iosr-Jm   Volume-7 ~ Issue-4

Abstract: Here an attempt is made to find positive numbers a,b and c such that an+bn=cn. Two cases were considered. an+bn is not divisible by (a+b)2 and an+bn is divisible by (a+b)2. From this a solution for the case n=3 is obtained as (9+5)3 + (9-5)3 = 123. A condition for finding such numbers for any n is also reached.

Key words: Fermat's Last Theorem, Number Theory, Mathematics, Diophantine equation, Pythagorean triples

[1]. Stewart, I. and Tall, D. Algebraic Number Theory . (Chapman & Hall, 1994)
[2]. www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html. Accessed on 02-08-2004. [online].

Abstract: Writing a standard scientific researchable paper be it project, technical report, dissertation or journal that reveal the application of science and technology to millennium challenges seems to be an herculean task among some scholars and students. Based on the above assertion, this paper critically examined the core aspect of a standard paper that is, methodology. Methodology is the framework and live-wire of any scientific paper. Hence, this paper takes a step to x- ray the components of methodology such as research design, study population, operational variables, sampling technique, and sample size determination, method of date collection and method of data processing/analysis with a view to harnessing the components together in their application for science and technology to millennium challenges.

Key words: research design, study population, sample size, data collection, variable(s) and methodology.

[1]. Creswell, J.W (2003). Research Design: Sage publications, London
[2]. Miles, M.B et al (1974). Quantitative Data Analysis 2nd edition: Sage publications, London
[3]. Altman, D.G (1991). Practical Statistics for Medical Research: Chapman and Hall, London.
[4]. Colton, T. (1974). Statistics in Medicine: Boston Little, Brown and Company 9INC.
[5]. Corlien, M.V; Indra Pathmanathan and Ann Brownlee (2003). Designing and conducting health systems research projects: Volume 1 Proposal development and fieldwork: KIT/IDRC
[6]. Degu, G and Tessema, F. Biostatistics for Health Science Students: lecture note series: Department of Community Health, Jimma Institute of Health Sciences.
[7]. Development in primary health care (1998). Ethical considerations in research focus ESTC-EPHA/CDC PROJECT (2004), Training modules on health research.
[8]. Fletcher, M (1992), Principles and Practice of Epidemiology. Department of Community Health, Faculty of medicine, Addis Ababa University.
[9]. Manktelow, B; Hewitt, M.J and Spiers, N (2002). An introduction to Practical Statistics Using SPSS: Trent Focus, Gondar (1995). Manual for field training Jimma (1996). Manual for student research project
[10]. Mathers, Nigel; Howe, Amanda and Hunn, Amanda. Trent focus for research and The Carter Center 9EPHTI), Addis Ababa.

Abstract: The main thrust of this paper is to study the biquadratic equation with four unknowns ( ) 1 2 x  y  z  w  xyzw . We present six different infinite families of positive integral solutions to this equation.

[1] L.E.Dickson, History of Theory of Numbers, Chelsea Publishing company, New York, Vol.11, (1952).

[2] L.J.Mordell, Diophantine equations, Academic Press, London(1969).

[3] Andreescu, T.Andrica, D., An Intrduction to Diophantine Equations, GIL Publishing house, 2002.

[4] Andreescu ,T., A note on the equation x  y  z  xyz 2 ( ) , General Mathematics Vol.10, No.3-4 ,17-22,(2002).

[5] M.A.Gopalan, S.Vidhyalakshmi, A.Kavitha, "observations on x  y  z  xyz 2 ( ) , accepted in IJMSA.

Abstract: The condition for which ) = 0 (2 ) 1 ) ( (2 1) 1 ( 2 =1 2 n=1 n Z n n Z      where Z is a complex number reveals those points Z for which the functions ) (2 ) 1 ) ( (2 1) 1 ( n=1 Z n=1 n Z i n      and ) (2 ) 1 ) ( (2 1) 1 ( n=1 Z n=1 n Z i n      have zeroes.Finally,by direct analysis we can find zeroes of Riemann zeta funtion .

[1] KONRAD KNOPP.THEORY OF FUNCTIONS,5TH Edition.NEW YORK.DOVER PUBLICATIONS.
[2] JOHN B.CONWAY.FUNCTIONS OF ONE COMPLEX VARIABLE,2nd Edition.NAROSA PUBLISHING HOUSE PVT LTD.
[3] J.N.SHARMA.FUNCTIONS OF A COMPLEX VARIABLE,4th Edition.Krishna Prakashan Media(P) Ltd.

