iosr-Jm   Volume-9 ~ Issue-2

Abstract: The surface instability of Kelvin-Helmholtz type in a couple stress fluid layers bounded above by a porous layer and below by a rigid surface is investigated using linear stability analysis. A simple theory based on fully developed flow approximations is used to derive the dispersion relation for the growth rate of KHI in presence of couple stress fluid. In order to observe the effect of boundary layer applying the Beavers-Joseph (BJ) slip condition. The dispersion relation is derived using suitable boundary and surface conditions and the results are discussed through graphically. The couple stress fluid is found to be stabilizing effect and the influence of the various parameters of the problem on the interface stability is thoroughly analyzed.

Key Words: Couple-stress fluid, KHI, B-J Condition, dispersion relation, porous media.

[1]. L. Kelvin(1910), Hydrokinetic solutions and observations , On the motion of free solids through a liquid, 69-75, " Influence of wind and capillary on waves in water superposed frictionless, 76-85, Mathematical and Physical Papers IV, Hydrodynamics and General Dynamics, Cambridge, England.
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[10]. T. R. Benjamin and T. J. Bridges(1997), Reappraisal of the K-H problem, Part-2: Introduction of the Kelvin-Helmholtz superharmonic and Benjamin-Feir instabilities, J. Fluid Mech., 333, 327.

Abstract: In this paper we study an epidemic model with immigration and non-monotone incidence rate under limited resources for treatment is proposed to understand the effect of the capacity for treatment. It is assumed that the treatment rate is proportional to the number of patients as long as this number is below a certain capacity and it becomes constant when that number of patients exceeds this capacity. Global analysis is used to study the stability of the disease free equilibrium and endemic equilibrium. It is shown that this kind of treatment rate leads to the existence of multiple endemic equilibria where the basic reproduction number plays a big role in determining this stability.

Keywords: Endemic, Global Stability, Non-monotone incidence rate, Reproduction number, Treatment rate.

[1] Alexander M.E. and Moghadas S.M. (2004). Periodicity in an epidemic model with a generalized non-linear incidence, J.Math. Biosci., 189, 75-96.

[2] Capasso V. and Serio G., (1978), A Generalization of the Kermack-Mckendrick Deterministic Epidemic model, J.Math. Biosci. 42, 43-61.

[3] Derrick W. R. and Van den Driessche P., (1993), A disease transmission model in a non constant population.,J. Math. Biol., 31, 495-512.

[4] Esteva L. and Matias M., (2001), A model for vector transmitted diseases with saturation incidence, Journal of Biological Systems, 9(4), 235-245.

[5] Hethcote H.W. and Van den Driessche P., (1991), Some epidemiological models with non-linear incidence, J. math. Boil., 29, 271-287.9.

[6] Jasmine D. and Henry Amirtharaj E.C., (2013), Modeling and Stimulation of Modified SIR Epidemic Model with Immigration and Non-monotonic Incidence Rate under Treatment, Indian Journal of Applied Research, 3(7), 43-44.

[7] Jasmine D. and Henry Amirtharaj E.C.,(2013), Global Analysis of SIR Epidemic Model with Immigration and Non-Monotone Incidence Rate, International Journal of Applied Research and Statistical Sciences, Vol. 2(5).,83-92.

[8] Leung G. M., The impact of community psychological response on outbreak control for severe acute respiratory syndrome in Hong Kong, J. Epidemiol. Community Health, 57 (2003), 857-863.

[9] Liu W. M., Hethcote H. W. and Levin S., (1987), A. Dynamical Behavior of Epidemiological Models with Nonlinear Incidence Rates, J. Math. Biol., 25, 359-380. [10] Liu W.M., Levin S.A., and Iwasa Y., (1986), Influence of Nonlinear Incidence Rates upon the Behavior of SIRS Epidemiological Models, J. Math. Biol., 187-204.

Abstract: This paper is concerned with the development of non-polynomial spline function approximation method to obtain numerical solution of ordinary and partial differential equations. The parametric spline function which depends on a parameter p  0, is discussed which reduced to the ordinary cubic spline [1] when the parameter p  0. The numerical method is tested by considering an example. Keywords : Cubic spline function, Parametric spline function, finite difference method

[1] J.Ahlberg, E.Nilson, J.Walsh, The Theory of Splines and Their Applications, Academic Press, New York (1967).
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(1995).
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problems, Computers Math. Appl.,27(11),45-48(1994).
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143(1981).

Abstract: We considered the equation of population dynamics and interacting growth of Clarias glariepinus (catfish) in a concrete pond. We proposed a logistic model using a combination of Euler and Runge- Kutta methods as the "best" approach in approximating the increases in the yield of the fish according to time. The results obtained showed that it allowed a choice of optimal regimes of aeration, feeding and fertilization of the fish for different climatic conditions in order to maximize the yield. We concluded that these approaches were the best in determining maximum yields of the fish in a concrete pond when tested for growth, stocking densities and harvesting processes.

