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Abstract: In this paper, an analysis is presented to find the numerical solutions of the second order linear and nonlinear differential equations with Robin, Neumann, Cauchy and Dirichlet boundary conditions. We use the Legendre piecewise polynomials to the approximate solutions of second order boundary value problems. Here the Legendre polynomials over the interval [0,1] are chosen as trial functions to satisfy the corresponding homogeneous form of the Dirichlet boundary conditions in the Galerkin weighted residual method. In addition to that the given differential equation over arbitrary finite domain [a,b] and the boundary conditions are converted into its equivalent form over the interval [0,1].

Numerical examples are considered to verify the effectiveness of the derivations. The numerical solutions in this study are compared with the exact solutions and also with the solutions of the existing methods. A reliable good accuracy is obtained in all cases. Keywords: Galerkin Method, Linear and Nonlinear VBP, Legendre polynomials. Bhatti and P.

Bracken, 'Solutions of Differential Equations in a Bernstein Polynomial Basis,' Journal of Computational and Applied Mathematics, Vol. 205, No.1, 2007, pp.272-280. Lashien and W. Zahra, 'Polynomial and Nonpolynomial Spline Approaches to the Numerical Solution of Second Order Boundary Value Problem,' Applied Mathematics and Computation, Vol.184, No. 2, 2007, pp.476- 484.doi:10.1016/j.amc.2006.06.053. Usmani and M.

Sakai, 'A Connection between Quatric Spline and Numerov Solution of a Boundary value Problem,' International Journal of Computer Mathematics, Vol. Arshad Khan, 'Parametric Cubic Spline Solution of Two Point Boundary Value Problems,' Applied Mathematics and Computation, Vol. 1, 2004, pp.175-182. Al-Said, 'Cubic Spline Method for Solving Two Point Boundary Value Problems,' Korean Journal of Computational and Applied Mathematics, Vol. Al-Said, 'Quadratic Spline Solution of Two Point Boundary Value Problems,' Journal of Natural Geometry, Vol.

12, 1997, pp.125-134. Abstract: In this work error estimation for numerical solution of Diffusion equation by finite difference method is done. The Explicit centered difference scheme is described to find the numerical approximation of the Diffusion equation. The numerical scheme is implemented in order to perform the numerical features of error estimation.

To get analytic solution, we present the variable separation method. We develop a computer program to implement the finite difference method in scientific programming language. An example is used for comparison; the numerical results are compared with analytical solutions. Keywords: Analytic solution, Diffusion equation, Finite difference scheme, Initial value problem (IVP), Relative error. Richmond (2006, July), 'Analytical solution of a class of diffusion problems', International journal of mathematical education in Science and Technology, Vol.15, issue 5, p. Mahdy ( 2012, Jan.), 'Crack Nicolson finite difference method for solving time-fractional diffusion equation', Journal of Fractional Calculus and Application, Vol.

(1960), 'The Numerical Treatment of Differential Equation', 3rd ed., Springer- Verlag, Berlin. LeVeque (1992), 'Numerical methods for conservation laws', Second edition, Springer. John A.Trangestein (2000), 'Numerical Solution of Partial Differential Equation', Durham. L.S.Andallah (2008), 'Finite Difference Method-Explicit Upwind Difference Scheme', lecturer note, Department of Mathematics, Jahangirnagar University. Harary, Graph Theory (Addison Wesley, Massachusetts, 1972). Hedetniemi, and P. Slater, Fundamentals of domination in graphs (Marcel Dekker, New York, 1998).

Berge, Theory of Graphs and its Applications (Methuen, London, 1962). Ore, Theory of graphs, Amer. 38, (1962), 206-212.

Cockayne and S.T. Hedetniemi, Independence graphs, Congr.

Numer., X, (1974), 471-491. Cockayne and S.T.

