Abstract
One of the most beneficial techniques for studying partial differential equations of the parabolic and elliptic types is the use of the maximum and minimum principles. They enable the acquisition of specific solution attributes without the need for knowledge of the solutions’ explicit representations. Despite the fact that the maximum principle for fractional differential equations has been studied since the 1970s, a particular interest in this field of study has just lately arisen. In the present study, a maximum principle for the one-dimensional time fractional diffusion equation with memory is formulated and established. The proof of the maximal principle is based on a maximum principle for the Caputo fractional derivative. The initial boundary value problem for the time-fractional diffusion equation with memory has at most one classical solution, and the maximum principle is then used to show that this solution is continuous depends on the initial and boundary conditions.