iosr-Jm   Volume-11 ~ Issue-3 ~ May-June 2015

Abstract: The classical linear stability analysis is used to analyze the effect of thermal radiation on the onset of Darcy electroconvection in a horizontal porous layer heated from below. The boundaries are assumed to be black bodies and the optical properties of the transparent dielectric fluid are independent of the wavelength of radiation. The principle of exchange of stabilities is shown to be valid. The critical values pertaining to the stationary instability are obtained by means of the higher order Galerkin method. It is found that basic temperature profile becomes exponential and symmetric as the radiative parameters increase and that the effect of thermal radiation is to delay the onset of Darcy electroconvection.

Keywords: Thermal Radiation, Electroconvection, Integro-differential equations, Milne- Eddington approximation

[1]. R.M. Goody, The influence of radiative transfer on cellular convection, J. of Fluid Mech., 1, 1956, 424-435.
[2]. E. Spiegel, The convective instability of a radiating fluid layer, Astrophys. J., 132, 1960, 716-728.
[3]. M.P. Murgai and P.K. Khosla, A study of the combined effect of thermal radiative transfer and a magnetic field on the gravitational
convection of an ionized fluid, J. Fluid Mech., 14, 1962, 433-451.
[4]. P.K. Khosla and M.P. Murgai, A study of the combined effect of thermal radiative transfer and rotation on the gravitational stability
of a hot fluid, J. Fluid Mech., 16, 1963, 97-107.

Abstract:In this paper, we show that with a few modifications and techniques the Adomian Decomposition Method (ADM) for solving a class of hyperbolic equations with nonlocal conditions can be applied to obtain the exact solutions to this type of problems. The clue of this one consists in transforming the boundary value nonlocal problem into a classical problem whose solution is obtained.
Keywords: Hyperbolic equation, Nonlocal conditions, Adomian decomposition method

[1]. G. Adomian, Nonlinear stochastic operator equations (Academic Press, Orlando, FL 1986).
[2]. G. Adomian, Solving frontier problems of physics; the decomposition method (Boston: kluwer Academic Publisher, 1994).
[3]. G. w. Batten, Jr., Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for
parabolic equations, Math. Comp., 17 (1963), 405-413.
[4]. S. A. Belin, Existence of solutions for one-dimensional wave equations with nonlocal conditions, Electronic Journal of differential
equations, 2001(2001), No. 76, pp. 1-8.
[5]. L. Bougoffa, Weak solution for hyperbolic equations with a non-local condition, J. The Australian Journal of mathematical Analysis
and Applications (AJMAA) Vol. 2, Issue 1, pp. 1-7, 2005.

Abstract: In this paper, we solve numerically initial value problems for ordinary differential equations by Euler method .The proposed method is quite efficient and is practically well suited for solving these problems. Several examples are presented to demonstrate the accuracy and easy implementation of the proposed method. Our approximate solutions are compared with the exact solutions. The approximate solutions converge to the exact solutions monotonically. Finally we investigate and compute the error of proposed method for different step size.
Key words: Initial value problem, Euler's method, Error analysises.

[1]. Ogunrinde R. Bosede,Fadugba S.Emmanuel, Okunlola J. Temitayo ,2012,On Some Numerical Methods for Solving Initial value Problems in ordinary Differential equations, IOSR Journal of Mathematics , volume 1, Issue 3, PP 25-31.
[2]. S.M. Eaqub Ali, 2006, A Text Book of Numerical Methods with Computer programming, Beauty publication, Khulna-9100, Bangladesh.
[3]. Akanbi M A, Propagation of Errors in Euler Method, Scholars Research library, Archives of Applied Science Research, 2010, PP 457-469.
[4]. Kockler,N. 1994. Numerical Method for Ordinary Systems of Initial value Problems. John Wiley and Sons: New York, Ny.
[5]. Lambert, J.D. 1973.Computational Methods in ordinary differential equations, Wiley.

Abstract: The authors introduced the concept of ρ-continuity between a topological space and a non empty set where ρϵ {L,M,R,S} in [ρ- continuity between a topological space and a non empty set where ρϵ {L,M,R,S} , International Journal of mathematical sciences 9(1-2)(2010), 97-104.]. In this paper, the concept of α-ρ-continuity is introduced and its properties are investigated.Recently, Navpreet singh Noorie and Rajni Bala introduced the concept of f# function to characterize the closed, open and continuous functions. In this paper, α-ρ-continuity is further characterized by using f# functions.

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Abstract: In this communication, we propose a new generalized fuzzy entropy measure using segment decomposition and effective range and study its particular cases. Also some fuzzy coding theorems have been established.

Keywords:- effective Range, segment Decomposition, Fuzzy set, Average Mean Length, Average Fuzzy Mean Length, Generalized Fuzzy entropy, Information Bounds. AMS Subject Classification: 60E15, 62N05, 90B25, 94A17, 94A24.

