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आयतन 9, मुद्दा 5 (2020)


Modelling and Optimal Control of Toxicants on Fish Population with Harvesting

Zachariah Noboth, Estomih S. Massawe* Daniel O. Makinde, Lathika .P

Toxic in water bodies is a worldwide problem. It kills fish and other aquatic animals in water. Human beings are affected by this indirectly through eating affected fish.

In this paper, a model for controlling toxicants in water is formulated and analysed. Boundedness, positivity and analysis of the model are examined where four steady

states are determined by using Eigen-value analysis and found to be locally stable under some conditions. The optimal control strategies are established with the help

of Pontryagin’s maximum principle. The simulations for the model with control show that when control is applied the results reveals that the amount of toxic is reduced

and hence there is an increase in fish population for both prey and predator populations. It is recommended that the government has to introduce laws and policies

which ensure that the industries treat waste water before they are discharged into water bodies and to develop a system for waste recycling


The Fundamental Solution of the One Dimensional Elliptic Operator and its Application to Solving the Advection Diffusion Equation

Ronald Mwesigwa, GodwinKakuba and David Angwenyi

The advection–diffusion equation is first formulated as a boundary integral equation, suggesting the need for an appropriate fundamental solution to the elliptic operator.

Once the fundamental solution is found, then a solution to the original equation can be obtained through convolution of the fundamental solution and the desired right

hand side. In this work, the fundamental solution has been derived and tested on examples that have a known exact solution. The model problem here used is the

advection–diffusion equation, and two examples have been given, where in each case the parameters are different. The general approach is that the time derivative

has been approximated using a finite difference scheme, which in this case is a first order in �??t, though other schemes may be used. This may be considered as the

time-discretization approach of the boundary element method. Again, where there is need for finding the domain integral, a numerical integration scheme has been

applied. The discussion involves the change in the errors with an increase in �??x. Again, for small solution values, considering relative errors at selected points along

the domain, and how they vary with different choices of �??x and �??t. The results indicate that at a given value of x, errors increase with increasing �??x, and again as R�??

increases, the magnitudes of the errors keep increasing. The stability was studied in terms of how errors from one time step do not lead to high growth of the errors in

subsequent steps.


Solution of Ordinary Differential Equation with Variable Coefficient Using Shehu Transforms

Mulugeta Andualem

Shehu transform is a new integral transform type used to solve differential equations as other integral transforms. In this study, we will discuss the Shehu transform

method to solve ordinary differential equation of variable coefficient. In order to solve, first we discussed the relationship between this new integral transform with

Laplace transform


Revised Methods for Solving Nonlinear Second Order Differential Equations

Lemi Moges Mengesha

In this paper, it has been tried to revise the solvability of nonlinear second order Differential equations and introduce revised methods for finding the solution of nonlinear

second order Differential equations. The revised methods for solving nonlinear second order Differential equations are obtained by combining the basic ideas of

nonlinear second order Differential equations with the methods of solving first and second order linear constant coefficient ordinary differential equation. In addition to

this we use the property of super posability and Taylor series. The result yielded that the revised methods for second order Differential equation can be used for solving

nonlinear second order differential equations as supplemental method.


Mathematical Model Of Ingested Glucose In Glucose-Insulin Regulation

Suparna Roy Chowdhury

Here, we develop a mathematical model for glucose-insulin regulatory system. The model includes a new parameter which is the amount of ingested glucose. Ingested glucose is an external glucose source coming from digested food. We assume that the external glucose or ingested glucose decays exponentially with time. We establish a system of three linear ordinary differential equations with this new parameter, derive stability analysis and the solution of this model.

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