analysis of time-dependent fractional integro-differential equations under the maximum principle

This paper establishes continuous and discrete maximum principles for a time-dependent fractional integro-differential equation involving the caputo derivative. we show that when the right-hand side preserves a constant sign, the solution cannot attain an interior local extremum of that sign. a numerical discretization combining the l1 scheme and trapezoidal rule is also introduced, and a discrete maximum principle is proved, ensuring qualitative consistency between the continuous and numerical models.