Abstract
Our main goal is to define a fuzzy solution for problems involving diffusion. To this end, the solution of fuzzy diffusion-reaction-advection equation will be defined as Zadeh’s extension of deterministic solution of the associated problem. Important aspects such as unity and stability of these solutions will also be studied. Graphical representations of these solutions will be presented.
1. Introduction
In 1965, Zadeh introduced the concept of fuzzy sets [1]. Therefore, much research has been developed botkh from a theoretical and practical [2–11]. One area of great interest was the modelling uncertain phenomena by means of differential equations. When subjectivity is random, the nature of uncertainty can be treated as a tool for stochastic differential equations; however, the fuzzy variational models can include various types of uncertainties beyond the stochastic.
Depending on the choice of state variable and/or model parameters, respectively, demographic fuzziness state variables are modelled by fuzzy sets and/or environmental fuzziness only when the parameters are considered fuzzy. In biological phenomena in general, both types are present fuzziness [2].
The fuzzy variational equations have been studied by different methods. The first attempt to include subjectivity of the nonrandom systems was with the variational derivative of Hukuhara [12, 13], but this formulation is not able to reproduce the rich behaviour of deterministic differential equations such as periodicity, stability, and bifurcation.
In 1999, [14] proposed another way of looking at subjectivity not by random differential inclusions. The fuzzy solutions obtained by this approach are able to present frequency, stability, and bifurcation; however, from a technical standpoint, to obtain solutions through a family of differential inclusions can be extremely difficult.
When the variables are uncertain, a way to have a more adequate treatment variational models for all uncertainty is to pass parameters to the mathematical model. In this case, an alternative method that is used is that which is to make fuzzy solutions of a deterministic model, using Zadeh’s extension [15].
Zadeh’s extension is the way we produce a fuzzy transformation for a given function . Zadeh’s extension has been studied and applied by many authors, including [12, 16–18], in the study of fuzzy fractals and Nguyen [7] in set-representation of fuzzy sets.
Our main objective in this paper is to explore properties such as uniqueness and stability of the fuzzy solution of a fuzzy differential equation associated with classical advection-diffusion-reaction equation (see Figure 9), using Zadeh’s extension [19].
2. Basic Concepts
2.1. Fuzzy Sets
In this section we present the basic fundamentals necessary for the proper understanding and application of fuzzy set theory in the following sections. For more information, the reader may consult [2, 19]. The fuzzy subsets of a space are characterized by functions of space in the interval . Therefore we have the following.
Definition 1. Let be a nonempty set. A fuzzy subset of is a subset nonempty of for some function .
The function is called the membership function of and the value assumed for each is the degree of relevance of in . The set consists of all fuzzy subsets of a set that will be denoted by .
Directly follow that, when defining that any subset random is also a fuzzy subset of , since this set is well determined by the characteristic function and therefore satisfies the definition above. In fact, if subset with characteristic function , then the set is a subset of with membership function for all and hence . To a better distinction, the fuzzy subset will be referred to herein as set crisp of .
Given a fuzzy subset in defined, for each , the set is the set of elements of such that the degree of relevance in is at least . The set is called -level of and, mathematically, is defined as
The zero level of a fuzzy subset is defined by where is the support of fuzzy subset .
Two fuzzy sets are equal when the characteristic functions are equal, that is,
The equality between fuzzy sets can also be characterized through -levels. In this case, the sets are equal when -level matches for every .
In many cases it may be necessary to extend the domain of a application for the fuzzy sets in . Note that for each subset an application defines a subset . Assuming now that is a fuzzy subset , that is, , then we need to determine how the image induced the application over . The way this picture is can be characterized by principle of Zadeh’s extension as defined below.
Definition 2. Let be a applying a fuzzy subset . Zadeh’s extension whose image is applied has membership function:
As mentioned earlier, a subset determines fuzzy set of whose membership function is a characteristic function of . The image of by extension that is a function coincides with the fuzzy set defined by . That is, . In fact, the above definition ensures that has membership function: which is the characteristic function of . Therefore, . In particular, for all it is worth .
Theorem 3. If is continuous, then Zadeh’s extension is continuous and satisfies for all .
By the Hausdorff metric , for the compacts of , we can define a metric for the set . Thus, given two points , the distance between and is given by where
Particular type of fuzzy set that will prove useful is the call number fuzzy, which is an attempt to generalize real numbers in the fuzzy context.
