iosr-Jm   Volume-12 ~ Issue-2 ~ Mar-Apr 2016

Abstract:In this paper we obtain some inequalities as sufficient conditions for the harmonic multivalent function G(z) to be in classes
* m S H () . Inequalities for convolution multiplier of two harmonic multivalent functions f and G are also obtained. Also it shown that these inequalities are necessary and sufficient for the function 1 G (z) . Further, the necessary and sufficient conditions for the functions 2 G (z) to be in classes * m S H () are obtained.

[1]. Ahuja, O.P., Jahangiri, J.M., and Silverman,H., Convolutions for special classes of Harmonic Univalent Functions, Appl. Math.
Lett, 16(6) (2003), 905-909.
[2]. Ahuja,O.P. and Silverman,H., Extreme Points of families of Univalent Functions with fixed second coefficient. Colloq. Math. 54
(1987), 127-137.
[3]. Ahuja,O.P. and Jahangiri,J.M., Harmonic Univalent Functions with fixed second coefficient, Hakkoido Mathematical Journal, Vol.
31(2002) p-431-439.
[4]. Ahuja,O.P. and Silverman,H., Inequalities associating hypergeometric Functions with planar harmonic mappings, Vol 5, Issue 4,
Article 99, 2004.
[5]. Ahuja,O.P. and Jahangiri,J.M., Multivalent Harmonic starlike Functions, Ann. Univ Mariae Curie – Sklodowska, Section A, 55 (1)
(2001), 1-13.

Abstract:In this paper, we have considered the unsteady MHD free convection flow of an incompressible electrically conducting second grade fluid through porous medium bounded by an infinite vertical porous surface in the presence of heat source and chemical reaction in a rotating system taking hall current into account. The momentum equation for the fluid flow through porous medium is governed by Brinkman's model.....
Key words: Convection flows, Hall effects, heat and mass transfer, MHD flows, infinite vertical plates, porous medium, rotating channels, second grade fluids.

[1]. Anand Rao. G. Hydromagnetic flow and heat transfer in a saturated porous medium between two parallel porous wall in a rotating
system, Proceedings of the Eighth National heat and mass transfer conference, Assam, India, 1985.
[2]. Ariel. P.D. On exact solution of flow problems of a second grade fluid between two parallel porous wall, Int. J. Eng. Sci., 40 (2002)
913-941.
[3]. Bég.O.A, Takhar.H.S., Singh. A.K. Multi-parameter perturbation analysis of unsteady, oscillatory magnetoconvection in porous
media with heat source effects, Int. J. Fluid Mech. Res., 32(5)(2005) 635-661.
[4]. Chen C-H. Combined heat and mass transfer in MHD free convection from a vertical surface with Ohmic heating and viscous
dissipation, Int. J. Eng. Sci., 42(7)(2004) 699-713.
[5]. Cramer. K.R., Pai S-I, Magneto fluid dynamics for engineers and applied physicists, New York, USA: McGraw-Hill (1973).

Abstract: This paper deals with the estimation of the parameters, reliability and hazard rate functions of the mixture of two Weibull distributions (MTWD), with a common shape parameter, based on the generalized order statistics (GOS). The maximum likelihood and Bayes methods of estimation are used for this purpose with standard errors and credible intervals........

Keywords: Mixture of two Weibull distributions, generalized order statistics (GOS ), Maximum likelihood estimation, Bayesian estimation, Markov Chain Monte Carlo

[1]. B.S.Everitt and D.J. Hand,Finite Mixture Distributions(University Press, Cambridge,1981).
[2]. D.M. Titterington, A.F.M.Smith and U.E. Makov,Statistical Analysis of Finite Mixture Distributions. (Wiley, New York, 1985).
[3]. S.A. Adham, On finite mixture of two-component Gompertz lifetime model, Master's Thesis, King Abdulaziz University, Jeddah, Saudi Arabia,1996.
[4]. G.J.McLachlan and K.E. Basford,Mixture Models( Inferences and Applications to Clustering. Marcel Dekker, New York,1988)
[5]. A.S. Papadapoulos, and W.J.Padgett, (1986). On Bayes estimation for mixtures of two exponential-life-distributions from right-censored samples. IEEE Trans. Relia., Vol. 35(1),1986, 102-105.

Abstract: The unsteady flow of an incompressible Mhd free convection flow of Visco-elastic Kuvshinshiki fluid through a porous medium with simultaneous heat and mass transfer near an infinite vertical oscillating porous plate under the influence of uniform transverse magnetic field has been discussed. The governing equations of the flow field are solved by a regular perturbation method for small elastic parameter.......

Keywords:Heat and mass transfer, Mhd flows, porous medium, unsteady flows and visco-elastic fluids.

[1]. Azzam, G. E. A. (2002). Radiation effects on the MHD-mixed free-forced convective flow past a semi-infinite moving vertical plate
for high temperature differences. Phys. Scr., 66, pp. 71–76.
[2]. Bejan A and Khair K R. (1985). Mass Transfer to Natural Convection Boundary Layer Flow Driven by Heat Transfers, ASME J. of
Heat Transfer, Vol. 107, pp. 1979-1981.
[3]. Benenati R F and Brosilow C B Al (1962). Ch. E.J., Vol, 81, pp. 359-361.
[4]. Chamkha, A. J. (2004). Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with
heat absorption. Int. J. Eng. Sci., 42, 217–230.
[5]. Cheng. P and Lau K H (1977). In Proc., 2nd Nation's Symposium Development, Geothermal Resources, pp. 1591-1598.

