iosr-Jm   Volume-2 ~ Issue-1

Abstract: Analytical solutions are obtained for one-dimensional advection-diffusion equation with variable coefficients in longitudinal semi-infinite homogeneous porous medium for uniform flow. The solute dispersion parameter is considered temporally dependent while the velocity of the flow is considered uniform. The first order decay and zero-order production terms are considered inversely proportional to the dispersion coefficient. Retardation factor is also considered in present paper. Analytical solutions are obtained for two cases: former one is for uniform input point source and latter case is for increasing input point source where the solute transport is considered initially solute free. The Laplace transformation technique is used. New space and time variables are introduced to get the analytical solutions. The solutions in all possible combinations of increasing or decreasing temporally dependence dispersion are compared with each other with the help of graph. It is observed that the concentration attenuation with position and time is the fastest in case of decreasing dispersion in accelerating flow field.
Keywords: Advection, Diffusion, First-order Decay, Zero-order Production, Retardation Factor, Homogeneous Medium.

[1] G I Taylor, Dispersion of soluble matter in solvent flowing slowly through a tube, Proceedings of Royal Society of London, A219, 1953, 186-203.
[2] A E Scheidegger, The Physics of Flow through Porous Media (University of Toronto Press, 1957).
[3] R R Rumer, Longitudinal dispersion in steady and unsteady flow, Journal of Hydraulic Division, 88, 1962, 147-173.
[4] R A Freeze and J A Cherry, Groundwater (Prentice-Hall, New Jersey, 1979).
[5] M Th van Genuchten and W J Alves, Analytical solutions of the one-dimensional convective-dispersive solute transport equation (Technical Bulletin No 1661, US Department of Agriculture, 1982).
[6] F T Lindstrom and L Boersma, Analytical solutions for convective-dispersive transport in confined aquifers with different initial and boundary conditions, Water Resources Research, 25, 1989, 241-256.
[7] A Ogata, Theory of dispersion in granular media, US Geol. Sur. Prof. Paper 411-I, 34, 1970.
[8] M Marino, Flow against dispersion in non-adsorbing porous media, Journal of Hydrology, 37, 1978, 149-158.
[9] A Ogata and R B Bank, A solution of differential equation of longitudinal dispersion in porous media, U. S. Geol. Surv. Prof. Pap. 411, A1-A7, 1961.
[10] D R F Harleman and R R Rumer, Longitudinal and lateral dispersion in an isotropic porous medium, Journal of Fluid Mechanics, 16(3), 1963, 385-394.

Abstract: Effect of magnetic field on transient free convection flow past an electrically conducting fluid over an oscillating vertical plate with chemical reaction is studied here. Exact solutions obtained by Laplace Transform methods are presented graphically for different values of physical parameters. It is observed that chemical reaction parameter and magnetic parameter influence the velocity and concentration profiles significantly.
Keywords: Free Convection, MHD, Oscillating Plate, Chemical Reaction AMS 2000 subject classification: 76R10, 76W05, 80A20, 80A32

[1] Abramowitz B. M. and Stegum I. A. : Handbook of Mathematical Functional function, Dover Publications, NewYork, (1965).
[2] Chaudhary R. C. and Jain A. : MHD heat and mass diffusion flow by natural convection past a surface embedded in a porous medium, Theoret. Appl. Mech., 36(1)(2009),1-27
[3] Das U. N., Deka R. K. and Soundalgekar V. M. : Effects of mass transfer on flow past an impulsively started vertical infinite plate with constant heat flux and chemical reaction, Forschung in Ingenieurwesen, 60(1994), 284-287.
[4] Das U. N., Deka R. K. and Soundalgekar V. M. : Effect of Mass Transfer on Flow Past an Impulsively Started Infinite Vertical Plate With Chemical Reaction, The Bulletin, GUMA, 5(1998), 13-20
[5] Das U.N., Deka R.K. and Soundalgekar V.M. : Transient free convection flow past an infinite vertical plate with periodic temperature variation, Journal of Heat Transfer, Trans. ASME, 121(1999), 1091-1094.
[6] Deka R. K. and Neog B. C. : Combined effects of thermal radiation and chemical reaction on free convection flow past a vertical plate in porous medium, Adv. Appl. Fluid Mech., 6-2(2009),181-195.
[7] Deka R. K. and Neog B. C. (2009): Unsteady MHD Flow past a vertical Oscillating Plate with Thermal Radiation and Variable Mass Diffusion, Cham. J. Math, 1(2009), 79-92.
[8] Neog B. C. : Unsteady MHD Flow past a vertical Oscillating Plate with Variable Temperature and Chemical Reaction, J. As. Aca. Math., 1(2010), 97-109.
[9] Gebhart B. : Heat Transfer, Tata McGraw Hill, (1971).
[10] Gebhart B. and Pera L. : The nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion, Int. J. Heat and Mass Transfer, 14(1971), 2025-2050

