iosr-Jm   Volume-14 ~ Issue-5 ~ Sep-Oct 2018

Abstract: There are many methods for solving a nonlinear algebraic equation. Here a recurrence iteration for-mula for two-roots finding is derived based on the quadratic expansion of Taylor series. The general formula of quadratic equation is obtained using the derived formula. A family of iteration functions is derived from the derived formula. This family includes the Newton,Patrik, Halley, and Schroder'smethods.All methods of the family are cubically convergent for a simple root(except Newton's which is quadratically convergent).A simple general formula is derived and proved to be one of the familyof Halley-like method..

Key Words: Simple roots, Nonlinear equations, Halley method, Taylor expansion

[1]. E. Halley, A new, exact and easy method of finding the roots of equations generally and that without any previous reduction, Philos. Trans. R. Soc.London 18 (1694) 136–148.
[2]. Hansen, Eldon, and Merrell Patrick. "A family of root finding methods."NumerischeMathematik 27.3 (1976): 257-269.
[3]. Neta, Beny. "Several new methods for solving equations." International Journal of Computer Mathematics 23.3-4 (1988): 265-282.
[4]. Scavo, T. R., and J. B. Thoo. "On the geometry of Halley's method." The American mathematical monthly 102.5 (1995): 417-426.
[5]. Thoo, J. B. "Some derivatives of Newton's method." Problems, Resources, and Issues in Mathematics Undergraduate Studies 12.2 (2002): 165-180..

Abstract: In this study, certain independence and domination properties of generalized Fibonacci graphs are considered. Domination, upper domination, total domination, upper total domination, independent domination and connected domination numbers of generalized Fibonacci graphs are calculated. Several illustrative examples are given.

Keywords: Heston partial differential equation, Heston stochastic volatility model, Elzaki transform method, reckless interest rate.

[1]. M. Korenblit and V. E. Levit, Mincuts in generalized Fibonacci graphs of degree 3, Journal of Computational Methods in Sciences and Engineering, 11(5,6), 2011, 271–280.
[2]. M. Korenblit and V. E. Levit, The 𝑠𝑡-connectedness problem for a Fibonacci graph, WSEAS Transactions on Mathematics 1(2), 2002, 89–93.
[3]. H. Akyar and E. Akyar, Certain properties of generalized Fibonacci graphs, Suleyman Demirel University Journal of Natural and Applied Sciences, 22(2), 2018, 661–666.
[4]. H. Akyar, On the Cartesian product of generalized Fibonacci graphs, International Journal of Mathematics and its Applications, 6(2-A), 2018, 63–70.
[5]. S. El-Basil, Theory and computational applications of Fibonacci graphs, Journal of Mathematical Chemistry, 2(1), 1988, 1–29..

Abstract: In this paper, the mathematical and stability analyses of the SIR model of malaria with the inclusion of infected immigrants are analyzed. The model consists of SIR compartments for the human population and SI compartments for the mosquito population. Susceptible humans become infected if they are bitten by infected mosquitoes and then they move from susceptible class to the infected class. In the similar fashion humans from infected class will go to recovered class after getting recovered from the disease. A susceptible mosquito becomes infected after biting an infected person and remains infected till death. The reproduction number 𝑅0 of the model is calculated using the next generation matrix method. Local asymptotical stabilities of the steady states are discussed using the.........

Keywords: Infected immigrants, Reproduction number, Steady states, Local stability, Lyapunov function.

[1]. Gbenga J. Abiodun, P. Witbooi and Kazeem O. Okosun. Modelling the impact of climatic variables on malaria transmission. Hacettepe Journal of Mathematics and Statistics. Volume 47 (2) (2018), 219 – 235.
[2]. Bakary Traor´e et al. A Mathematical Model of Malaria Transmission with Structured Vector Population and Seasonality. Journal of Applied Mathematics. Volume 2017, Article ID 6754097,https://doi.org/10.1155/2017/6754097, 1-15.
[3]. R. Ross. The prevention of malaria, London, John Murray, 1911.
[4]. A. George Maria Selvam, A. Jenifer Priya: Analysis of a Discrete SEIR Epidemic Model. International Journal of Emerging Technologies in Computational and Applied Science, 12(1-5), (2015) Pp. 73-76.
[5]. Abid Ali Lashari, Shaban Aly, Khalid Hattaf, Gul Zaman, Hyo Jung, and Xue-Zhi Li. Presentation of Malaria Epidemics Using Multiple Optimal Controls. Journal of Applied Mathematics. Volume 2012, Article ID 946504, 1-17 .doi:10.1155/2012/946504...

