iosr-Jm   Volume-16 ~ Issue-2 ~ mar – apr 2020

Abstract: During the history of the earth different kinds of processes by which living organisms are believed to have developed from earlier forms experience a change of state abruptly at certain moments of time. These processes act like short term perturbations and its duration is negligible in comparison with the total duration of the process. That is differential equations with impulse effects called impulsive differential equations. In recent years there have been intensive studies on the behavior of the solution of impulsive differential equations. It has wide applications in physics, population dynamics, ecology, biology, industrial robotics, biotechnology, optimal control theory etc. The Lyapunovs second method is used as a tool to obtain the stability of impulsive differential equations.

Key Word: impulsive differential equations, Lyapunov's second method, asymptotic stability

[1]. A. Dishliev and D.D. Bainov, "Dependence upon initial conditions and parameters of solutions of impulsive differential equations with variable structure", International Journal of Theoretical Physics, 29, (1990), 655-676.
[2]. A. I. Dvirnyi* and V. I. Slyn'ko, "Application of Lyapunov's Direct Method to the Study of the Stability of Solutions to Systems of Impulsive Differential Equations", vol. 96, No. 1, pp. 22–35., 2014
[3]. A. M. Samoilenko and N. A. Perestyuk, "Impulsive Differential Equations", vol.14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, Singapore, (1995).
[4]. A. S. Abdel-Rady, A. M. A. El-Sayed, S. Z. Rida, and I. Ameen, "On some impulsive differential equations",Math. Sci. Lett. 1 No. 2, 105-113 (2012)
[5]. Fulai Chen a ,Xianzhang Wen b, "Asymptotic stability for impulsive functional differential equation", J. Math. Anal. Appl. 336 (2007)

Abstract: A function T(s) whose numerator polynomial is the derivative of the denominator polynomial is designated as NDD function. It is shown that a class of non-NDD (NNDD) functions (a ratio of finite length
polynomials) can be converted into an NDD function by multiplying both the numerator and denominator by a suitable function. Expression for this function is derived. Transformation procedure is developed and illustrated
with several examples. Its application in obtaining partial fraction expansion is given and compared with the routine method. How an NDD function can be transformed into an NNDD function is mentioned. Finally a
property of NDD function is stated and proved.

Key Word: NDD functions, Partial fraction expansion, RPPFC Theorem, PFRPC theorem

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Abstract: Background: In this research, we studied the harvesting strategies for fish farming in Gashua. We considered three logistic growth models namely constant harvesting, periodic harvesting and proportional harvesting model. Even though fish farming has been locally commercialized, the use of mathematical models in determining harvesting strategies has not been widely applied in Bade. Logistic model is appropriate for population growth of fishes when overcrowding and competition for resource are taken into consideration. Our aim is to estimate the highest continuing yield from fish harvesting strategies implemented. The study predicted the optimum quantity for harvesting that can ensure the fish supply is continuous. We compare the results obtained between the three strategies and observed the best harvesting strategy for the selected fish farm is periodic (seasonal) harvesting. Our finding can assist fish farmers in Bade (Gashua), Yobe State, North East Nigeria, to increase fish supply to meet its demand and positively affect the economic growth of the area.

Key Word: Biomathematics, Fishery Management, Logistic Growth Models, Harvesting, Periodic

[1]. Adams, S. B. (2007). Direct and Indirect Effects of channel catfish (IctalurusPunctatus) on Native Crayfishes (Cambaridae) in Experimental Tanks. The American Midland Naturalist, 158(1): 58-95. Retrieved June 20, 2018 from https://naldc.nal.usda.gov/download/13465/PDF.
[2]. Asmah, R. (2008). Development potential and financial viability of fish farming in Ghana, University of Stirling, Stirling, UK (Doctoraldissertation),RetrievedJuly23,2017,fromhttp://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.845.2582&rep=rep1&t ype=pdf
[3]. Dubey, B., Chandra, P. and Sinha, P. (2003). A Model for fishery resource with reserve area, Nonlinear Analysis: Real World Application, Vol. 4: 625-637. Retrieved June 22, 2018, from http://home.iitk.ac.in/~peeyush/pdf/nla_bd_pc_ps.pdf
[4]. Berryman, A. A. (1992). The Origins and Evolution of Predator-Prey Theory. The Ecological Society of America, Vol. 73, No. 5: 1530-1535. Retrieved June 22, 2018, from http://homepages.wmich.edu/~malcolm/BIOS6150- Ecology/Discussion%20References/Berryman-Ecology1992.pdf.
[5]. Cooke, K. L. and Nusse, H. (1987). Analysis of the complicated dynamics of some harvesting models. Journal of Mathematical Biology, Vol. 25, pp 521-542.