Keywords: Population Dynamics and Growth, Clarias glariepinus, Runge- Kutta method, Euler's method, Logistic model.

[1]. Aluko, P. I.; Nlewadim, A.A and Aremu, A. (2001); Observation of fry cannibalism in clarias gariepinys (Burchell, 1822), Journal of aquatic science 16:1-6
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[7]. Hogendoorn, H. (1997); Controlled propagation of African Catfish, (clarias glariepinus). Feeding and growth of fry aquaculture 21:233 – 241
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Abstract: In this paper the concept of fuzzy rg -super irresolute mappings have been introduced and explore some of its basic properties in fuzzy Topological Space.

Keywords: fuzzy topology, fuzzy super closure, Fuzzy Super Interior fuzzy rg-super closed sets and fuzzy rg-super open sets, fuzzy rg-super continuous and fuzzy rg -super irresolute mappings.

[1]. B. Ghosh, Semi-continuous and semi-closed mappings and semi-connectedness in fuzzy setting,Fuzzy Sets and Systems 35(3) (1990), 345–355.
[2]. C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968), 182–190.
[3]. C.W. Baker on Preserving g-super closed sets Kyungpook Math. J. 36(1996), 195-199.
[4]. G. Balasubramanian and P. Sundaram, On some generalizations of fuzzy continuous functions,Fuzzy Sets and Systems 86(1) (1997), 93–100.
[5]. G. Balasubramanian and V. Chandrasekar, Totally fuzzy semi continuous functions, Bull. CalcuttaMath. Soc. 92(4) (2000), 305–312.
[6]. G. Balasubramanian, On fuzzy pre-separation axioms, Bull. Calcutta Math. Soc. 90(6) (1998),427–434.
[7]. K. K. Azad, On fuzzy semi continuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl. 82(1) (1981), 14–32.
[8]. K. M. Abd El-Hakeim, Generalized semi-continuous mappings in fuzzy topological spaces, J. Fuzzy Math. 7(3) (1999), 577–589.
[9]. L. A. Zadeh, Fuzzy sets, Information and Control 8 (1965), 338–353.
[10]. M.K. Mishra et all on " Fuzzy super continuity" International Review in Fuzzy Mathematics vol-2,No-3,July –December2012,143-146.

Abstract: In this paper we have tried to develop second countability in frames parallel to that in classical topology.

Keywords: B L 2 frame, Dense Sublocale , L-base, Locales, Lower frame

[1] Steven Vickers,Topology via logic(Cambridge Tracts in Theoretical Computer Science,Cambridge University Press1989) [2] Peter.T.Johnstone,Stone Spaces(Cambridge University Press,1982) [3] Jorge Picardo, Ales Pultr ,Frames and Locales –Topology without points(springer Basel AG 2012)

Abstract: This paper presents a review of the Algorithmic method of Hamming codes techniques for detection and correction of computational errors in a binary coded data, analysis in an integer sequence A119626. This integer sequence is obtained from computing the difference in the number of inverted pairs of the first and the last cycles which in turn is obtained from a special (123)-avoiding permutation pattern. The computation in this paper was restricted to values of n = 1, 2, 3, 4, and 5 respectively. This paper simply considered the execution time (T) and rate (R) for any given time t of the algorithmic method of analysis based on the number of iterations (steps) involved in the general procedure of encoding, detection and correction of errors for all the values of n.

Key words: Algorithmic method, binary coded data, execution time (T) and rate (R), Hamming codes and integer sequence A119626.

[1] I. Koren, Computer arithmetic algorithms (Natrick MA): A. K. Peters, 2002. [2] R. W. Hamming, Bell system Technology, Journal of Error Detecting and Correcting Codes vol. 29, April, 1950, pp. 147-160. [3] A. A. Ibrahim, Mathematics Association of Nigeria, the journal On the Combinatorics of A Five-element sample Abacus of vol. 32, 2005, No. 2B: 410-415.
[4] Moon & K. Todd, Error correction coding (http://www.neng.usu.edu/ece/faculty/tmoon/eccbook/book.html). (New Jersey: John Wiley, 2005 and sons ISBN 978-0-471-64800-0). [5] B. Sklar, Digital communication: fundamentals and applications (Second Edition: Prentice-Hall, 2001).

Abstract: In this paper, we will compare between Adomian decompositionmethod(ADM) and Homotopyperturbation method(HPM)for obtaining the numerical solutions of higher-order linear fractional integro-differential equations with boundary conditions. Numericalexamples arepresentedtoillustratetheefficiencyandaccuracy oftheproposed methods.

Keywords: Adomiandecompositionmethod, Homotopy perturbationmethod, Boundary valueproblems,Fractionalintegro-differentialequations,Caputofractional derivative.

