On type-2 fuzzy partial differential equations and its applications

Article type: Research Article

Authors: Bandyopadhyay, Abhirup | Kar, Samarjit^{; *}

Affiliations: Department of Mathematics, National Institute of Technology, Durgapur, India

Correspondence: [*] Corresponding author. Samarjit Kar, Department of Mathematics, National Institute of Technology, Durgapur 713 209, India. E-mail: [email protected].

Abstract: Type-2 fuzzy sets are generally used to describe those membership values of fuzzy sets that are imprecise. This paper attempts to develop the theory of type-2 fuzzy partial differential equations (T2FPDE) using type-2 fuzzy initial condition. The theory of type-2 FPDEs could be used with type-2 fuzzy initial values, type-2 fuzzy boundary values and type-2 fuzzy parameters. Some natural phenomena can be modelled as dynamical systems whose initial conditions and/or parameters might be imprecise in nature. This imprecision of initial values and/or parameters are generally modelled by fuzzy sets. In this paper the concept of generalized H2-differentiability is applied. This concept is based on the enlargement of the class of differentiable type-2 fuzzy mappings which are commonly known as Hukuhara derivatives. Some theorem is presented to show the solution of a fuzzy type-1 and type-2 PDE could be obtained by solving the corresponding embedded systems based on fuzzy differential inclusions. An algorithm is also developed to simulate of type-2 fuzzy partial differential equations and obtain its solution numerically. Some illustrative examples have also been provided for different type-2 FPDE models.

Keywords: Type-2 FPDE, Type-2 Hukuhara partial derivatives, Type-2 fuzzy differential inclusion, numerical solutions, Type-2 fuzzy heat equation, Type-2 fuzzy wave equation, Type-2 fuzzy advection equation

DOI: 10.3233/JIFS-17175

Journal: Journal of Intelligent & Fuzzy Systems, vol. 34, no. 1, pp. 405-422, 2018