Abstract: The attack of measles is endemic and dangerous to human existence; hence, this paper focused onmodeling of the transmission and dynamics with control of measles. Basic mathematical differential equation is used to model the rate of spread and possible control of the disease was discussed and computed. Data used was sourced secondarily. In this study, analysis revealed that the transmission of measles is significantly rising with about 5000 new individuals yearly. In addition, it was discovered that as time goes on (say t = 2025, I(t) = 70,582) the number of people that will be infected by measles if preventive or curative measure are not taken in metropolis will grow increasingly and will one day cover up the entire population.However, the study developed a model for the control and the result appear to significantly reduce the growth at which measles spread in the population given the parameters included in the model. It was recommended therefore that since the model shows that the spread of a disease largely depend on the number of people infected; therefore, the National Measles Control Programme should emphasize on the improvement in early detection of measles cases so that the disease transmission can be minimized.

Key Word: Transmission, Dynamics, Control, Model and Epidemics

[1]. Adegoke, R. (2005). Mathematical model for malaria transmission dynamics on human and mosquito population with nonlinear forces of infectious disease. International journal of pure and applied mathematics, Vol. 88, Issues 4.
[2]. Carabin, U., Kris, H. and Freeman, K. O. (2002). Mathematical model of malaria,Malaria Journal, Vol. 3 Issues 2.
[3]. Etuk, I., Obuyp, H. and Vitrea, G. (2015). Malaria model with stage- structured mosquitoes. Mathematical Biosciences and Engineering, Vol. 8, Issues 4.
[4]. Fatiregun, T., Yu, G. and Grat, U. (2014). The dynamics of multiple species and strains of malaria. Letters in Biomathematics, Vol. 16,Issues 9.
[5]. Michel, G. (2011). Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology,Vol.5, Issues 9.

Abstract: Singular two-point boundary value problems arise in different areas of applied sciences such as engineering, physics and thermal management. Numerous methods like DTM, HAM and MVIM have been applied to determine solutions of these problems that require additional computational work since all boundary conditions are not included in the canonical form. This research investigated solutions for the problems in a direct way both numerically and analyticallyusing the modifications of the decomposition method.
Symbolic programming was employed to handle linear and nonlinear STPBVPs both analytically and numerically. Examples were solved and analyzed using tables and figures for better elaborations where appreciable agreement between the approximate and exact solutions was observed. All the computations were performed using MATHEMATICA and MATLAB.

Key Words: MATHEMATICA, MATLAB, STPBVPs, ADM

[1]. Abdelrazec Ahmed, H. M. 2008. Adomian decomposition method: Convergence analysis and numerical approximations. McMaster University, Hamilton, Ontario.
[2]. Adomian, G. 1988. A Review of the Decomposition Method in Applied Mathematics.Academic Press, Inc, 135:501-544.
[3]. Adomian, G. 1994. Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers.
[4]. Almazmumy, M., Hendi, F.A, Bakodah, H.O and Alzumi, H. 2012.Recent modifications of Adomian decomposition method for initial value problem in ordinary differential equations. American Journal of Computational Mathematics, 2(1):228-234.
[5]. Arpaci, S. Vedat 1966. Conduction heat transfer. Addison Wesley Publishing Company,Massachusetts..

Abstract: Background:Many numerical techniques have been developed for solving first orderordinarydifferentialequationswithan initial condition (initial value problems : IVP)such as Euler'smethod, Huan'smethod, Runge-Kutta'smethod. Thesemethodsare used for engineer, scientistandappliedmathematicianwhoneeds an algorithm to solve IVPeasilyand the mostefficient. Materials and Methods: In this purposed paper a new methodimproves Euler's method and Heun's method by increasing Taylor's series order (usually 1st order) and using total derivative. Some examplesare presented to compare the numerical solutions of Euler's method and Heun's method solutionsby approximation graphs and absolute error graphs..........