Abstract: This paper studies about the dynamics of three population species interactions in biological ecology. Here, an interaction among two mutualistic preys and one predator populations has been considered. The population interaction areas are classified into two: free area and refuge area. In free area only the second prey and predator population species exist and interact while in a refuge area only the first prey population species exists. In the refuge area the predator population species cannot enter and attack the prey species. However, in the refuge area the two preys can interact and help each other. Additionally, in this model proportional harvesting function and functional responses are considered among these population interactions. Based on the unique and positive equilibrium points, local and global stability can be determined analytically and numerically. Simulation results supporting the analytical part are considered.

Keywords: Mutualism, Functional response, Local stability, Global stability, Harvesting function, Boundedness and Positivity.

[1]. Lotka A. J. Elements of Physical Biology, Williams and Wilkins, Baltimore, USA, 1925.
[2]. Volterra V. Fluctuations in the abundance of species considered mathematically, Nature, 1926, 118, 157-158.
[3]. Hastings A. Population Biology, Springer, Berlin, Germany, 1997.
[4]. G. Butler, H. Freedman, P. Waltman. Uniformly persistent systems in: Proceedings of the American Mathematical Society, 1986, 96: 425–430.
[5]. J. Collings. Bifurcation and stability analysis of a temperature dependent mite predator-prey interaction model incorporating a prey refuge. Bulletin of Mathematical Biology, 1995, 57: 63–76...

Abstract: This paper collects data on 9 zonings in Anyang, China, and comprehensively considers the impact of natural factors, living factors and industrial factors on each zoning. The input factors of the pollution levels of the 9 districts of Anyang City from 2008 to 2017 are input into the network. The zoning environmental quality index is the network output, and the BP-MIV (Back-Propagation neural network-Mean impact value) model is established to explore the non-equilibrium effect of various factors on the innovation level, and the pollution of the city in the next three years. Forecast at the level. The results show that the order of influencing factors of pollution level is industrial level, living level and natural level. This result is consistent with the actual pollution situation of each division

Keywords: Pollution level, BP-MIV, Influence factors, Environmental quality

[1]. JohnWarren. Statistical Methods for Environmental Pollution Monitoring[J]. Technometrics, 1987, 30(3):348-348.
[2]. Yang Y H, Bao J L, Wen J, et al. Analysis on Influencing Factors of Atmospheric Environmental Quality in Tianjin[J]. Advanced Materials Research, 2014, 864-867:1582-1585.
[3]. Kang Y L, Zhu G M. Development Trend of China's Automobile Industry and the Opportunities and Challenges of Steels for Automobiles[J]. Iron & Steel, 2014.
[4]. Sowlat M H, Gharibi H, Yunesian M, et al. A novel, fuzzy-based air quality index (FAQI) for air quality assessment[J]. Atmospheric Environment, 2011, 45(12):2050-2059.
[5]. Doel R V D, Knaap A G A C, Sangster B. Poison control centre and environmental pollution health care and risk assessment[J]. Clinical Toxicology, 2015, 26(1-2):89-102...

Abstract: The Laplace Transformation is one of the most widely and frequently used transformation in sciences and Engineering. Its application in solving initial value Problems (IVP) of ordinary differential equations (ODE's) is well known to scholars. In this paper we reviewed the traditional algebraic method (i.e. the Laplace Transformation Method) of solving system of linear Ordinary Differential Equations with constant coefficients and now show how the newly established Residue Inversion Formula can best be applied directly in obtaining the Inverse Laplace Transform when solving system of linear ode's with constant coefficients hence, simplify the traditional method. This new Residue approached eliminates computational stress and resultant time wastage by circumventing the rigor of resolving..............