Definition 4. A fuzzy set is called fuzzy number if satisfies the following: (a)the -level are compact, connected, and nonempty for all ;(b)there is a unique such that .
Definition 5. Let be the -level associated with a function fuzzy and , functions such that and then define a diameter of as
2.2. Diffusion Equation
2.2.1. Diffusion-Reaction-Advection Equation in and
Diffusion models have been extensively employed to investigate dispersal and have yielded considerable insight into the dynamics of animal movement in space and time. Diffusion models can be written in the simplest form as where the operator denotes the spatial gradient, is time, is the local population density in the spatial variables and , is the coefficient of diffusion, and is the reaction-advection term describing the net population change due to birth, death, and direction of travel.
Most of the phenomena involving diffusion are described by models that suggest a dynamic in and is what will be treated below. Consider the initial value problem given by where is a real function, is a diffusion coefficient, is a number of individuals at the initial time, and is a Dirac function.
Thus, the classical solution to problem (12) exists and is unique for values of in bounded domains and is given by
If incorporate into (13) the parameters of reaction and advection will have a problem of the form whose solution is given by
If we consider , then we have where , are real functions, with being the solution to
3. Fuzzy Diffusion-Reaction-Advection Equation
3.1. Fuzzy Solution in
Most of the phenomena involving diffusion are described by models that describe a particular dynamic in and and that is what will be done next. Consider a classical diffusion equation (15), where is a diffusion constant and is the number of individuals at the initial time.
Thus, we interpret a problem that works as follows. We know that the phenomenon occurs by diffusion, but its initial condition is not well determined. Thus, we can consider the initial condition as a fuzzy number and thereafter apply the principle of Zadeh’s extension in initial condition. So we have the solution to the fuzzy initial value problems: and their -levels are given by which is plotted in Figures 1, 2, 3, and 10. An important note about these graphs is the fact that the deterministic solution for fuzzy initial value problem associated with the chart was kept deliberately fuzzy solution for the appropriate comparisons. About the -levels and their degree of membership a simulation of the evolution of the solution with time is made. A video also was available on https://www.youtube.com/watch?v=sHSfVfW67M4. The evolution of -levels is plotted in Figures 5, 6, 7, and 8.
It is easily verified that there is stability and convergence to zero (see Figure 4) in cases where the process is described only by diffusion and advection. For this, note that Thus, when time progresses we see that the fuzzy solution converges to the deterministic.
4. Fuzzy Diffusion-Reaction-Advection Equation
In two-dimensional case, we have the same behavior found in one-dimensional case. Of course, there is an inability to plot the graphs relating to two-dimensional fuzzy solution, but we can plot the graph of the spatial distribution for values of fixed, to make a more detailed study of these solutions. In fact, we have that -levels of the two-dimensional solution to the fuzzy problem: which is given by where . So we can look at some cuts in fuzzy two-dimensional flow graph that give us an idea of fuzzy population distribution, that is, preserving uncertainty of the initial condition.
Just as in the one-dimensional case, we can observe the stability of fuzzy solution for this using the concept across, and this will be explored in the next section with dimension .
5. Stability and Uniqueness of the Diffusion Equation Fuzzy
There is the possibility of defining a solution to the diffusion equation dimensional. Thus, we find significant results that generalize the properties of stability and unity of interest within the classical theory. To this end, consider the initial value problem given by equation where , are a real functions, with . In this case, consider , which characterizes our diffusion equation.
In terms of models that simulate reality, we want to play the above problem as follows: we have knowledge of the law governing the growth of a certain variable, the equation partial differential diffusion, but the initial condition is not well determined and can be known only partially loading with it a degree of uncertainty. Although the law is the classical solution to load the initial condition uncertainty over time, so let us consider the problem with initial condition uncertainty associated with the classical model given in (23): where is a fuzzy real function.
We know that when the function , then this solution to the classical problem exists for all and and is unique to values in limited domain . Thus we have the classical solution given by
Thus, the solution of the problem is a continuous function in . After we define Zadeh’s extension of solution as Zadeh’s extension of function , this solution will be denoted by and with and the -levels are given by
Then, we can define a fuzzy infinitely differentiable function with continuous derivatives of all orders as follows.
Definition 6. One says that a fuzzy function is when all -levels of are .
