# Generalized FitzHugh-Nagumo equations with Caputo gH-differentiability: A novel fuzzy fractional approach to digital memristor networks

## TL;DR

This paper introduces a new mathematical approach using fuzzy fractional calculus to solve complex equations relevant to biological systems and digital circuits.

## Contribution

The novel contribution is the analytical solution of fuzzy fractional generalized FitzHugh-Nagumo equations using Caputo gH-differentiability and fuzzy Laplace transform.

## Key findings

- Closed-form solutions for FFGFH-NDEs are derived using fuzzy Laplace transform and multivariate Mittag-Leffler functions.
- The proposed method is applied to digital memristor networks with graphical analysis of uncertain behavior.
- Solutions are presented for both homogeneous and nonhomogeneous cases of the model.

## Abstract

The fuzzy fractional generalized FitzHugh-Nagumo differential equations (FFGFH-NDEs) is a well-known and generalized model that plays a significant role in biological systems, including complex synchronization in brain networks, cardiac dynamics, propagation of signals through nerve impulses, and digital circuit theory. The analytical study of the FFGFH-NDEs is more complex and difficult to deal with. An effective and efficient technique is required to solve FFGFH-NDEs analytically. This article introduces and investigates the analytical fuzzy solutions of FFGFH-NDEs using fuzzy fractional Caputo generalized Hukuhara (FFCgH)-differentiability. The closed-form solutions of FFGFH-NDEs for various cases and types of FFCgH-differentiability are extracted for the homogeneous and nonhomogeneous case of the concerned model. The potential solutions are determined using fuzzy Laplace transform (FLT) and are presented in terms of multivariate Mittag-Leffler functions (MLFs). To highlight the innovation of this work, the digital memristor networks problem is designed and solved as an application of the proposed study including the graphical analysis to understand the uncertain behavior of the proposed model.