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In this paper, a class of fractional Sturm–Liouville advection–dispersion equations with instantaneous and noninstantaneous impulses is considered, in particular, the nonlinearities discussed here include Caputo fractional derivatives. Since the nonlinear terms contain fractional derivatives, this problem does not directly have variational structure, we need to combine critical point theory and an iterative method to deal with such problems. Finally, the existence of at least one nontrivial solution is proved by the mountain pass theorem and the iterative method. At the same time, an example is given to illustrate the main result. [ABSTRACT FROM AUTHOR]