Mathematical Problems in Engineering
Volume 5 (1999), Issue 3, Pages 193-221

A river water quality model for time varying BOD discharge concentration

Seth F. Oppenheimer,1 Donald Dean Adrian,2 and Akram Alshawabkeh3

1Department of Mathematics and Statistics, Mississippi State University, P.O. Drawer MA, 39759, MS, USA
2Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge 70803, LA, USA
3Department of Civil and Environmental Engineering, Northeastern University, Boston 02115, MA, USA

Received 8 February 1999; Revised 3 May 1999

Copyright © 1999 Seth F. Oppenheimer et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a model for biochemical oxygen demand (BOD) in a semi-infinite river where the BOD is prescribed by a time varying function at the left endpoint. That is, we study the problem with a time varying boundary loading. We obtain the well-posedness for the model when the boundary loading is smooth in time. We also obtain various qualitative results such as ordering, positivity, and boundedness. Of greatest interest, we show that a periodic loading function admits a unique asymptotically attracting periodic solution. For non-smooth loading functions, we obtain weak solutions. Finally, for certain special cases, we show how to obtain explicit solutions in the form of infinite series.