Generalizing the meaning of derivatives and integrals of any order differential equations by fuzzy-order derivatives and fuzzy-order integrals

Abstract

This paper develops the correlation between fuzzy numbers and order of differential equations and overcomes the limitation in the existence of fractional order in the formulation of equation. In the view of fractional calculus, a new logic called fuzzy order by generalizing the meaning of derivatives and integrals of any order as fuzzy-order derivatives and fuzzy-order integral. We discuss Dα, where Dα is derivative of order α and α may be a triangular fuzzy number or trapezoidal fuzzy number, and propose to rewrite Dαy(x)=gx,y(x), when α=A,B,C and A,BandC∈N (where N is the set of natural numbers) and rewrite Riemann-Liouville integral, Riemann-Liouville derivative and Caputo fractional derivatives with respect to this new logic of fuzzy order. The proposed approach also covers multi cases, where the order is either integer or fractional. At the end, three numerical examples are presented to demonstrate the application of new logic, when the order of derivatives and integrals are given as triangular fuzzy numbers. These include time fractional heat equation represented as a time fuzzy-order heat equation and the time-fractional diffusion wave equation represented as a time-fuzzy-order diffusion wave equation. Keywords: Fractional calculus, Fuzzy numbers, Riemann-Liouville fuzzy definitions, Caputo fuzzy definitions