In this study, we propose the conditions on which a class of boundary value problems, presented by fractional $q$-differential equations, is well-posed. First, under the suitable conditions, we will prove the existence and uniqueness of solution. Then, the stability of solution will be discussed under the perturbations of boundary condition, function exists in the problem and the fractional order derivative. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.
