**1. Introduction**

It is worthwhile mentioning, since last few decades, the theory of fractional calculus has gained significant importance in almost every branch of science, for having the capability to consider integrals and derivatives of any arbitrary order. The characteristic feature of generalizing the classic integer‐order differentiation and n‐fold integration to arbitrary fractional order have broadened its application in modeling several phenomena of physics, mathematics, and engineering. The differential models of fractional order, due to the nonlocal properties of fractional operator, are excellent instruments for providing information about the current as

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well as the historical state of the system. For these reasons, it is intensively developed and advanced, and existence of its solution is studied by well‐known authors, Euler, Laplace, Liouville, Riemann, Fourier, Abel, Caputo, etc., to further widen its scope in describing various real‐world problems of science, for instance see [1–6]. Another wide‐spreading exploration of mathematics is theory of fuzzy calculus, which has a lot of interesting applications in physics, engineering, mechanics, and many others. It is the theory of a particular type of interval‐ valued functions, in which mapping is made in such a way that it takes all the possible values in 0, 1 and not only the crisp values as found in usual interval‐valued functions. After the inception of fuzzy set theory by Zadeh [7], its attributes have been extended and established to overcome impreciseness of parameters and structures in mathematical modeling, reasoning, and computing [8–12].

Advanced development of mathematical theories and techniques has gained very high standard. On the basis of classical theories, new theories are pioneered by undergoing its inadequacies and widening its scope in many disciplines. In a similar manner, the aforemen‐ tioned theories have been brought together in modeling different aspects of applied sciences, to analyze the change in the respective system at each fractional step with the uncertain parameters. Agarwal et al. [13] initiatively incorporated uncertainty into dynamical system, modeled fractional differential equations with uncertainty, and studied its possible solutions. Ahmad et al. [14] described the situation of impreciseness of initial values of fractional differential equations and discussed its solutions by utilizing Zadeh's extension principle. In [15, 16], authors considered the concept of Caputo and Riemann fractional derivative, respec‐ tively, together with the Hukuhara differentiability and demonstrated the fuzzy fractional differential equations and a lot of others [17–23].

In light of noteworthy applications of above‐mentioned theories, in this chapter, we demon‐ strate fractional order dynamical models in fuzzy environment to depict unequivocal frac‐ tional differential equations of dynamical system. Moreover, we investigate its numerical solutions using the well‐known Grünwald‐Letnikov's fractional definition. This definition is widely applicable as a numerical scheme to solve linear and nonlinear differential equations of fractional order [24–26]. It is considered as an extended form of the classical Euler method. Here it will be utilized, for the first time, to solve fractional differential equations of imprecise functions. Sequentially, this chapter features description of fuzzy theory and fuzzy‐valued functions for the explanation of impreciseness, modeling of system of nonlinear fractional order differential equations with imprecise functions, deliberation of Grünwald‐Letnikov's fractional approach in conjunction with its truncation error for the proposed system, tabulated and pictorial investigations of some examples, and conclusive remarks of the undergone experiments and findings of the whole manuscript.