Hedetniemi, Towards a theory of domination in graphs, Networks, 7, (1977), 247-261. Goddard and M. Henning, Independent domination in graphs: A survey and recent results, Discrete Mathematics, 313, (2013), 839-854. Abstract: This study presents a co infection deterministic model defined by a system of ordinary differential equations for HIV/AIDS, malaria and tuberculosis. The model is analyzed to determine the conditions for the stability of the equilibria points and investigate the possibility of backward bifurcation. The study shows that the local disease free equilibrium is stable when the reproduction number is less than unity but the global stability of the disease free equilibrium is not guaranteed.

The model exhibits the phenomenon of backward bifurcation which posses a challenge to the design of effective control measures. Keywords: Bifurcation, Counseling, HIV/AIDS - TB and Malaria, Stability,Treatment. Abu-Raddad, P.Patnaik, and J.

Kublin, 'Dual infection with HIV and malaria fuels the spread of both diseases in Sub-Saharan Africa', Science, 314(5805), (2006), 1603-1606. [2] E, Allman and J. Rhodes, 'An introduction to Mathematical models in Biology', Cambridge University press: New York, (2004). Anderson and R. May, 'Infectious Diseases of Humans: Dynamics and Control', Oxford University Press: United Kingdom, (1993).

Onwujekwe, C. Onyewuche, C.

Idigbe, 'Impact of co infections of tuberculosis and malaria on the C D4+ cell counts of HIV patients in Nigeria', Annals of African Medicine, (2005), 4(1): 10-13. Baryama, and T. Mugisha, 'Comparison of single - stage and staged progression models for HIV/AIDS models', International Journal of Mathematics and Mathematical sciences.(2007), 12(4):399 - 417.

Garira and Z. Mukandavire, 'Modeling HIV/AIDS and Tuberculosis Co infection', Bulletin of Mathematical Biology, (2009), 71: 17451780. Abstract: In this paper a fuzzy inventory model is developed for deteriorating items with power demand rate.

Shortages are allowed and partially backlogged. The backlogging rate of unsatisfied demand is assumed to be a decreasing exponential function of waiting time. The cost components are considered as triangular fuzzy numbers. The objective of this paper is to develop an inventory model in a fuzzy environment, minimize the total cost and thereby derive optimal ordering policies. The total cost is defuzzified using Graded mean representation, signed distance and centroid methods.

The values obtained by these methods are compared with the help of numerical examples. The convexity of the cost function is depicted graphically. Sensitivity analysis is performed to study the effect of change of some parameters. Keywords: Centroid Method, Defuzzification, Deterioration, Graded mean representation method, Inventory, Partial backlogging, Power Demand, Shortages, Signed Distance Method, and Triangular Fuzzy Number. Chang, J.S Yao, Huey M Lee, Economic reorder point for fuzzy backorder quantity, European Journal of Operational Research, 109, 1998, 183-202. Ouyang, Fuzzy mixture inventory model involving fuzzy random variable lead-time demand and fuzzy total demand, European Journal of Operational Research, 169(1), 2006, 65-80. Ramer, 'Backorder fuzzy inventory model under function principle,' Information Sciences, 95, pp.

71-79, November 1996. Pal, Order level inventory system with power demand pattern for items with variable rate of deterioration, Indian J.Pure Appl. 19(11), 1988, 1043–1053. Dave and L.K. Patel, (T, S) policy inventory model for deteriorating items with time proportional demand, Journal of the Operational Research Society, 32, 1981, 137-142.

Dy, A note on An EOQ model for items with Weibull distributed deterioration, shortages and power demand pattern, Int.J.Inf.Manag.Sci.15 (2), 2004, 81–84. Ghare, and G.F. Schrader, A model for an exponentially decaying inventory, Journal of Industrial Engineering, 14, 1963, 238-243. Abstract: In this paper, a study of the empirical performance of the universal portfolios generated by certain reciprocal functions of the price relatives is presented.

