12 and 13 we get the following differential system, satisfying the initial conditions (45): Assume that this implies where is solution of system (48) under (45). Thus(c)Case : If is (ii)-differentiable with respect to and (i)-differentiable with respect to , then we get the following differential system, satisfying the initial conditions (45): Assume that this implies where is solution of system (51) under (45). Therefore(d)Case : If is (ii)-differentiable with respect to and , then we get the following differential system, satisfying the initial conditions (45): Assume that this leads to where is solution of system (54) under (45). Hence

5. Numerical Examples

Example 1. ConsiderCase 1. If is (i)-differentiable with respect to and , then by Laplace transform we getThis differential system satisfies the following initial conditions:Solving (58) under (59), we getBy the inverse Laplace transform we deduceThe lengths of , , and are, respectively, given bySo, this solution is valid for all and .
Case 2. If is (i)-differentiable with respect to and (ii)-differentiable with respect to , then analogouslywhere is the unit step function or the Heaviside function:Therefore, this solution is valid only over .
Case 3. If is (ii)-differentiable with respect to and (i)-differentiable with respect to , then similarlyAs in Case , one can verify that this solution is valid only over .
Case 4. If is (ii)-differentiable with respect to and , thenOne can verify that function is not (ii)-differentiable with respect to either or .
So, no solution exists in this case.

Example 2. ConsiderCase 1. If is (i)-differentiable with respect to and , then, by application of the algorithm above, one obtains The lengths of , , and are, respectively, given bySo, this solution is valid for all , : , .
Case 2. If is (i)-differentiable with respect to and (ii)-differentiable with respect to , therefore Then, this solution is valid for all , : , .
Case 3. If is (ii)-differentiable with respect to and (i)-differentiable with respect to , then analogouslyHence, this solution is valid for all , : , .
Case 4. If is (ii)-differentiable with respect to and , then similarlySo, this solution is valid for all , : , .

6. Conclusion

Theorems of continuity and differentiability for a fuzzy function defined via a fuzzy improper Riemann integral are proved which are used to prove some results concerning fuzzy Laplace transform. Then, using Laplace transform method, the solutions for some linear fuzzy partial differential equations (FPDEs) of first order are given. For future research, one can apply this method to solve FPDEs of high order.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.