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In this paper, we consider a new setting of fuzzy delay differential equations by employing the concepts of granular differentiability. Firstly, we establish some basic properties of complete granular metric space. Then we study the local and global existence and uniqueness of integral fuzzy solution. For the first task, we use successive approximation technique combined with Gronwall's lemma. The second task will be solved by Banach contraction principle in weighted metric space defined by granular distance. Additionally, we demonstrate the theoretical results by solving some real-world models in both analytical method and numerical method. The first model is often referred to the exponential growth, namely time-delay growth Malthusian model. The second model is one type of dynamical systems, which is used for studies involving large amounts of tissues, called Ehrlich ascites tumor model with delay. Furthermore, we introduce the Mackey-Glass model which is considered as a useful tool for times series prediction in the neutral network and fuzzy logic fields. Finally, we represent the surface of fuzzy solution for visible identification mark-on. Keywords Granular differentiability • Fuzzy delay differential equations • Malthusian growth model • Ehrlich ascites tumor model • Mackey-Glass time series model Mathematics Subject Classification 34A07 • 35R13 Communicated by Anibal Tavares de Azevedo.

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