Abstract: Patterns of Pythagorean triangle, where, in each of which either a leg or the hypotenuse is a pentagonal pyramidal number and Centered hexagonal pyramidal number, in turn are presented.

Keywords: Pythagorean triangles, pentagonal pyramidal,centered hexagonal pyramidal.

[1]. L.E.Dickson, History of Theory of Numbers, Vol.2, Chelsea Publishing Company,New York, 1971.

[2]. L.J.Mordell, Diophantine Equations, Academic Press, New York, 1969.

[3]. S.B.Malik, "Basic Number Theory", Vikas Publishing House Pvt. Limited,New Delhi,1998.

[4]. B.L.Bhatia and Suriya Mohanty,Nasty Numbers and their Characterization, Mathematical Education,1985 pp.34-37.

[5]. M.A.Gopalan and S.Devibala, Pythagorean Triangle: A Treasure House, Proceeding of the KMA National Seminar on Algebra,Number Theory and Applications to Coding and Cryptanalysis, Little Flower College, Guruvayur.Sep.(2004) 16-18.

[6]. M.A.Gopalan and R.Anbuselvi,A Special Pythagorean Triangle, Acta Cienia Indica,XXXI M (1) (2005) pp.53.

[7]. M.A.Gopalan and S.Devibala, On a Pythagorean Problem,Acta Cienia Indica,XXXIM(4) (2006) pp.1451.

[8]. M.A.Gopalan and S.Leelavathi,Pythagorean Triangle with Area/Perimeter as a Square Integer, International Journal of Mathematics, Computer Science and Information Technology 1(2) (2008) 199-204.

[9]. M.A.Gopalan and S.Leelavathi,Pythagorean Triangle with 2Area/Perimeter as a Cubic Integer,Bulletin of Pure and Applied Sciences 26 E(2) (2007) 197-200.

[10]. M.A.Gopalan and A.Gnanam,Pairs of Pythagorean Triangles with Equal Perimeters,Impact J.Sci.Tech 1(2) (2007) 67-70.

Abstract: The multinomial distribution has found applications in many practical situations that deal with a discrete random variable with k possibilities (k > 2). Interval estimation for the proportion parameter in a special case k = 2, the binomial distribution has been studied extensively in literature. However for k > 2, studies have focused the performance of estimation procedure mainly on coverage probabilities and yet there are other important aspects such as propensity of aberration in the limits of confidence intervals and computational issues. The present paper makes an attempt to look beyond coverage probabilities by marshalling the existing procedures in classical and Bayesian approaches for k > 2. To alleviate the computational issues, a comprehensive R program is also made available to facilitate the implementation of the procedures in both classical and Bayesian statistical paradigms.

Keywords: Aberrations, Bayesian, Contingency Tables, Multinomial, Zero Width Intervals.

[1] M. Subbiah, Bayesian inference on sparseness in 2 × 2 contingency tables and its applications, Unpublished Ph.d Thesis., University of Madras, Chennai, 2009.

[2] R.G. Newcombe, Interval estimation for the difference between independent proportions: Comparison of eleven methods in Statistics in Medicine, 17 (1998) 873-890. Mathematical Analysis of propensity of aberration on the methods for interval estimation of the

[3] J.M. Sweeting, J.A. Sutton, & C.P. Lambert, What to add to nothing? Use and avoidance of continuity corrections in meta-analysis of sparse data. Statistics in Medicine, 23 (2004) 1351 –1375.

[4] C.P. Queensberry, & D.C. Hurst, Large Sample Simultaneous Confidence Intervals for Multinational Proportions, Technometrics, 6 (1964) 191-195.

[5] L.A. Goodman, On Simultaneous Confidence Intervals for Multinomial Proportions, Technometrics, 7 (1965) 247-254.

[6] S. Fitzpatrick & A. Scott, Quick simultaneous confidence interval for multinomial proportions, Journal of American Statistical Association, 82 (1987) 875-878.

[7] P.C. Sison & J. Glaz, Simultaneous Confidence Intervals and Sample Size Determination for Multinomial Proportions, Journal of the American Statistical Association, 90 (1995) 366-369.

[8] J. Glaz & P.C. Sison, Simultaneous confidence interval for multinomial proportions, Journal of Statistical planning and inference, 82 (1999) 251-262.