[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-
Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
[2] V. Lakshmikantham, S. Leela, and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific, 2009.
[3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience
Publication, JohnWiley & Sons, New York, NY, USA, 1993.
[4] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego,
Calif, USA, 1999.
[5] K. Diethelm and A.D. Freed, "On the solution of nonlinear fractional order differential equations used in the modeling of
viscoelasticity," in Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics, Reaction Engineering and
Molecular Properties, F. Keil, W. Mackens, H. Voss, and J. Werther, Eds., pp. 217–224, Springer, Heidelberg, Germany, 1999.
[6] R. Metzler, W. Schick, H.-G. Kilian, and T. F. Nonnenmacher, "Relaxation in filled polymers: a fractional calculus approa ch,"
Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995.
[7] L. Gaul, P. Klein, and S. Kemple, "Damping description involving fractional operators," Mechanical Systems and Signal
Processing, vol. 5, no. 2, pp. 81–88, 1991.
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differential equations and inclusions," Acta ApplicandaeMathematicae, vol. 109, no. 3, pp. 973–1033, 2010.

Abstract: A vector square fuzzy transportation problem is a special type of transportation problem of the network optimization problems has the special data. Transportation problem with fuzzy supply values of the suppliers and with fuzzy demand values of the receivers. In this paper we are changing in interval form then solving a vector fuzzy transportation problem. A solution concept is attractive from the standpoint of feasibility and efficiency is specified. An investigation of the stability set of parameters corresponding to one α - efficient solution of the ordinary problem – α- VTP to a vector fuzzy transportation problem is presented. We used ranking technique for solve trapezoidal fuzzy number. An illustrative example is given.

Keywords: Vector fuzzy transportation problem, square Trapezoidal fuzzy numbers, α- efficiency, Optimal solutions, interval number.

[1]. Ammar E.E. and A. Kozae (2011) "A study on fuzzy vector transportation problem with fuzzy data" journal of nature science and mathematics, vol. 5, no. 1, pp 27-40.
[2]. Balachandran.V, and Thompson.G.L (1975 ) "An operator theory of parametric programming for the generalized transportation problem I basic theory" Nov. Res.Log. Quart,22,79-100.
[3]. Bellman.R.E, and Zadeh.L.A. (1970) "Decision-making in a fuzzy environment management Science", Journal Vol 17 B141 CB164.
[4]. Charnes.A,and klingman.D. (1977) "The more-for-less paradox in the paradox in the distribution models", cahiers du centre di.Etudes recherché operationnelle , Journal Vol 13 pp 11-32.
[5]. Chanas, S., Kolodziejczk, W. and Machaj A. (1984); "A fuzzy approach to the transportation problem" FSS 13,211-221.
[6]. Chanas, S. Kuchta, D. (1996) "A concept of the optimal solution of the transportation problem with fuzzy cost coefficient" FSS, 82 (3)pp 299-305.
[7]. Chen.S.H. (1985) "Operations on fuzzy numbers with function principle" Tamkang Journal of Management Sciences, Journal Vol 6 pp13-25.
[8]. Hitchock (1978) "Distribution of product from several sources to numerous localities" Journal of Math Physics, Volume 12 No.3.
[9]. R.Jahirhussain , P.Jayaraman fuzzy optimal transportation problem by improved zero suffix method via robust rank techniques International Journal of Fuzzy Mathematics and Systems (IJFMS). Volume 3 (2013) pp 303-311
[10]. Jiménez F., Verdegay J.L. (1999) "Solving fuzzy solid transportation problems by an evolutionary algorithm based parametric approach" European Journal of Operation Research (117) 3 485-510.

Abstract: In this paper, we present Optimal Equi-scaled families of Jarratt method for computing zeros of system of nonlinear equations numerically. In this paper, we extending the idea of the proposed families of Jarratt method to system of nonlinear equations .It is proved that the above said families have second order of convergence. Numerical tests are performed, which conform theoretical results. Form the compersion with known methods it is observed that present method shows good stability and roubustness. Keywords: System of Nonlinear equations, Optimal Order of Convergence, Halley's method, Schroder's method, Jarrat method.

[1] A.M., Ostrowski, (1960). Solutions of Equations and System of Equations, Academic Press, New York.
[2] Ostrowski, A. M., (1966). Solution of equations and system of equations. Academic Press NY, London.
[3] Ortega, J. M. and Rheinboldt, W. C.,(1970). Iterative solution of non-linear equations in several variables, Academic Press, Inc.
[4] Traub, J. F., (1982). Iterative method for the solution of equations, Chelsca Publishing Company, New York.
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[7] H.T. Kung,J.F. Traub,(1974), Optimal order of one-point and multipoint iteration, Journal of the ACM 21, pp.643-651.
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[9] E.Schroder's, (1870), Uber unendlichviele Algorithm zur Auffosung der Gleichungen, Ann. Math.,2,pp.317-365.
[10] R. Bhel, V. Kanwar and K. Sharma,(2012), optimal equi-scaled families of Jarratt's method, International Journal of Computer Mathematics.

