Forward Modelling of River Pollutant Transport: Analytical and Numerical Solutions of the Advection–Diffusion Equation with Temporally Varying Coefficients

Accurate prediction of pollutant transport is the foundation on which any inverse source-identification scheme is built, yet most river-pollution studies assume constant or, at most, spatially varying flow parameters and so cannot represent the unsteady conditions of natural rivers. This paper develops, derives, and validates a forward transport model based on the advection–diffusion equation (ADE) with temporally varying velocity and dispersion coefficients, in both one and two spatial dimensions. The ADE is derived from first principles through conservation of mass and the divergence theorem. For the one-dimensional case, a closed-form analytical solution is obtained by a sequence of variable transformations that reduce the variable-coefficient ADE to a constant-coefficient diffusion equation solved by Laplace transforms; a numerical solution is constructed using the Forward Time Central Space Centered Scheme (FTCSCS), whose von Neumann stability condition is derived explicitly. For the two-dimensional case, an unconditionally stable, second-order Alternating Direction Implicit (ADI) scheme is developed, with the governing equation split into an x-sweep and a y-sweep, each reduced to a tridiagonal system solved by the Thomas algorithm. The analytical and numerical 1D solutions agree closely, with RMSE decreasing from 0.028 at Pe ≪ 1 to 0.004 at Pe ≫ 1. Simulations reveal that pollutant concentration is highest near the source and decays downstream; that concentration grows with time at any fixed point; and that the longitudinal distribution becomes increasingly advection-skewed as the Peclet number rises. The effect of four temporal coefficient regimes is quantified, showing that concentration is lowest when both dispersion and velocity increase with time and highest when dispersion increases in a decelerating flow. The framework provides a physically faithful, computationally efficient forward solver suitable for generating the concentration fields required by inverse source-identification methods. Keywords: Advection–diffusion equation; Temporally varying coefficients; Alternating direction implicit method; Laplace transform; von Neumann stability; Pollutant transport. Title: Forward Modelling of River Pollutant Transport: Analytical and Numerical Solutions of the Advection–Diffusion Equation with Temporally Varying Coefficients Author: Constance A. Ojwando, Mark Kimathi, John Awino International Journal of Recent Research in Mathematics Computer Science and Information Technology ISSN 2350-1022 Vol. 13, Issue 1, April 2026 - September 2026 Page No: 33-42 Paper Publications Website: www.paperpublications.org Published Date: 11-July-2026 DOI: https://doi.org/10.5281/zenodo.21319969 Paper Download Link (Source) https://www.paperpublications.org/upload/book/Forward%20Modelling%20of%20River%20Pollutant%20Transport-11072026-2.pdf