Abstract: In this paper, we use Chebyshev polynomials method of the first kind of degree n to solve linear Fredholm weakly singular integro- differential equations (LFWSIDEs) of the second kind.This techniques transform the linear Fredholm weakly singular integro-differential equations to a system of a linear algebraic equations .Application are presented to illustrate the efficiency and accuracy of this method.

Keywords: Linear Fredholm integro-differential equations, Weakly singular kernel, Chebyshev polynomials, Trapezodial rule.

[1]. P.J.Davis,P.Rabinowitz, "Methods of Numerical integration", Academic Press,London,1984.
[2]. T.Okayama, T.Matsuo, M.Sugihara,"Sinc-collocation methods for weakly singular Fredholm integral equation of the second kind",J.Comput.Appl.Math.234, pp1211-1227,2010.
[3]. Chen,Y,Tang,T, "Spectral methods for weakly singular Volterra integral equations with smooth solutions",J.Comput.Appl.Math.233,pp 938-950,2009.
[4]. P.Uba, "ACollocation Method With Cubic Splines for multidimensional Weakly Singular Non Linear Integral Equations" ,J.Integral Equations Appl.6,pp 257-266.
[5]. H.Brunner, A.Pedas,G.Vainikko, "Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels" ,SIAM J.Numer.Anal.39,pp 957-982,2001.

Abstract:The harmonic index H(G) of a graph G is defined as the sum of weights ( ) ( ) 2 dG u  dG v of all edges uv of G, where dG(u) is the degree of a vertex u in G. In this paper we obtained the harmonic index of some class of trees and further developed an algorithmic technique to find harmonic index of any graph.

Keywords: Algorithm, Degree of a vertex, Harmonic Index, Tree.

[1]. R. Chang, Y. Zhu, On the harmonic index and the minimum degree of a graph, Romanian J. Inf. Sci. Tech., 15 (2012), 335-343.
[2]. H. Deng, S. Balachandran, S. K. Ayyaswamy, Y. B. Venkatakrishnan, On the harmonic index and the chromatic number of a graph,
Discrete Appl. Math., 161 (2013), 2740-2744.
[3]. O. Favaron, M. Maheo, J. F. Sacle, Some eigenvalue properties in graphs (conjectures of Graffiti -II), Discrete Math., 111 (1993),
197-220.
[4]. I. Gutman, O. E Polansky. (1986) Mathematical Concepts in Organic Chemistry, Springer – Verlag, Berlin
[5]. Y. Hu, X. Zhou, On the harmonic index of the unicyclic and bicyclic graphs, Wseas Tran. Math.,12 (2013), 716-726.

Abstract:In the pr esent paper ,we prove exi stence of f ixed point and cont ract ion mapping in Hi lber t space s by i retates .

Keywords: Hi lber t space, f ixed point ,Cont ract ion,Cauchy sequence

[ 1 ] . Browd e r ,F.E. :Fix ed p oin t th eor ems f or n on l in ea r s emi con t r a c t iv e map p in gs in Ban a ch sp a c e ,Ar ch ,Ra t ,Mech , An a l , 2 1 , 2 5 9 -2 6 9 , (1 96 5 - 66) .
[ 2 ] . 2 . Browd er ,F.E. an d Pet r ysh yn W.V. : Con t r a c t i on o f f i x ed p oin t s of n on l in ea r map p in gs in Hi lb er t sp a c e , J .Ma th . An l . Ap p l .2 0 ,1 9 7 -2 2 8 , (19 6 7 ) .

[3]. Hichs,T.L.and Huffman,Ed.W. : Fixed point theorems of generalized Hilbert space , J.Math Anal,Appl ,64 (1978).
[4]. Jungck G. : Compatible mappings and common fixed points ,Internet J. Math. and Math. Sci. 9 (4) (1986), 771-779.

Abstract: Dedicated to research conditions of existence of solution quasi-linear differential equations with measurable coefficients, in other words, we study limitations imposed on the nonlinearity of equation or system and under the condition that system will have a unique solution and that solution belong to certain class of functions........

Keywords: Quasi-linear differential equation, elliptical equation , monotone weakly compact operators, weak solution

[1]. Вишик М.Й. О сильно эллиптических системах дифференциальных уравнений / М.Й. Вишик // Матем. сб. - 1951. - Т.
29(71), №3. - С. 615-676.
[2]. Kato T. Perturbation theory for linear operators / T. Kato. – Berlin-Heidelberg-New York: Springer-Verlag, 1980. – 578 p.
[3]. Kato T. Non-linear semigroups and evolution equations / T. Kato // J. Math. Soc. Japan. – 1967. – V. 19. – P. 508 – 520.
[4]. Komura Y. Differentiability of nonlinear semigroups / Y. Komura // J. Math. Soc. Japan. –1969. – V. 21. – P. 375–402.
[5]. Komura Y. Nonlinear semi-groups in Hilbert space / Y. Komura // J. Math. Soc. Japan. – 1967. – V. 19. – P. 493 – 507.