Evaluating the Impact of the linear Distance on Biochemical Oxygen demand and the Concentration of dissolved Oxygen of a River for fixed initial Conditions using a Computational Approach

This paper has examined the impact of the linear distance on the concentration of biochemical oxygen demand (BOD) and the concentration of dissolved oxygen (DO) of a river for fixed initial conditions using the method of a numerical scheme called ODE45. Our results are presented and discussed quantitatively.


INTRODUCTION
The primary aim of this study is to evaluate the effect of varying the linear distance on the concentration biochemical oxygen demand (L) and the concentration of dissolved oxygen(C) of a river for fixed initial conditions. Since mathematical formulations that describe the growth of L and C depend on several factors such as the linear distance, average flow velocity, deoxygenation coefficient (k1) and reoxygenation or reaeration coefficient (k2) [4] In this present analysis, we are interested to measure numerically the effect of varying the linear distance on L and C with fixed initial conditions. Several other researchers have studied other aspect of modelling the growth of L and C using different independent variables. See Tadeusz et.al [5], Borsuk and Stow [6],Tyagiet.al [1],Kaushik et. al [2],Runnel [3],Adrian and Sanders [7][8].

II.
MATHEMATICAL FORMULATIONS Under some simplifying assumptions and following Bank [4], we have considered the steady flow of a river with an average velocity, 0 u defined by the following system of first order ordinary differential equations: Here, s represents the linear distance along the river, C represents the concentration of dissolved oxygen (mg/l), L represents the concentration of the biochemical oxygen demand, D represents the difference between the equilibrium concentration of oxygen(mg/l), Cs and the concentration of the dissolved oxygen C, k1 represents deoxygenation coefficient and k2 represents the reoxygenation or reaeration coefficient. For the purpose of this analysis, we have considered the following parameter values; k1 = 0.25/day,k2 = 0.5/day,u0 = 25km/day and s=20km.

III.
METHOD OF ANALYSIS We have adopted a computational approach in our investigation using MATLAB function, ODE45 being more computationally efficient than ODE23, ODE23TB and ODE15s.

IV.
RESULTS AND DISCUSSION The full results of implementing the method above are presented and discussed here as follows In Table 1, apart from the initial condition data having the effect of zero, as the linear distance ranges from 0.1 to 2.0, the biochemical oxygen demand(BOD) data increases monotonically from 16.801 to 16.833 approximately whereas the new biochemical oxygen demand data when the linear distance is 2km ranges from 16.0001 to 16.8033. On the basis of these calculations, a small value of the linear distance has dominantly predicted a relatively small increase in the original BOD data. For the same value of the linear distance, the concentration of dissolved oxygen (DO) dominantly tend to increase from the percentage effect of 1.797 to 42.673 approximately. Therefore, when the linear distance (s) is decreased from its original value of 20km to a smaller value of 2km, the BOD suffers some sort of depletion which mimics biodiversity loss whereas the concentration of DO also suffers some sort of depletion which depicts biodiversity gain. As presented in Table 2, when the linear distance is 4km, the DO concentration dominantly suffers a depletion value of 4.125 percent whereas the DOB suffers a depletion level of 14.786 percent. As presented in Table3, both BOD and DO data corresponding to the scenario when the linear distance is 19 km are both vulnerable to the ecological risk of depletion. In the scenario when the value of the linear distance is 24km, the BOD data and DO data both predict dis-similar outcomes of biodiversity gain at the magnitudes of 4.08 and 1.69 approximately. In Table 5, a similar observation has been made when the linear distance is 28km Our present cutting-edge contribution to knowledge compliments the earlier formulations and analysis provided by Bank [4] and also has moved the frontier of knowledge forward to evaluate the effect of varying the linear distance on the BOD and DO data which were not previously considered by Bank [4].

V. CONCLUSION
We have successfully utilised a computationally efficient numerical scheme known as the Runge-Kutta MATLAB ODE 45 function to predict a biodiversity loss and a biodiversity gain on the BOD and DO data due to a variation of the linear distance along a river. In our future investigation we will be interested to study the impact of varying the k1 and k2 on the BOD and DO dependent variables. Our present predictions have shown the response of BOD data and DO data when the linear distance increases. An alternative mathematical model of interaction between BOD and DO can be constructed to tackle this challenging problem which we did not consider in this pioneering paper.