[9] M. Jhun & H.C. Jeong, Applications of bootstrap methods for categorical data analysis, Computational Statistics & Data Analysis, 35 (2000) 83-91.

Abstract: A double acceptance sampling plan for the life test truncated at the pre-assigned time to decide on acceptance or rejection of the submitted lots is considered. This double acceptance sampling plan uses only zero and one failure scheme. The probability model of the lifetime of the product is specified as Kumaraswamy-log-logistic distribution which contains several sub models. The minimum sample sizes of the first and second samples necessary to ensure the specified median life are obtained at the given consumer's confidence level. The operating characteristic values are analyzed with various ratios of the true median lifetime to the specified lifetime of the product. The minimum ratios of the median life to the specified life are also obtained so as to lower the producer's risk at the specified level. Numerical examples are provided to explain the application of sampling plan.

Keywords: Consumer's confidence level; double sampling plan; Kumaraswamy-log-logistic distribution; Median life; Poisson distribution.

[1] M. Aslam, and C.H. Jun, A double acceptance sampling plan for generalized log-logistic distributions with known shape parameters, Journal of Applied Statistics, 37(3), 2010, 405-414.
[2] G. M. Cordeiro, and M. de Castro, A new family of generalized distributions, Journal of Statistical Computation and Simulation, 81, 2011, 883-898.
[3] A.J. Duncan, Quality Control and Industrial Statistics, 5th ed., Richard D. Irwin, Homewood, Illinois, 1986.
[4] B. Epstein, Truncated life tests in the exponential case, Ann. Math. Statist. 25, 1954, 555-564.
[5] H.P. Goode, and J.H.K. Kao, Sampling plans based on the Weibull distribution, In Proceeding of the Seventh National Symposium on Reliability and Quality Control, Philadelphia, 1961, 24-40.
[6] S.S. Gupta, Life test sampling plans for normal and lognormal distributions, Technometrics 4, 1962, 151-175.
[7] S.S. Gupta, and P.A. Groll, Gamma distribution in acceptance sampling based on life tests, J.Am. Statist. Assoc. 56, 1961, 942-970.
[8] P. Kumaraswamy, A generalized probability density function for double-bounded random processes, Journal of Hydrology 46, 1980, 79-88.

Abstract: In this paper we prove the theorems on absolute weighted mean | , |k A  -summability of orthogonal series. These theorems are generalize results of Kransniqi [1].

Keywords: Orthogonal Series, Nörlund Matrix, Summability.

[1] Kransniqi, Xhevat Z.; On absolute weighted mean summability of orthogonal series, Slecuk J. App. Math. Vol. 12 (2011) pp 63-70.
[2] Nantason, I.P.; Theory of functions of a real variable (2 vols), Frederick Ungar, New York 1955, 1961.
[3] Okuyama, Y.; On the absolute generalized, Nörlund summability of orthogonal series, Tamkang J. Math. Vol. 33, No. 2, (2002) pp.
161-165.
[4] Sarigöl, M.A.; On absolute weighted mean summability methods, Proc. Amer. Math. Soc. Vol. 115 (1992).
[5] Sarigöl, M.A.; On some absolute summability methods, Proc. Amer. Math. Soc. Vol. 83 (1991), 421-426.

Abstract: In this paper, we attempt to present a short argument, different from that of the original proofs by that of Hawking, for a theorem stated that no closed timelike curves can exist. In a later paper, we apply this to quantum gravity and relate the curvature of spacetime to this theorem. Also, we present this paper as a preliminary introduction to the complete argument of this, and we also provide a preliminary notion of the concepts which will be narrated in the later papers. We also use this as a starting basis for a true theory of everything for a theory of everything. We use the notation of [1] and of [2].

Keywords: Theoretical Physics, Quantum Gravity, Mathematical Physics, General Relativity, Quantum Mechanics

[1] de A. Gomes, Henrique, The Dynamics of Shapes, Ph.D., University of Nottingham, 2011
[2] R.Penrose, The Road to Reality (Vintage Books, 2004).

Abstract: This paper proposes to develop a ties adjusted or corrected approach to the usual Mann-Whitney U test that structurally makes provision for the correction of the U statistic and its variance for the possible presence of ties observation between the two sampled populations. The modification makes it unnecessary as is the case with the ordinary Mann-Whitney U test for the populations to be continuous with this method data analysis may precede without any problem when the populations are measurements on as low as the ordinal scale and may not be numerical. The method which is based on both magnitude and direction enables the estimation of the probabilities that a randomly selected subject from one population performs or scores higher, as well as or lower than a randomly selected subject from the other population. The procedure is illustrated with some sample data and shown to be at least as powerful as the modified median test and the usual Mann-Whitney U test.

