Journal of Applied & Computational Mathematics

ISSN: 2168-9679

Open Access

Current Issue

Volume 9, Issue 5 (2020)

    Research Pages: 1 - 10

    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.

    Research Pages: 1 - 8

    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

    Research Pages: 1 - 12

    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.

    Research Pages: 1 - 4

    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.

    Research Pages: 1 - 4

    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

    Volume 9, Issue 4 (2020)

      Research Pages: 1 - 6

      Assessing the Impact of Age –Vaccination Structure Models on the Dynamics of Tuberculosis Transmission

      Solomon T. Kwao*, Francis T. Batsa, Samuel Naandam and Gaston M. Kuzamunu

      Vaccination has been the only preventive mechanism of tuberculosis (TB) yet due to inconsistencies in the efficacy of the mostly used vaccine, Baccille Calmette-Guerin (BCG), re-vaccination has been deemed to be ineffective. In this work, we sought to assess the impact of age-vaccination and re-vaccination on the transmission dynamics of TB. We developed an age-vaccination re- vaccination model to explore the disease transmission dynamics and the impact of re-vaccination on the disease transmission. By applying the vaccine within ten year intervals, we noticed that, there is no significant difference when the vaccine is administered once or many times for people less than 45 years of age. However, re-vaccination can prove to be effective when it is applied either before or immediately after the waning of the first vaccine.

      Research Pages: 1 - 9

      A New One-Dimensional Finite Volume Method for Hyperbolic Conservation Laws

      José C. Pedro*, Mapundi K. Banda and Precious Sibanda

      In this paper, a new one-dimensional Finite Volume Method for Hyperbolic Conservation Laws is presented. The method consists in an improved numerical inter-cell flux function at the element interface. To back theoretically the method, necessary components for convergence are presented. Therefore, it is proved that the method is consistent with the P.D.E and that it is monotone with respect its variables. Moreover, to validate the approach and show its efficiency, we compute several one-dimensional test problems with discontinuous solutions and we make comparisons with traditional methods. The results show an improvement on the non-oscillatory shock-capturing properties based on the new approach.

      Research Pages: 1 - 4

      Numerical Approach for Determining Impact of Steric Effects in Biological Ion Channel

      Abidha Monica Gwecho*, Wang Shu and Onyango Thomas Mboya

      Flow through biological ion channel is understandably complex to support numerous and vital processes that promote life. To account for the biological evolution, mathematical

      modelling that incorporates electrostatic interaction of ions and effects due to size exclusion has been studied, conceivably with element of difficulty and inaccuracy. In this

      paper the Nernst-Planck(NP) equation for ion fluxes that uses Lennard Jonnes(LJ) potential to incorporate finite size effects in terms of hard sphere repulsion is examined.

      To minimize emerging numerical intricacy, the LJ potential is modified by a band limit function with a cut-off length to eliminate troublesome high frequencies in the integral

      function. This process is achieved through Fourier transform to simplify and hence render the mPNP equation solvable with precision. The resultant modified NP and Poisson

      equation representing electrostatic potential are then coupled to form system of equation which describes a realistic transport phenomena in ion channel. Consequently,

      to discretize the 2D steady system of equations, mixed finite element approach based on Taylor hood eight node square referenced elements is adopted. In the method,

      Galerkin weighted residual approach help obtain sparse matrix and finally Picard Method applied to the nonlinear terms in the algebraic equations to linearize them and

      improve rate of convergence. Iterative solution for the system of equations then obtained and concentration profiles of ion species under varied steric effects for mPNP are

      computed and analysed

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