[1]. Abramson, N. [1963]: Information theory and coding; McGraw Hill, New York.
[2]. Autar, R and Soni, R.S. [1975]: Inaccuracy and coding theorem; Journ. Appl. Prob. Vol.12, pp. 845-851.
[3]. Beckenback, E.F. and Bellman, R. [1961]: Inequalities; Springer, Berlin.
[4]. Bernard, M.A. and Sharma, B.D. [1988]: Some combinatorial results on variable length error correcting codes; ARS combinatorial, 25B, pp. 181-194.
[5]. Bernard, M.A. and Sharma, B.D. [1990]: A lower bound on average code word length of variable length error correcting code; EEE trns. Information theory, 36(6), pp. 1474-1475.
[6]. Campbell, L.L. [1965]: A coding theorem and Renyi's entropy; Information and control, vol. 8, pp.423-429.

Abstract: In this paper, we considerthe unsteady flow within a double-sided symmetric thin liquid film with negligible inertia. We apply the Navier-Stokes equations in two dimensional flows for incompressible fluid.The similarity method is used in which the explicit time dependence can be isolated and thus the shape of the film will depend on one variable only.The differential equation of the film thickness is obtained analytically and the solution of equation thatrepresents the film thickness is obtained numerically by using Rung-Kutta method.

Keywords: Thin Liquid Films, Navier-Stokes equations, continuity equation.

[1]. Mysels, K.J., Shinoda, K. and Frankel, S., (1959), Soap Films, Studies of Their Thinning and a Bibliography, Pergamon press, London.
[2]. Abdulahad J. G. and Derbaz S. A.,(2014), Steady flow of thin liquid films on an inclined solid surface, IOSR Journal of engineering, Vol.4, Issue 4,pp.41-46.
[3]. Abdulahad J. G. and Derbaz S. A.,(2014),The drainage of thin liquid films on an inclined solid surface, IOSR Journal of engineering, Vol.10, Issue 42, pp. 73-80.
[4]. Chen, J. L. and Hwang, C. C.,(1996), Nonlinear rupture theory of thin liquid film on a cylinder, Journal of Colloid and Interface Science 182, pp. 564-569.
[5]. Lin, T. S., Kondik, L. and Filippove, A., (2012), Thin films flowing down inverted substrate: three-dimensional flow, Physics of Fluids 24,022105, PP. 1-18.
[6]. Abdulahad J. G., (2010), Flow of Soap Films on Inclined Plane, Rafidain journal of computer sciences and mathematics,Vol.7, Issue 2, pp. 67-78.
[7]. Abdulahad J.G., (1989), Liquid Surface Condition, Ph.D. Thesis, University of Essex.

Abstract: stability and dynamics of a free double-sided symmetric thin liquid film are investigated by using the long-wave method. The flow in thin liquid film is considered in two dimensions for Newtonian liquid with constant density and dynamic viscosity. The Navier-stokes equations is used with appropriate boundary conditions of zero shear stress and also of normal stress on the bounding free surfaces with non-dimensional variables to obtain an equation that governs such flow.

Keywords: Thin Liquid Films, Navier-Stokes equations, continuity equation.

[1]. Hajar, F. I.,"Stability Analysis of Inclined Free Thin Liquid Films". M.Sc. thesis, University of Zakho, (2011).
[2]. Sharma, A. and Khanna, R.,"Pattern formation in unstable thin liquid films under the influence of antagonistic short- and long-range forces", Journal of Chemical Physics, Vol. 110, No. 10, (1999), pp. 4929-4936.
[3]. Jameel, A. T. and Atieh, M. A.,"Nonlinear simulation of thin liquid film flow on inclined planes", The 6th Saudi Engineering Conference, KFUPM, Dhahran, Vol. 5, (2002), pp. 607-621.
[4]. De Wit, A. and Gallez, D.,"Interfacial instabilities in thin liquid films with insoluble surfactants", Advances in multi-fluid flows, (1996), pp. 112-130.
[5]. De-Souza, E. R., Anteneodo, C., Gallez, D. and Bisch, P. M.,"Long-scale evolution of thin liquid films bounded by a viscous phase with diffusing charged surfactants", Journal of Colloid and Interface Science 244, (2001), pp. 303-312..

Abstract: Steady two dimensional MHD free convective boundary layer flow of an electrically conducting Newtonian nanofluid past an impulsively started infinite vertical plate is investigated numerically. Nanofluids are well-dispersed (metallic) nanoparticles at low-volume fractions in liquids. They may enhance the mixture's properties among them, thermal conductivity over the base fluid values. A magnetic field can be used to control the motion of an electrically conducting fluid in micro-scale systems used for the transportation of fluids. The momentum and energy equations along with the boundary conditions are first converted into dimensionless form by dimensional analysis.