[1]. Argesti A.(1992).Analysis of Ordinal Paired Comparison data. Appl Statist 41:287-297.
[2]. Gibbons, JD (1971). Nonparametric Statistical Inferences. McGraw Hill, New York.
[3]. Hay, William (1973).Statistics for the social sciences: Holt, Rinchart and Winston Inc New York, PP 778-780.
[4]. Munzel U, Brunner E.(2002).An exact paired rank test. Biometrical Journal 44:584-593.
[5]. Oyeka I.C.A (2009): An Introduction to Applied Statistical Methods, Nobern Avocation Publishing Company, pp 530-533.
[6]. Okeh,UM.(2009).Statistical analysis of the application of Wilcoxon and Mann-Whitney U test in medical research studies. Biotechnology and Molecular Biology Reviews Vol. 3 (5), pp. 105-110.
[7]. Oyeka I.C.A and Okeh U.M.(2012). Intrinsically Ties Adjusted Sign Test by Ranks. J Biom Biostat 2012, 3:8
[8]. Spiegel M.R (1998): Theory and Problems of Statistics, McGraw-Hill Book Company, New York. Pp 261-264.

Abstract: This paper proposes a generalized and structured method for use for rank determination in rank-order statistics. The sampled populations may be measurements on as low as the ordinal scale and need not be continuous or numeric. The proposed method would readily enable the researcher assign ranks to sample observations without the need to first arrange the observations in some form as is often the case with the traditional approach in the ranking of observations. The method also provides expressions that are intrinsically and structurally formulated to enable one easily break ties and assign appropriate ranks to any tied observations if need be in a straight forward manner. The proposed method is illustrated with some sample data and shown to be often easier to use in practice than the traditional method and of more generalized and wider applicability than some other existing formulations which are often limited in their use.

Keywords: ordinal scale, positive integers, rank order statistics, intrinsically and structurally

[1]. Gibbons, J. D.: Non- Parametric Statistical. An Introduction; Newbury Park: Sage Publication 1993
[2]. Freund, R.J.1972.Some observations on regressions with grouped data.Amer.Statist.25(3):29-30.
[3]. Hollander, M. and Wolfe, D.A.(1999): Non-Parametric Statistical Methods (2nd Edition). Wiley Interscience, New York Freund, R.J.1971.Some observations on regressions with grouped data.Amer.Statist.25(3):29-30.
[4]. S. Siegel, "Non-Parametric Statistics for the Behavioral Sciences," McGraw-Hill Series in Psychology, New York,1956
[5]. C. A. Oyeka, C. E. Utazi,C.R.Nwosu,P.A.Ikpegbu G. U. Ebuh, H. O. Ilouno and C. C. Nwankwo, "Method of Analysing Paired Data Intrinsically Adjusted for Ties," Global Journal of Mathematics, Vol. 1, No. 1, 2009, pp. 1-6.

Abstract: In this paper we prove a stability theorem for large solutions of three dimensional incompressible Magnetohydrodynamic equations under suitable initial & boundary conditions and appropriate boundedness assumptions on the initial data. We use Sobolev spaces as function space for various quantities.

Keywords: Magnetohydrodynamic equations, stability of solutions, strong solution,Sobolev spaces, generalized Gronwall's inequality.

[1] E.Sanchez Palencia, some results of existence & uniqueness for MHD non- stationary flows, Jour de Mecaniqne , Vol.8, No.4 Dec 1969, 509-541.

[2] M.Sermange & R.Temam ,Some mathematical questions related to the MHD equations, Commu.Pure & Appl Math VolXXXVI,635-664.

[3] E.Zeidler , Non-linear Functional Analysis & its Applications (Volume II/A and II/B Springer:1990.)

[4] J.Priede, S. Aleksandrova & S. Molokov: Linear Stability of MHD flow in a perfectly conducting rectangular duct, arXiv: 1111.0036 V2 [physics. Flu- dyn] 6 Sun 2012

[5] G.Ponce, R . Racke, T.C. Sideris, E.S. Titi ,Global stability of large solutions to the 3D Navier Stokes Equations Common, Math. Phys.159,329-341 (1994).

[6] K.J.Ilin & V.A. Vladimirov, Energy principle for MHD flows.