Abstract: We define a Cap-coset a ^ S of a subset S in an Artex space A over a Bi-monoid M and a Cup-coset a v S of a subset S in an Artex space A over a Bi-monoid M. We prove a ^ S need not be equal to S even when S is a SubArtex Space of A and a ϵ S. We prove if a ϵ S, then a ^ S C S. We prove that for a,b ϵ S a ≤ b implies a ^ S C b ^ S. We also prove that a v S need not be equal to S even when S is a SubArtex Space of A and a ϵ S. We prove if a ϵ S, then a v S C S. We prove that for a,b ϵ S a ≤ b implies b v S C a v S.
Keywords: Artex spaces, Cap-cosets, Cup-cosets, SubArtex Spaces.

[1] K.Muthukumaran and M.Kamaraj, "Artex Spaces Over Bi-monoids", "Research Journal of Pure Algebra", 2(5),May 2012, Pages 135-140.
[2] K.Muthukumaran and M.Kamaraj, "SubArtex Spaces Of an Artex Space Over a Bi-monoid", "Mathematical Theory and Modeling", An USA Journal of "International Institute for Science, Technology and Education", Vol.2, No.7, 2012, pages 39 – 48.
[3] K.Muthukumaran and M.Kamaraj, "Bounded Artex Spaces Over Bi-monoids and Artex Space Homomorphisms", "Research Journal of Pure Algebra", 2(7), July, 2012, pages 206 – 216.
[4] K.Muthukumaran and M.Kamaraj, "Some Special Artex Spaces Over Bi-monoids", "Mathematical Theory and Modeling", An USA Journal of "International Institute for Science, Technology and Education", Vol.2, No.7, 2012, pages 62 – 73. .
[5] K.Muthukumaran and M.Kamaraj, "Boolean Artex Spaces Over Bi-monoids", "Mathematical Theory and Modeling", An USA Journal of "International Institute for Science, Technology and Education", Vol.2, No.7, 2012, pages 74 – 85..
[6] J.P.Tremblay and R.Manohar, Discrete Mathematical Structures with Applications to Computer Science, Tata McGraw-Hill Publishing Company Limited, New Delhi, 1997.
[7] John T.Moore, The University of Florida /The University of Western Ontario, Elements of Abstract Algebra, Second Edition, The Macmillan Company, Collier-Macmillan Limited, London,1967.
[8] Garrett Birkhoff & Thomas C.Bartee, Modern Applied Algebra, CBS Publishers & Distributors,1987.
[9] J.Eldon Whitesitt, Boolean Algebra And Its Applications, Addison-Wesley Publishing Company, Inc.,U.S.A., 1961.

Paper Type	:	Research Paper
Title	:	On The Integrated Formulation of the Tau Method Involving At Most Two Tau Parameters for Ivps in Odes
Country	:	Nigeria
Authors	:	A.I.Ma'Ali, R.B.Adeniyi
	:	10.9790/5728-0163951

Abtract: An error estimation of the integrated variant of the tau method for ordinary differential equations is hereby considered for the class of equations characterized by m+s ≤ 2 where m and s are, respectively, the order and the number of overdetermination of the differential equation. Some general results are obtained and applied to test problems. Numerical evidences show that the estimate adequately captures the order of the tau approximation.
Key Words: Tau Method, Overdetermination, perturbation, class, Error estimation, Variant

