Existence and Uniqueness of a Weak Solution of a Time Fractional Reaction-Diffusion System of Fitzhugh-Nagumo Type

In the mathematical theory of nerve impulse propagation, the Fitzhugh-Nagumo Reaction-Diffusion System has attracted a great deal of attention. The Fitzhugh-Nagumo Reaction-Diffusion System provides a prototype for chemical and other nerve conduction and biological systems. In this paper, we define two types of weak solutions of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System, namely (1) -weak solutions and (2) -weak solutions, and demonstrate the existence and uniqueness of these weak solutions. First, we have obtained a generalization of [1, Lemma 1] in Lemma 2.1 and using Lemma 2.1 and Galerkin’s approximation sequence, we have found the existence of (1)-weak solutions and (2)-weak solutions. We also obtained a generalization of the result of [10, Lemma 6] to Hilbert spaces in Lemma 2.2, and using this result we proved the uniqueness of the (2)-weak solution. Lemma 2.1 and Lemma 2.2 of this paper are results that can be effectively used to show the existence and uniqueness of weak solutions of time fractional partial differential equations. And the existence and uniqueness results of the weak solution of the time fractional Fitzhugh-Nagumo Reaction-Diffusion System can be used in the numerical solutions of this reaction-diffusion system. Also, we can be used in the optimal control problems described in this system.