What is the best way to use loop statements on a quantum computer?

I am interested in solving a time dependent linear partial differential equation of the form $Ax=b$ which, in classical computing, would amount to looping over solutions of $Ax=b$ where $b$ is updated each step.

My question is, is there a quantum analog to the looping procedure that I have described? If not, does that imply that to solve this one would need to write a loop on the classical system which calls the quantum solver each step? In case it is relevant, I am using the IBMQ system.

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