Singularly perturbed fuzzy initial value problems

Abstract

In this work, we have firstly introduced singularly perturbed fuzzy initial value problems (SPFIVPs) and then we have given an algorithm for the solutions of them by using the extension principle given by Zadeh. We have presented some results on the behaviour of the $\alpha$-cuts of the solutions. To show the robustness of the given algorithm, we have fuzzified some examples given in the literature and then we have applied the algorithm.

Introduction

Singular perturbations were firstly introduced by Prandtl (1905). But, the term “singular perturbation” was first given by Friedrichs and Wasow (1946). The solution of a singular perturbation problem (SPP) has boundary layers. The concept of the boundary layer has gained a more general use after Wasov’s work (Wasow, 1942). In SPPs for differential equations, the highest order derivative have a small positive $ϵ$ coefficient. This kind of problem has multiscale solution. This means that there are layers in the solution such that in the thin transition layers the solution changes quickly and the other parts of the solution acts regularly and varies slowly (Alquran and Doğan, 2010, Doğan et al., 2010, Doğan et al., 2012, Doğan et al., 2011, Doğan et al., 2013, Neyfeh, 1973, O’Malley, 1974, O’Malley, 1991). These narrow regions, generally, are seen at the boundaries of the domain. These changes at solutions in applied problems are seen as transition points, shock layers, boundary layers, skin layers, edge layers etc. Kumar et al. (2010). These problems are seen in very different applied problems (Kumar et al., 2010). In recent years, new studies have been carried out on the solutions of singularly perturbed initial value problems and attract the attention of researchers. The following studies can be given as examples of some of these studies. A parameter-uniform implicit scheme have constructed for a class of parabolic singularly perturbed reaction–diffusion initial–boundary value problems with large delay in the spatial direction (Kumar & Kumari, 2020). Two computational approaches on the basis of the reproducing kernel Hilbert space method were proposed for solving singularly perturbed 2D parabolic initial–boundary-value problems (Fardi & Ghasemi, 2022). Gie et al. have implemented their new semi-analytic time differencing methods, applied to singularly perturbed non-linear initial value problems (Gie et al., 2022). Shishkin and Shishkina have considered an initial–boundary value problem for the singularly perturbed transport equation. They proposed a new approach to constructing the difference scheme based on a special decomposition of solution into the sum of a regular and a singular components (Shishkin & Shishkina, 2022).

The fuzzy set theory was firstly defined by Zadeh (1965). But, the definition of “fuzzy differential equation” was first imported to the literature by Byatt and Kandel (1978) in 1978. There are different suggestions for obtaining the solution of a fuzzy initial value problem (FIVP). One of these suggestions is the Zadeh’s extension principle approach applied by Mizukoshi et al., 2007, Oberguggenberger and Pittschmann, 1999. For solving a FIVP by using the extension principle given by Zadeh, firstly the corresponding crisp initial value problem’s solution is obtained, then this solution is fuzzified by the extension principle. Recently, several studies have used the concept of interactivity to study fuzzy differential equations (Esmi et al., 2021, Santo Pedro et al., 2019). These interactivities may arise depending on the physical/biological restrictions of the phenomena under consideration (Fullér and Majlender, 2004, Wasques et al., 2020, Wasques et al., 2018). To find the solution of such fuzzy differential equations, a generalization of Zadeh’s extension principle called the sup-J extension principle can be used to obtain fuzzy solutions (Esmi et al., 2021). The sup-J extension principle is another method for extending classical functions to the fuzzy domain, which considers interactivity as stated by Carlsson and Fullér (2004). In science and engineering, FIVPs are important as a tool to explain vagueness in models. Hence, many researches were carried out in applications of fuzzy set theory such as population models (Akin and Oruç, 2012, Barros et al., 2000), civil engineering models (El Naschie, 2005), computational biology models (Casasnovas and Rossell, 2005, Hassan et al., 2012) and etc. In addition, many researchers have recently been interested in the solutions of FIVPs. Salgado et al. have presented some properties of the fuzzy Laplace transform with the notion of linearly correlated differentiability also called L-differentiability. They have used these properties to solve a interactive fuzzy initial value problem (Salgado et al., 2021). Dallashi and Syam have presented an accurate numerical approach based on the reproducing kernel method (RKM) for solving second-order fuzzy initial value problems (FIVP) with symmetry coefficients such as symmetric triangles and symmetric trapezoids (Dallashi & Syam, 2022). Çitil have solved the fuzzy initial value problem with negative coefficient by using fuzzy Laplace transform and generalized differentiability (Çitil, 2020).

In this paper, we used the approach by Akın and Bayeğ, 2019, Akin et al., 2016 to develop an algorithm to obtain $\alpha$-cuts for the solutions to the singularly perturbed first order differential equations with fuzzy initial values. Akın and Bayeğ, 2019, Akin et al., 2016 proposed an algorithm based on Zadeh’s extension principle to find the solution of second order linear differential equations with fuzzy initial values. To prevent the switching of end points of $\alpha$-cuts for the fuzzy solutions, the heaviside step function is used in the algorithm. With this implementation, there is no need to consider “interval condition” as stated by Buckley and Feuring (2001) or decomposition of the domain with respect to the signs of the unknown function and its derivative as considered by Akın et al. (2013). Buckley and Feuring (2001) proposed two methods based on Zadeh’s extension principle to find the solution of fuzzy initial value problems. However, one need to check if the obtained solution satisfies the interval condition or not. If the solution satisfies it, the solution is said to be a solution to the fuzzy initial value problem. Hence, an algorithm based on the method proposed by Buckley and Feuring (2001) may not guarantee a fuzzy solution on the whole interval of solution. Similarly, Akın et al. (2013) proposed a method to find the local solutions for fuzzy differential equations. However, an algorithm based on the method proposed by Akın and Bayeğ, 2019, Akin et al., 2016 guarantees the fuzzy solution. That is why, we developed an algorithm based on the method given by Akın and Bayeğ, 2019, Akin et al., 2016.