[1]. Akharinia A, Abdolzadeh M, Laur R. (2011): critical investigation of heat transfer enhancement using nanofluids in micro-channels
with slip and non-slip flow regimes. Journal of applications of thermal engineering 31: 556-565.
[2]. Aziz A, Khan WA (2012) Natural convective boundary layer flow of a nanofluid
[3]. Boungiorno J. (2006) Convective transport in nanofluids. Journal of heat transfer 128: 240-250.

Abstract: An influence function matrix for autocorrelation and partial autocorrelation were constructed for some economic series and it is used as a tool for the detection of outliers and model order, depending on the lagging effects of observation in the correlation coefficient and size of the observations. A suitable model was fitted for all the data judging from the behaviour of both the plots on ACF and PACF and the influence function matrix and forecast were made for model fitted for each data with the presence and removal of outliers from the model. Our findings reveals the detection of 4 outliers for Real GDP, 6 for GDP at current market prices and 1 for GDP at current factor cost (Nominal GDP), 6 for retail prices of diesel recorded by N.N.P.C. while there is no outlier detected for retail prices of PMS (petrol) it was also discovered that the confidence interval on the forecast values for the model without outliers overlap that of the model with outliers.

Keywords: Autocorrelation, Partial autocorrelation, Influence function matrix, Outliers and Autoregressive Moving Average (ARMA).

[1]. Anderson, OD. (1971). The Statistical Analysis of Time Series. Wiley, New York.
[2]. Box G.E.P. and Jenkins, G.M. (1976). Time Series Analysis, Forecasting and control. Holden-day Inc.
[3]. Chernick, M.R.; Dowring, D.J. and Piks, D.K (1982) "Detecting Outliers in Time Series Data ". Journal Amer. Stat. Vol. 77, p. 743-747.
[4]. Durbin, J. (1960). "The fitting of time series Models" Rev. Int. Inst. Of Stat. Vol. 28, p. 223-244.
[5]. Granger, C.W.J. and Morris, M.J.(1976). Modelling and Interpretation.

Abstract: The purpose of present problem is to study the effect of unsteady flow of an incompressible viscous fluid past an uniformly Accelerated Infinite Vertical Porous Plate through Porous Medium taking account of the presence of variable temperature, uniform mass diffuse with Heat and Mass Transfer. The dimensionless governing equations are solved by using Laplace Transform technique. The Velocity, temperature and concentration fields are studied for different physical parameters like Thermal Grashof Number, Modified Grashof Number, Permeability Parameter, Prandtl Number, Schmidt Number, and time. The effect of these parameters on the Velocity field, Temperature field and Concentration distribution have been studied and the results are presented graphically and discussed quantitatively.

Keywords: Accelerated infinite Vertical Plate, Heat Transfer, Mass Transfer, Porous Medium and mass diffusion.

[1]. Gupta.A.S, I.Soudalgekar.V.M "Free convection effects on the flow past an Accelerated vertical plate in an incompressible dissipative fluid". Rev, Roum Sci Techn-mec- Apl 24, pp 561-568, (1979).
[2]. Soudalgekar.V.M and Patel M.R "On flow past a vertical Oscillating plate with variable temperature". Journal of Heat and Mass Transfer (4) pp 143-148, (1980).
[3]. Singh.A.K and Singh.J "Mass Transfer Effects on the Flow past an Accelerated vertical plate with constant Heat flux" Astrophysics and Space Science. 97, pp 57 – 61, (1983).
[4]. Muthucumaraswamy.R and Vijayalakshmi.A "Effect of Heat and Mass Transfer on Flow past, an Oscillating vertical plate with variable temperature" Int.J.of. Appl Math and Mech 4(1) pp 59-65, (2008).
[5]. Basant kumar Jha. Ravindra Prasad "Free convection and Mass Transfer Effects on the Flow past an Accelerated vertical plate with Heat Source". Mechanics Research Communication 17, pp 143-148, (1990).

Abstract: We say a natural number n is abundant if (n)>2n, where (n) denotes the sum of the divisors of n. The aliquot parts of n are those divisors less than n, and we say that an abundant number n is pseudoperfect if there is some subset of the aliquot parts of n which sum to n. We say n is weird if n is abundant but not pseudoperfect. We call a weird number n primitive if none of its aliquot parts are weird. We find all primitive weird numbers of the form 2kpq (p<q being odd primes) for 1k14. We also find primitive weird numbers of the same form, larger than any previously published.

[1]. S. Benkoski and P. Erdős, "On weird and pseudoperfect numbers," Mathematics of Computation, 28, No. 126 (april 1974), 617–623.
[2]. R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer-Verlag (New York), 2001.
[3]. S. Kravitz, "A search for large weird numbers," Journal of Recreational Mathematics, 9, No. 2, (1976–77), 82–85.
[4]. S. Pajunen, "On primitive weird numbers," MRFS: A Collection of Manuscripts Related to the Fibonacci Sequence (18th Anniversary Volume), V. Hoggat, ed., Fibonacci Association, (1980), 162–166.