In the present paper, a novel numerical model based on the finite volume method is established to predict a time-dependent, one-dimensional, advection-diffusion equation with variable coefficients in a semi-infinite domain. The third-and fifth-order schemes are employed to solve the above-mentioned equation. Totally, two dispersion problems are used to simulate various conditions as follows: (i) solute dispersion along steady flow through inhomogeneous domain and (ii) solute dispersion along temporally dependent unsteady flow through inhomogeneous domain. The inhomogeneity of the domain is provided by spatially dependent flow. The uniform node distribution is considered to divide the problem domain into a collection of smaller parts. Analytical solutions proposed in the literature are employed to demonstrate the accuracy and reliability of the suggested model. Meanwhile, the results of the aforementioned approaches are compared with the performance of the quadratic upstream interpolation for convective kinetics scheme. Lastly, the accuracy of the implemented schemes in developed model are discussed and evaluated.