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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

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On type-2 fuzzy partial differential equations and its applications

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