Theorem 7. Suppose that , and define as (26). Then (i),(ii),(iii) for all .
Proof. (i) The function is infinitely differentiable with limited derivatives of all orders in for all . Thus each for each . Then .
(ii) Since and each is a solution of diffusion equation, then solves problem (24).
(iii) Consider where each is initial condition to
Thus fix , , and choose such that
Then, if , we have
But, we have
by (29). If and , then
Then . Consequently
Then we have and ; this way, we have for all -level and each ; then
Define Then we have that is a solution of the problem where denotes Dirac’s measure in per unit mass in point .
Note that if is bounded and continuous and then is positive for all -level and for all and .
Let’s focus our attention to the nonhomogeneous initial value problem:
To solve this problem the following question must be answered: what is the formula to solve the following problem? Consider
Consider the solution of the homogeneous problem. Note that the application is a solution of diffusion equation to given. Now for fixed, the function is solution of which is solution of (23) to and . Then is not solution of (41). But by Duhamel principle, we can consider Then, we have to , . Now consider the following theorem.
Theorem 8. Define given by (43) and with compact support. Then (i),(ii), , ,(iii) for each point .
Proof. (1) Since has a singularity at , first we change variables to white
As has a compact support and is smooth near , we compute
Thus , , , belong to .
(2) We then calculate
Now
Integrating by parts, we also find
since solves the heat equation. We have
(3) Note that .
We can of course combine Theorems 7 and 8 to discover that is given by -levels in the form
Suppose that (50) is a fuzzy solution of the fuzzy initial value problem
One way to study the behaviour of the -dimensional fuzzy flow to the diffusion equation is using the concept of a diameter of a -level of a fuzzy number; to this, we will prove the following proposition.
Proposition 9. Define as a fuzzy flow given by (26) and is of -level. Then, In particular, , that is, , converges uniformly to , when .
Proof. Since , then we can define . Then we have proven that It is easy to see because .
Theorem 10. Let be a bounded region; thus, if a problem of fuzzy initial condition is given by where and , then the fuzzy flow associated to problem (25) is unique for all fuzzy initial condition given.
Proof. Let and be fuzzy flows associated to problem (26), then define . Thus, we have that resolves the problem which is a classical problem and, by the maximum principle, we have the maximum value of which is assumed at the border; then and .
Now, we have an important tool in the study of uncertainties that involves the diffusion equation, because we know that that limited our domain conveniently; we have the unique solution to problems involving diffusion. Furthermore, the stability of this solution is guaranteed and classical functions retaining it is degree of uncertainty.
Theorem 11. Define as a solution of problem satisfying the estimate for and . Then
Proof. Suppose that
Then we have
for some . Fix , , and define
Then we have
Fix and , . By the theorem of the maximum
where is a boundary of .
Now if ,
and if , , then
Now, if then for some . Then
for selected sufficiently large. Thus,
for all , because . Let , then fails! We repeatedly apply the result above on the time intervals , , and so forth, for .
Theorem 12. Let , . Then there exists at most one solution of the initial-value problem satisfying the growth estimate for constants .
Proof. If and satisfy (68) and (69), define ; we apply Theorem 11 to and we have the result.
6. Conclusion
The study of fuzzy diffusion equations brings an alternative method of treating process modeling where we have diffusive parameters and/or uncertain initial conditions. In the case of models that simulate reality, we have an inaccuracy in almost all parameters. This article develops an alternative method for treatment parameters which may have some uncertainty, for this, the fuzzy initial condition and setting conditions for the fuzzy solution is unique and ensures that these solutions are stable.
Appendix
In this work we have used the system of RGB colours for the graphical representation of the fuzzy solution of a fuzzy differential equation. In the RGB colour system, each colour is expressed by a combination of three values which have been considered here to be in the interval . We may then consider each colour as an function with , where represents a black colour and represents a white colour.
Thus, given an , the colour was used to colour the region in the plane bounded by the -level ; then the region bounded by is coloured white, and if then the region bounded by is coloured black. Thus, the higher the degree of membership of a point is, the corker its colour will be.
In practice we partitioned on and, for each , we used the equality to determine the set as follows. The higher the value of , darker the color used to fill the region bounded by the set .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors express their gratitude to the Fundação de Amparo à Pesquisa de Mato Grosso, FAPEMAT (no. 332507/2012), for financial support of this research.