[1]. Adeniyi R.B. (2008): An improved error estimation of the tau method for Boundary value problems. Research journal of applied sciences. Vol.3, issue 6 pp 456-464.
[2]. Adeniyi R.B. (2000): optimality of error estimate of one-step Numerical International journal of computer Mathematics, Vol.75 issue 3 pp 283-295.
[3]. Adeniyi R.B, Onumanyi P. (1991): error estimation in the numerical solution of the ordinary differential equation with the tau method. Computer and mathematics with application Vol.21, pp 19 – 21 (Available online 4 sept 2002).
[4]. Adeniyi R.B. (1991): On the tau method for numerical solution of ordinary differential equation doctoral Thesis, University Ilorin.
[5]. Adeniyi R.B, Onumanyi P and Taiwo O.A (1990): A computational error estimation of the tau method for non-linear differential equation. Journal of Nigeria mathematics Soc. Vol.9, pp 21 – 32.
[6]. Adeniyi R.B, (1985): An error estimation technique for linear ordinary differential equations using the tau method. M. Sc dissertation (unpublished), University of Ilorin.
[7]. Banks H.T and Wade J.G (1991): Weak tau approximations for distributed parameter system in inverse problems. Numer. Funct. Anal. Optima, Vol.12 pp 1 -31.
[8]. Egbetade S.A (200): A computational error estimate of he tau method. Ife journal of Science, Vaol.82 pp 105 – 110.
[9]. Hussaini S.M (2009): The Adaptive operational tau method for system of ordinary differential equations. Journal of computernal and applied mathematics, Vol.3 issue 1, Sept 2009.
[10]. Namasivayam S. and Ortiz E.L (1993): Error analysis of the tau method. Dependence of the approximation error on the choice of perturbation term. Computer and mathematics with application, Vol.25, pp 89 – 104.

Abstract: This paper is concerned with error estimation of the integrated variant of the tau method for Initial Value Problems (IVPs) for the class of equations for which 𝑚+𝑠≤3 where m and s are, respectively, the order and the number of over determination. Some general results obtained are applied to some problems. The numerical evidences show that the order of the tau approximant is closely captured.
Key Words: Tau methods, error estimation, overdetermination, variant, order, Initial. Value Problem (IVP)

[1]. A. I. Ma'ali (2012): "A Computational error estimation of the Integrated Formulation of the Tau method for some class of Ordinary differential equation", Doctoral thesis, University of Ilorin; Ilorin Nigeria (Unpublished).
[2]. Adeniyi, R.B. Onumanyi, P. and Taiwo O.A, (1990). A Computational Error Estimate of the Tau Method for Non-linear ordinary differential equation, J. Nig. Maths soc. Pp. 21 – 32.
[3]. Adeniyi, R.B. and Onumanyi, P., (1991): Error estimation in the numerical solution of ordinary differential equations with the tau method com. Maths-Applies, Vol. 21, No.9, pp. 19 – 27.
[4]. Adeniyi, R. B. (1985): An error estimation technique for linear ordinary differential equations using tau method, M.Sc. dissertation (Unpublished).
[5]. Adeniyi, R. B. (1991): On the tau method for Numerical Solution of Ordinary Differential Equations. Doctoral Thesis, University of Ilorin, Ilorin.
[6]. Adeniyi, R. B. and Aliyu, A. I. M. (2008): On the Tau Method for a class of Non-overdetermined Second Order Differential Equations. Journal for the Advancement of Modeling and Simulation Techniques in Enterprises (A.M.S.E.), 45 (2): 27 – 44.
[7]. Adeniyi, R. B. and Aliyu, A. I. M. (2011): On the Integrated Formulation of the Tau Method Involving at most two Tau parameters. International Journal on Research and Development, Book of Proceeding 30: 104 – 114.
[8]. Adeniyi, R. B. and Onumanyi, P. (1991): Error Estimation in the Numerical Solution of the Ordinary Differential Equations with Tau Method Computer and Mathematics with applications. 21: 19 – 21 (Available on line) 4, Sept 2002.
[9]. Aliyu A. I. M. (2007): On the Tau Method for solutions of a class of second Order Differential Equations. M.Sc. Thesis, University of Ilorin, Ilorin (Unpublished).
[10]. Egbetade, S. A. (2000): A computational Error Estimate of the Tau Method. Ife Journal of Science, 82: 105 – 110.

