Maximum and Minimum Principles for the Generalized Fractional Diffusion Problem with a Scale Function-Dependent Derivative

In the paper, we prove the necessary condition for the extremum existence in terms of the generalized function-dependent fractional derivatives. By using these results we extend the maximum and minimum principles, known from the theory of differential equations and from diffusion problems with the Caputo derivative of constant or distributed order. We study the fractional diffusion problem, where time evolution is determined by the scale function-dependent Caputo derivative and show that the maximum or respectively minimum principle is valid, provided the source function is a non-positive or a non-negative one in the domain. As an application, we demonstrate how the sign of the classical solution is controlled by the initial and boundary conditions.