On numerical solution of general fuzzy type-1 and type-2 arbitrary order dynamical systems

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: kar_s_k@yahoo.com.

Abstract: This paper develops the mathematical framework and the solution of a system of type-1 and type-2 fuzzy fractional and arbitrary order dynamical systems. The theory of arbitrary order differential equation is developed with fuzzy initial values, fuzzy boundary values and fuzzy parameters. There are numerous natural phenomena which can be modelled as arbitrary order dynamical systems whose initial conditions and/or parameters may be imprecise in nature. The imprecision of initial values and/or parameters are generally modelled by fuzzy sets. Here, the concept arbitrary order dynamical systems is developed with the introduction of fuzzy fractional derivatives, fuzzy fractional partial derivatives, fuzzy stochastic process, fuzzy stochastic random variable and fuzzy Brownian motion. The generalized Lp-integrability, based on the extension of the class of differentiable and integrable fuzzy functions is applied and fuzzy arbitrary order differential equations are represented as fuzzy integral equations. A novel numerical scheme for simulations of numerical solution for fuzzy arbitrary order differential equations have also been developed. Some illustrative examples have been provided for different fuzzy type-1 and type-2 arbitrary order general dynamical systems models related to mathematical finance and problems in fluid flow.

Keywords: Fuzzy fractional calculus, fuzzy arbitrary order ODE, fuzzy arbitrary order PDE, fuzzy arbitrary order SDE, numerical solutions

DOI: 10.3233/JIFS-171203

Journal: Journal of Intelligent & Fuzzy Systems, vol. 34, no. 3, pp. 1847-1862, 2018