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Empirical investigations of solute fate and carrying in streams and rivers often contain inventive liberate of solutes at an upstream perimeter for a finite interval of time. An analysis of various worth references on surface-water-grade mathematical formulation reveals that the logical solution to the continual-parameter advection- dispersion problem for this type of boundary state has been generally missed. In this work, we study the q-fractional homotopy analysis transform method (q-FHATM) to find the analytical and approximate solutions of space-time arbitrary order advection-dispersion equations with nonlocal effects. The diagrammatical representation is done by using Maple package, which enhance the discretion and stability of family of q-FHATM series solutions of fractional advection-dispersion equations. The efficiency of the applied technique is demonstrated by using three numerical examples of space- and …