[7] Roger Temam, Navier - Stokes equations Theory & Numerical Analysis.

[8] J.G.Heywood, The Navier Stokes equation on the existence ,regularity &decay of solutions, Indiana Univ.Math.J.29 no.5 (1980) 639-681.

[9] Sever Silvestru Dragomir, Some Gronwall type Inequalities and Applications, November 7 ,2002.

[10] W. Rusin , Navier – Stokes equations, stability & minimal perturbations of global solutions , Feb 2011.

Abstract: Fuzzy Graphs are having numerous applications in problems like Network analysis, Clustering, Pattern Recognition and Neural Networks. The analysis of properties of fuzzy graphs has facilitated the study of many complicated networks like Internet. In this paper we study the structures of complement of many important fuzzy graphs such as Fuzzy cycles, Blocks etc. The complement of fuzzy graphs with a certain structural property is studied in this paper.

Keywords: Fuzzy Relations, Fuzzy Graphs, Complement of fuzzy graph, Fuzzy cycle, Fuzzy Blocks, Connectivity in fuzzy graphs.

[1] Bhutani, K. R., Rosenfeld, A., 2003. Fuzzy end nodes in fuzzy graphs. Information Sciences, 319 – 322.

[2] George.J.Klir, B. Y., 2005. Fuzzy Sets and Fuzzy Logic Theory and Ap- plications. Prentice Hall India.

[3] J.N.Mordeson, P.S.Nair, 2000. Fuzzy Graphs and Fuzzy Hyper graphs.PhysicaVerlag.

[4] Mathew, S., M.S.Sunitha, 2009.Types of arcs in fuzzy graphs. Information Sciences (179), 1760 – 1768.

[5] M.S.Sunitha, A.Vijayakumar, 1999. A characterization of fuzzy trees. Information Sciences (113), 293 – 300.

[6] M.S.Sunitha, A.Vijayakumar, 2002. Complement of a fuzzy graph. Indian Journal Of Pure and Applied Mathematics 9 (33), 1451 – 1464.

[7] M.S.Sunitha, A.Vijayakumar, December 2005. Blocks in fuzzy graphs. Indian Journal Of Fuzzy Mathematics 13, 13–23.

[8] Rosenfeld, 1975. Fuzzy sets and their applications to cognitive and decision processes. Academic Press, 75 – 95.

[9] Zadeh, L., 1965. Fuzzy sets, information and control.8, 338–353. 6

Abstract: The transcendental equation with five unknowns represented by the diophantine equation 4 2 2 2 2 2 2 2 5 x y z w (k 1) R n      is analyzed for its patterns of non-zero distinct integral solutions.

Keywords: Transcendental equation,integral solutions,surd equation M.Sc 2000 mathematics subject classification: 11D99 NOTATIONS tm,n : Polygonal number of rank n with size m m Pn : Pyramidal number of rank n with size m mn Cp : Centered Pyramidal number of rank n with size m

[1] L.E.Dickson,History of Theory of numbers,Vol.2,Chelsea publishing company,Newyork,1952.
[2] L.J.Mordel, Diophantine Equations, Academic press,Newyork,1969.
[3] Bhantia.B.L and Supriya Mohanty '' Nasty numbers and their characterizations'' Mathematical Education,Vol -II,No.1 Pg.34-37,
[1985]
[4] M.A Gopalan, and S.Devibala, ''A remarkable Transcendental Equation'' Antartica.J.Math.3(2), 209-215, (2006).
[5] M.A.Gopalan,V.Pandichelvi ''On transcendental equation
z  3 x  By  3 x  By ''Antartica . J.Math,6(1), 55-
58,(2009).
[6] M.A.Gopalan and J. Kaliga Rani, ''On the Transcendental equation x  g x  y  h y  z  g z ''International
Journal of mathematical sciences,Vol.9, No.1-2, 177-182, Jan-Jun 2010.
[7] M.A.Gopalan and V.Pandichelvi, ''Observations on the Transcendental equation
z  2 x  3 kx y2 '' Diophantus
J.Math.,1(2), 59-68, (2012).
[8] M.A.Gopalan and J.Kaliga Rani, ''On the Transcendental equation x  x  y  y  z  z ''
Diophantus.J.Math.1(1),9-14, 2012.
[9] M.A.Gopalan,Manju Somanath and N.Vanitha, ''On Special Transcendenta Equations''Reflections des ERA-JMS, Vol.7,issue
2,187-192,2012.
[10] V.Pandichelvi, ''An Exclusive Transcendental equations
3 2 2 3 2 2 2 2 x  y  z  w  (k 1)R '' International
Journal of Engineering Sciences and Research Technology,Vol.2,No.2,939-944,2013.