The portfolios are obtained from the zero-gradient sets of specific logarithmic objective functions containing the estimated daily growth rate of the investment wealth. No solution of the zero-gradient equations is available and hence the pseudo Lagrange multiplier is used to generate the portfolios. Keywords: Empirical performance, investment wealth, pseudo Lagrange multiplier, reciprocal functions of price relatives, universal portfolio. Cover, Universal Portfolios, Mathematical Finance, vol.1, no.1, pp.1-29, Jan.1991. Ordentlich, Universal portfolios with side information, IEEE Transactions on Information Theory, vol.42, no.2, pp. Cambridge Ielts 6 Listening Mp3 Free Download. 348-363, Mar.1996. Singer and M. Warmuth, On-line portfolio selection using multiplicative updates, Mathematical Finance, vol.8, no.4, pp.325-347, Oct.1998.

Ordentlich, T. Cover, The cost of the achieving the best portfolio in hindsight, Math of Oper. Res.23, 960-982, 1998. Tan, Performance bounds for the distribution-generated universal portfolios, Proc. 59th ISI World Statistic Congress, Hong Kong, 5327-5332, 2013. Lim, An additive-update universal portfolio, Canadian J. Comp in Math., Nat.

2, 70-75, 2011. Lim, Universal portfolios generated by the Mahalanobis squared divergence, East-West J. Of Math., 225-235, (special volume 2012). Abstract: Population statistics in recent decades have revealed that the population of Nigeria is on the increase annually with about 2 to 3% growth rate. This is becoming troublesome in recent years. Among other solutions that may be proffered to curb this high population growth, is the need to adequately model the population of Nigeria, this will effectively help in population planning as well as a check on unprecedented growth of population without resources to match up the growth. This study attempts to compare various population growth model to determine the most best model for the Nigerian population.

From the findings, the study concludes that the exponential model is the best growth model for the Nigerian population. This model is closely followed by the linear model. Effect of population on economic development in Nigeria: a quantitative assessment.

International Journal of Physical and Social Sciences, vol. & Olayiwola, A.

M.(2013) An Assessment of the Growth of Ile-Ife, Osun State Nigeria, Using Multi-Temporal Imageries, Journal of Geography and Geology Vol. Eniayejuni, A. T., Agoyi, M (2011) A Biometrics Approach to Population Census and National Identification in Nigeria: A Prerequisite for Planning and Development, Asian Transactions on Basic & Applied Sciences, Vol.1(5) [4]. Folorunso, O., Akinwale, A., T. And Adeyemo, T. (2010) Population prediction using artificial neural network, African Journal of Mathematics and Computer Science Research Vol.

Gee, Ellen M. (1999) Population Growth Retrieved on 19th January, 2013, fromhttp://www.deathreference.com/Nu-Pu/population-Growth.html#b#ixzz2ICoYU8el. Abstract: In this paper, we have proposed and analyzed a mathematical model to study the simultaneous effect of two toxicants on a biological population, in which a subclass of biological population is severely affected and exhibits abnormal symptoms like deformity, fecundity, necrosis, etc.

On studying the qualitative behavior of model, it is shown that the density of total population will settle down to an equilibrium level lower than the carrying capacity of the environment. In the model, we have assumed that a subclass of biological population is not capable in further reproduction and it is found that the density of this subclass increases as emission rates of toxicants or uptake rates of toxicants increase.t. Agrawal AK, Sinha P, Dubey B and Shukla JB, Effects of two or more toxicants on a biological species: A non-linear mathematical model and its analysis, In Mathematical Analysis and Applications.

Dwivedi (Ed), Narosa Publishing House, New Delhi, INDIA, 2000, 97 – 113. Agrawal AK, Shukla JB, Effect of a toxicant on a biological population causing severe symptoms on a subclass, South Pacific Journal of Pure and Applied Mathematics, 1 (1), 2012, 12 – 27. DeLuna JT and Hallam TG, Effect of toxicants on population: a qualitative approach IV. Resource - Consumer Toxicant models, Ecol.

Modelling 35, 1987, 249 – 273. Freedman HI and Shukla JB, Models for the effect of toxicant in single species and predator-prey systems, J. 30, 1991, 15 – 30.