Abstract: In this paper, we applied the method of Double Laplace Transform for solving the one dimensional Boundary Value Problems. Through this method the boundary value problem is solved without converting it into Ordinary Differential equation, therefore no need to find complete solution of Ordinary Differential equation. This is the biggest advantage of this method. The scheme is tested through some examples & the results demonstrate reliability.

[1] D. G. Duff, Transform Methods for solving Partial Differential Equations, Chapman and Hall/CRC, Boca Raton, F. L. 2004.

[2] A. Estrin & T. J. Higgins, The Solution of Boundary Value Problems by Multiple Laplace Transformation, Journal of the Franklin Institute, 252 (2), 153 – 167, 1951.

[3] Adem Kilicman, Hassan Eltayeb, A note on defining singular integral as distribution and partial differential equations with convolution term, Elsevier, Mathematical & Computer Modelling, 49(2013), 327-336. [4] R. S. Dahiya, M. Vinayagamoorthy, Laplace Transform pairs of n- Dimensions &Heat conduction problem, Mathl Comput. Modelling, Vol. 13, No. 10, pp. 35- 50, 1990.

[5] A. Aghili, A. Motahhari, Multi- Dimensional Laplace Transform for non- homogeneous Partial Differential Equations, Journal of Global research in Mathematical Archives, Vol. 1, No. 1, January 2013. [6] H. Eltayeb & A. Kilicman, A Note on Double Laplace Transform and Telegraphic Equations, Abstract & Applied Analysis, Volume 2013.

Abstract: A numerical technique based on the Cauchy integral formula of complex analysis has been employed for evaluating the integrals and derivatives of fractional orders of an analytic function. The method of subtraction of singularity has been used to evaluate the contour integrals for a desired degree of accuracy. Subject Classification Primary (1999): 65D 25, 65D 30

Key words: Integrals and derivatives of fractional orders, Analytic function, Quadrature rules.

[1]. Acharya, M., Nayak, M. M., Acharya, B. P., Numerical Evaluation of Differintegrals of Analytic Function, J. Contp. Appl. Math., 1(2), 2011, 29-35.
[2]. Acharya, M., Mohapatra, S. N. and Acharya, B. P., On numerical evaluation of fractional integrals, Appl. Math. Sci., 5(29), 2011, 1401-1407.
[3]. Dalir, M. and Bashour, M., Applications of fractional calculus, Appl. Math.Sci., 4(21), 2010, 1021-1032.
[4]. Lether, F. G., On Birkhoff-Young quadrature of analytic functions , J. Copm. Appl. Math., 2, 1976, 81-84.
[5]. Liu, J.L. & Patel, J. Certain properties of multivalent functions associated with an extended fractional diferrintegral operator, J. Appl. Math. Comput. 203, 2008, 703- 713.
[6]. Miller, K.S. Ross, B., An introduction to the fractional calculus and fractional differential equation, John Wiley & Sons, New York, 1993.

Abstract: This paper deals with describing application of q-theory in different fields of mathematics and future areas where its use can be extended .

Keywords: q-analogue, q-function, q-hypergeometric function

[1] G. Gasper and M. Rahman, Basic Hypergeometric Series, CambridgeUniversity Press, Cambridge, 1990.

[2] Thomas Ernst, A method for q-Calculus, Journal of Nonlinear Mathematical Physics, 2003

[3] Prashant Singh, Pramod Kumar Mishra and R.S.Pathak ,q-Iterative Methods, IOSR Journal of Mathematics (IOSR-JM), Volume 8, Issue 6 (Nov. – Dec. 2013), PP 01-06

Abstract: This paper is devoted to derivation of q-analogues of Truncation Errors associated with numerical differentiation . To analyse the error in numerical differentiation Taylor polynomial with remainders are useful and we have used different approximations to calculate errors. It also deals with comparison of accuracy of q-analogue of Taylor's Series using both single and double q parameters and Truncation Errors related to it.

Keywords: Basic, q-special function, q-Truncation Error, basic analogue, T.E.(Truncation Error)

[1] G. Gasper and M. Rahman, Basic Hypergeometric Series, CambridgeUniversity Press, Cambridge, 1990.

[2] Thomas Ernst, A method for q-Calculus, Journal of Nonlinear Mathematical Physics, 2003

[3] Prashant Singh, Pramod Kumar Mishra and R.S.Pathak ,q-Iterative Methods, IOSR Journal of Mathematics (IOSR-JM), Volume 8, Issue 6 (Nov. – Dec. 2013), PP 01-06