Keywords: Residue Inversion Formula, Initial Value Problems of Simultaneous Ordinary Differential equations, Table of Laplace Transform, Partial fractions

[1]. Churchill R.V, Brown J.W and Verhey R.F (1974). Complex Variables and Applications, 3rd Edition, Mac Graw-hill book.co
[2]. Sambo Dachollom, Abashi, Bulus Ayuba and Dongs, Dung Pam. (2015). Application of Cauchy Residue Theorem in the solution
of Some Integrals, Proceeding of International Conference on Humanities, Science and Sustainable Development, 8 (10) pp 93-
103..
[3]. Dass, H.K (2008). Advanced Engineering Mathematics, 21st Revised Edition, pp 927-931. S.Chand and Company Ltd. New Delhi
[4]. Oko Nlia and Sambo Dachollom. (2015). Application of Residue Inversion Formula for Laplace Transformation of Initial Value
Problems of Linear ode's. IOSR Journal of Mathematics (IOSR-JM). Vol.11(6), pp. 56- 59
[5]. Mark J. Ablowitz, Athanassios S. Fokas: Complex Variable; Introduction and Application. Cambridge University Press. ISBN 0-
521-48523-1.

Abstract: The Infinite Series is ubiquitous in mathematics. It pops up everywhere- Euler's number, trigonometric ratios, logarithms etc. One of the most important infinite series is the Geometric Series. Traditionally its calculation requires concepts of Algebra, Limits, and Convergence. However, today I will show an elegant method based on the simple concepts of division and geometry to help you visualise it better

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Abstract: This research discussed the modification of second derivative linear multistep method (LMM) using Enright's approach,which focused in solving second order ordinary differential equations (ODEs).The newly constructed method satisfied the basic requirements for the analysis of Linear Multistep methods (LMM). The methods displayed better accuracy when implemented with numerical examples than the existing method with which we compared our results.

Keywords: Modification, Stiffly differential equation, LMM, second derivative.

[1]. Curtiss, C. F., Hirschfelder,J. O. Integration of Stiff Equations, Proc Natl. Acad. Sci. USA, (1952), pp. 235-243.
[2]. Hairer, E., Wanner, G. Solving ordinary differential equations. II. Stiff and differential algebraic problems, 2nd edn. Springer.
(1996).
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(1975), 10-48.
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27.

Abstract: Human Immunodeficiency Virus/Acquired Immunodeficiency Syndrome and pneumonia co-infection is a combination of two infections. Here, an individual contracted with both Human Immunodeficiency Virus and Pneumococcal Carinii Pneumonia at the same time is involved. This paper focuses on the HIV/AIDS - pneumonia co-infection model which is shown to be positively bounded. In this model, we consider treatment for co-infection at both initial and final stages of development. The endemic states are considered to exist when the basic reproduction number for each disease is greater than one. The basic reproduction numbers are also used to see the impact of treating one disease on the co-infection. Numerical simulations indicate the effect of varying the treatment parameters on single disease and the co-infection dynamics. As we increase treatment rates, the infections decrease.

Keywords: CD4+ T-cells, HIV, AIDS, Pneumonia, Initial stage, Final stage, Stability, Co-infection, Sensitivity, Reproduction number.

[1]. T. Ahmad and T. Ahmad. Human immunodeficiency virus (hiv) is a serious health problem for nepal. 2016.
[2]. S. Alizon and C. Magnus. Modelling the course of an hiv infection: insights from ecology and evolution. Viruses, 4(10):1984–2013,
2012.
[3]. R. Aris. Mathematical modelling techniques. Courier Corporation, 2012.
[4]. Bhunu, W. Garira, and G. Magombedze. Mathematical analysis of a two strain hiv/aids model with antiretroviral treatment. Acta
biotheoretica, 57(3):361–381, 2009.

Abstract: In this study, we establish new sequences that are obtained by altering the Pell and Pell Lucas sequences. Unlike other altered sequences in the literature, these new altered sequences depend on two integer parameters. Further, the greatest common divisors properties of these altered sequences are investigated.

Keywords : Altered Pell and Pell Lucas numbers; Sequences (mod n)

[1] K. W. Chen, Greatest common divisors in shifted Fibonacci sequences. J. Integer. Seq., Vol:14, 2011, 11. 4-7.
[2] T. Koshy, Fibonacci and Lucas Numbers with Applications, (Wiley-Interscience, 2001).
[3] T. Koshy, Pell and Pell-Lucas Numbers with Applications. (Springer, Berlin, 2014).
[4] U. Dudley and B. Tucker, Greatest common divisors in altered Fibonacci sequences, Fibonacci Quart., 9, 1971, 89-91.
[5] W.L. McDaniel, The G.C.D. in Lucas sequences and Lehmer number sequences, Fibonacci Quart., 29, 1991, 24-29.