Key Words: Ordinary differential equations; Initial value problems; Euler's method; Heun'smethod.

[1]. AmirulMd, A comparative study on numerical solutions of initial value problems (IVP) for ordinary differential equations (ODE) with Euler and RungeKutta methods. American Journal of Computational Mathematics. 2015;8:393-404
[2]. Gadisa G, Garoma H, Comparison of higher order Taylor's method and RungeKutta methods for solving first order ordinary differential equations. Journal of Computer and Mathematical Sciences. 2017;8(1):12-23
[3]. KamruzzamanMd, Nath MC, Acomparative study on numerical solution of initial value problem by using Euler's method, modified Euler's method and RungeKutta method. Journal of Computer and Mathematical Sciences. 2018;9(5):493-500
[4]. Ochoche A, Improving the modified Euler method. Leonardo Journal of Sciences. 2007;10:1-8
[5]. Ochoche A, Improving the improved modified Euler method for better performance on autonomous initial value problems. Leonardo Journal of Sciences. 2008;12:57-66

Abstract: This work presents the application of the Laplace transform to solve mixed problems associated with Equations in Partial Derivatives (EDP). Laplace transform is defined, the condition of existence of Laplace transform is established, also is presented a table of transformation of some functions, linear property and the derivative transform that is very fundamental for the theme that we propose to develop . Finally, a practical example of how the Laplace transform is applied to a mixed problem associated with EDP.

Key Words: Laplace transform. Mixed problems. Applications

[1]. Denni zil G. Zil.(1997). Differential equations with modeling applications. Intemational Thomson Publishers, Mexico.
[2]. Elsgoltz, L.( 1977). Differential Equations and Variational Calculus, second edition, Mir Moscó.
[3]. Hinoja M. A, Laplace transform with applications Cuba: Ed. People and education.

Abstract: It is true that several probability distributions exist for modeling lifetime data; however, some of these lifetime data do not follow any of the existing and well known standard probability distributions (models) or at least are inappropriately described by them. Therefore, in this paper, we derived maximum likelihood estimate of the parameters of both Rayleigh and Burr distributions and compared their performances with Weibull distribution in order to find an alternative to the Weibull computation.Random samples of different sizes with different shape parameter settings were drawn from the Weibull distribution and the parameters are estimated using both Rayleigh, Burrwith Weibull serving as reference. The estimate of the parameters of the considered distributions alongside with the..........

Key words: Weibull distribution, Rayleigh distribution, Burr distribution, wind data

[1]. Arun K. R., Himanshu P. and Kusum L.S. (2017). Bayesian estimation of the parameter of the p- Dimensional size biased Rayleigh distribution, Journal of statistics, 56(88-91).
[2]. Afaq Ahmad, S.P. Ahmad and Ahmad,A. (2015). Characterization end estimation of transmuted Rayleigh distribution. Journal of statistical applications and probability 2(315-321).
[3]. Bourguignon, M.,Rodrigo,B. Silva and Gauss, M. Cordeiro.(2014). Weibull-G Family of Probability Distributions. Journal of Data Science 12(2014)53-68.
[4]. Faton, M. and Ibrahim E. (2015). Weibull Rayleigh distribution theory and applications, appl. Math. Inf.5, 1-11.
[5]. Fatma, G.A., Sukru, A.andBirdal, S, (2018). Estimation of the Location and the Scale Parameters of Burr Type XII Distribution 1(15) 2618-6470. Doi10:1501/communal- 0000000000.

Abstract: In this paper we consider the idea of projective and inductive limits of uniform spaces and show that If for each I, f : X Y , J        is a mapping from a set X into a topological space Y , J    , there is a weakest topology on X, called the projective limit topology, denoted by PJ  under which every f is continuous.

Keywords: Topological space, projective limits, inductive limits , uniformity, filter

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'Introduction to topology and Modern Analysis, Tata McgrawHill comp. 2004.
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