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This is a time fractional PDE，where $0<\alpha<1$. I would like to know how to describe this equation and where does this equation come from.enter image description here

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  • $\begingroup$ Whats the ${}_0$ "pre-subscript" in front of $\partial_t^\alpha$? $\endgroup$
    – Dan Doe
    Jan 8, 2022 at 10:01
  • $\begingroup$ @DanDoe "a" is the lower integral limit in the definition of Caputo derivative, in there a=0 $\endgroup$
    – Shawn D
    Jan 8, 2022 at 10:51

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This equation is a variation of the heat equation. The fractional derivative is a nonlocal operator, which implies that the equation has "memory": whereas in a traditional ODE (or time dependent PDE), the rate of change only depends on what is happening at the current time, in fractional ODEs, the rate of change also depends on the past.

I don't know where this specific model appears, but fractional ODEs and PDEs appear in models in which memory plays a role; examples are the growth of populations where an increase in population now does not immediately increase its birth rate, but only some time later when individuals start to procreate themselves. This is probably better modeled using "delay differential equations", but for very specific growth term memories, fractional derivatives can also be used.

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