Hallam TG and Clark CE, Nonautonomous logistic equation as models of population in a deteriorating environment, J. 93, 1982, 303 – 311. Hallam TG, Clark CE and Jordan GS, Effects of toxicants on populations: a qualitative approach II. First order kinetics, J.

18, 1983, 25 – 37. [1] Attanassov, Intutionistic Fuzzy Sets, Fuzzy Sets And Systems, 20. [2] Azam.F.A, A.A.Mamun, and F. Nasrin, Anti fuzzy ideals of a ring, Annals of Fuzzy Mathematics and Informatics, Volume 5, No.2, pp 349 – 360, March 2013. [3] Biswas.R., fuzzy subgroups and Anti-fuzzy subgoups, Fuzzy sets and Systems, 5, 121 – 124, 1990. Y, Fuzzy semi-ideal and generalized fuzzy quotient ring, Iranian Journal of Fuzzy Systems, Vol.

5, pp 87 - 92, 2008. [5] Gouguen J.A., L- fuzzy sets, Journal of Mathematics Analysis and Applications, 18, 145 - 174, 1967. [6] Kumbhojkar.H.V. And Bapat.M.S, Correspondence theorem for fuzzy ideals, Fuzzy Sets and System, 41, 213-219, 1991. [7] Malik.D.S and John N. Moderson, Fuzzy Prime Ideals of a Ring, Fuzzy Sets and System 37, 93-98, 1990.

Abstract: in this work, we examine a supply chain network consisting of manufacturer, retailer and the demand market, we study the behaviour of the various decision – maker and formulate an optimization problem based on the condition proposed in this decision maker and transcribed the optimization problem to variational inequality because of its handiness in solving equilibrium problem. And derived equilibrium condition that satisfy the manufacturer, retailers and the demand market, these conditions must be simultaneously satisfied so that no decision- maker has any incentive to alter his transaction. Key words: supply chain network, variational inequality. Equilibrium,Convex Optimization. What is the Right supply chain for your product? Harvard Business Review 75 (2 march – April) 105- 116 (1997).

L and Belington C. The evolution of supply chain management models and practice at Hewlett – Packard interface (1995) [3]. Kinderlehrerand G.Stampachia (1980). An introduction to Variational Inequalities and their application, Academic press, New York. R.Anupindi and Y,bassok (1996) Distribution channel,information system and virtual centralized proceedings of the manufacturing and services operation management of society conference 87-92.

Supply chain management strategy, planning and operation 2nded Prentice Hall. Simchi – Levi, D, Designing and managing the supply chain concepts, strategies and case studies. New York (2002). Abstract: The fundamental characteristic of the IVIFS is that the values of its membership function and non-membership function are intervals rather than exact numbers.

In this paper, we define degree, order and size of IVIFGs. Also constant Interval valued intuitionistic fuzzy graphs and totally constant IVIFGs are introduced and discussed some properties of constant IVIFG. Keywords: Intuitionistic fuzzy graph, Interval valued intuitionistic fuzzy set, Interval valued intuitionistic fuzzy graph, Strong IVIFG, Constant IVIFG, totally constant IVIFG. AMS Mathematics Subject Classification (2010): 03E72, 05C69, 05C72. K, Intuitionistic fuzzy sets.

Fuzzy Sets & Systems, 20 (1986), 87-96. K, Operations over interval valued fuzzy set, Fuzzy Sets & Systems, 64 (1994), 159 - 174. K and Gargov. G, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets & Systems.

31 (1989), 343 -349. More on intuitionistic fuzzy sets. Fuzzy Sets & Systems, 33 (1989), 37 - 45. 015-7 Some new identities 2739.

New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets & Systems, 61 (1994), 137 - 42. R, Some Operations on intuitionistic fuzzy sets, Fuzzy Sets & Systems, 114 (2000), 477 - 484. D and Sudharsan. S, An Interval-valued Intuitionistic Fuzzy Weighted Entropy (IVIFWE) Method for Selection of Vendor, International Journal of Applied Engineering Research, 9 (21) (2014), 9075 - 9083.