Type 2 Interval Valued Caputo Fractional Differential Equations

This paper introduces the Caputo fractional differential equation for interval valued functions. A fractional differential equation incorporates memory sense through an iterated kernel for describing dynamical systems. Impreciseness exists in the process of quantification and analysis of the involved variables influencing such physical processes. In other words, memory and imprecision may coexist, necessitating the study of an imprecise fractional differential equation. Interval numbers and interval valued functions are mathematical tools to manifest uncertainty due to the variance of decision parameters between ranges. In this paper, an imprecise fractional differential equation is studied under Type 2 interval uncertainty, a generalization of interval uncertainty. This paper analyzes conditions for the existence of a unique solution of the Type 2 interval valued Caputo fractional differential equations. Riemann–Liouville fractional integral equations and metric spaces for Type 2 interval numbers and interval valued functions are employed for establishing results. Examples of linear and nonlinear Type 2 interval valued Caputo fractional differential equations are discussed, ensuring a smooth extension of the interval fractional differential equation to a wider domain. Economic and biological models are hinted at as possible applications of this proposed theory.