Abstract: Restaurants would avoid losing their customers due to a long wait on the line. Some restaurants initially provide more waiting chairs than they actually need to put them in the safe side, and reducing the chairs as the time goes on safe space. However, waiting chairs alone would not solve a problem when customers withdraw and go to the competitor's door; the service time may need to be improved. This shows a need of a numerical model for the restaurant management to understand the situation better. This paper aims to show that queuing theory satisfies the model when tested with a real-case scenario. We obtained the data from a restaurant in India. We then derive the arrival rate, service rate, utilization rate, waiting time in queue and the probability of potential customers to balk based on the data using Little's Theorem and M/M/1 queuing model. The arrival rate at Z square during its busiest period of the day is 2.22 customers per minute (cpm) while the service rate is 2.24 cpm. The average number of customers in the restaurant is 122 and the utilization period is 0.991. We conclude the paper by discussing the benefits of performing queuing analysis to a busy restaurant.
Keywords: Queue; Little's Theorem; Restaurant; Waiting Lines

[1] T. Altiok and B. Melamed, Simulation Modeling and Analysis with ARENA. ISBN 0-12-370523-1. Academic Press, 2007.
[2] D.M. Brann and B.C. Kulick, "Simulation of restaurant operations using the Restaurant Modeling Studio," Proceedings of the 2002 Winter Simulation Conference, IEEE Press, Dec. 2002, pp. 1448-1453.
[3] S. A. Curin, J. S. Vosko, E. W. Chan, and O. Tsimhoni, "Reducing Service Time at a Busy Fast Food Restaurant on Campus," Proceedings of the 2005 Winter Simulation Conference, IEEE Press, Dec. 2005.
[4] K. Farahmand and A. F. G. Martinez, "Simulation and Animation of the Operation of a Fast Food Restaurant," Proceedings of the 1996 Winter Simulation Conference, IEEE Press, Dec. 1996, pp. 1264-1271.
[5] A. K. Kharwat, "Computer Simulation: an Important Tool in The Fast-Food Industry," Proceedings of the 1991 Winter Simulation Conference, IEEE Press, Dec. 1991, pp. 811-815.
[6] M.Laguna and J. Marklund, Business Process Modeling, Simulation and Design. ISBN 0-13-091519-X. Pearson Prentice Hall, 2005.
[7] J. D. C. Little, "A Proof for the Queuing Formula: L= λW," Operations Research, vol. 9(3), 1961, pp. 383-387, doi: 10.2307/167570.
[8] K. Rust, "Using Little's Law to Estimate Cycle Time and Cost," Proceedings of the 2008 Winter Simulation Conference, IEEE Press, Dec. 2008, doi: 10.1109/WSC.2008.4736323.
[9] T. C. Whyte and D. W. Starks, "ACE: A Decision Tool for Restaurant Managers," Proceedings of the 1996 Winter Simulation Conference, IEEE Press, Dec. 1996, pp. 1257-1263.
[10] computing system – time and system- length distributions for MAP/D/1 queue using distributional little's law, by G. Singh, M.L. Chauadhary and U.C.Gupta performance evaluation , 69 (2012).

Abstract: The stress distribution on elastic space due to nuclei of thermo elastic strain distributed uniformly on the circumference of a circle of radius R situated in the place z= λ of the elastic semi space of Hookean model has been discussed by Nowacki: The Force stress and couple stress have been determined . The fore stress reduces to the one obtained by Nowacki for classical elasticity.

[1]. Palov, N.A. Fundamental equations of the theory of asymmetric elasticity (in Russian). Prikil. Mat. Mekh. 28 (1964). 401.
[2]. Peterson, M.E., Rev. Geophys. 11, 355, 1973.
[3]. Paria, G., Advances in Applied Mechanics, Vol. 10, p. 73, Academic Press, New York (1967).
[4]. Puri, P. Int. J. Engng. Sci. 11, 735 (1973).
[5]. Paria, G.and Wilson, Proc. Camb. Phil. Soc. 58, 527 (1972).
[6]. Papadepolos, M. (1963) : The elastodynamics of moving loads, Jt. Australian Maths. Soc. 3., 79-92.
[7]. Robin, P.Y.F., AMER, Mineral. 59, 1286, 1974.
[8]. Rayleigh Lord, Proc. Lond. Math. Soc. 20, 225 (1888).
[9]. Roy Choudhuri and Loknath Debnath, Magneto thermo-elasticity plane waves in Rotating Media, Int. J. Engg. Sci. Vol. 21, No. 2, pp. 155-169, (1983).
[10]. Schaefer, H. : Versuch einer Elastizitatetheorie des sweidimensionalen Cosserat – Kontinuum, Miss. Angew. Math. Festschrift Tollmien, Berlin, 1962, Akademia Verlag.