Abstract: This article is focused on the numerical investigation of a steady two-dimensional mixed convection boundary layer flow along a moving semi-infinite vertical plate. The plate is assumed to move with a constant velocity in the direction of the flow. The influences of thermophoresis and the thermal radiation are also included in the analysis. The governing boundary layer equations for the flow, thermal and species concentration fields are transformed into a non-dimensional form by a group of non-similar transformations. The resulting system of differential equations is solved by an implicit finite difference scheme in coupled highly nonlinear partial combination with the quasi-linearization technique. The effects of various parameters on the velocity and species concentration profiles and the wall thermophoretic deposition velocity are presented in terms of graphs. The present results are compared with previously published work and are found to be in excellent agreement.

Keywords: Thermophoresis; mixed convection; parallel free-stream; thermal radiation; finite difference.

[1] S.L. Goren, Thermophoresis of aerosol particles in the laminar boundary layer on a flat surface, Journal of Colloid and Interface
Science 61 (1977) 77–85.
[2] S.A. Gokoglu, D.E. Rosner, Thermophoretically augmented mass transfer rate to solid walls across laminar boundary layers, AIAA
Journal 24 (1986) 172.
[3] H.M. Park, D.E. Rosner, Combined inertial and thermophoretic effects on particle deposition rates in highly loaded systems,
Chemical Engineering Science 44 (1989) 2233–2244.
[4] M.C. Chiou, Effect of thermophoresis on submicron particle deposition from a forced laminar boundary layer flow onto an
isothermal moving plate, Acta Mechanica 129 (1998) 219–229.
[5] V. K. Garg, S. Jayaraj, Thermophoresis of aerosol particles in laminar flow over inclined plates, International Journal of Heat and
Mass Transfer 31 (1988) 875–890.
[6] M. S. Alam, M. M. Rahman, M. A. Sattar, Effects of variable suction and thermophoresis on steady MHD combined free-forced
convective heat and mass transfer flow over a semi-infinite permeable inclined plate in the presence of thermal radiation,
International Journal Thermal Sciences 47 (2008) 758–765.
[7] A. Y. Bakier, R. S. R. Gorla, Effects of thermophoresis and radiation on laminar flow along a semi -infinite vertical plate, 47 (2011)
419-765.
[8] H. Blasius, Grenzchiechten in flussigkeiten rnit kleiner reibung, Z. Math. Phys. 56 (1908) 1- 37.
[9] B. C. Sakiadis, Boundary-layer behavior on continuous solid surfaces, II. The boundary layer behavior on continuous flat surface,
AIChE J 7 (1967) 221 – 225.
[10] F.K. Tsou, E.M. Sparrow, R.J. Goldstein, (1967) Flow and heat transfer in the boundary layer on a continuous moving surface, Int.
J. Heat Mass Transfer 10 (1967) 219-235.

Abstract: Different modifications in the Newton's method with cubic convergence have become popular iterative methods to find the roots of non-linear equations. In this paper, a modified Newton's method for solving a single nonlinear equation is proposed. This method uses harmonic mean while using Simpson's integration rule, thus replacing 𝑓 ′ 𝑥 in the classical Newton's method. The convergence of the proposed method is found to be order three. Numerical examples are provided to compare e the efficiency of the method with few other cubic convergent methods.

Keywords: Non-linear equation, Iterative Methods, Newton's Method, Third order convergence.

[1] M.K. Jain, S.R.K. Iyengar, R.K. Jain, Numerical Methods for Scientific and Engineering Computation, New Age International,6th edition, 2012.

[2] S. Weerakoon. T.G.I. Fernando, A variant of Newton's method with accelerated third-order convergence, Applied Mathematics Letters 13 (8)(2000) 87-93.
[3] V. I. Hasanov, I. G. Ivanov, G. Nedjibov, A new modification of Newton's method, Applied Mathematics and Engineering 27 (2002) 278 -286.

[4] G. Nedzhibov, On a few iterative methods for solving nonlinear equations. Application of Mathematics in Engineering and Economics'28, in: Proceeding of the XXVIII Summer school Sozopol' 02, pp.1-8, Heron press, Sofia, 2002.

[5] M. Frontini, E. Sormoni, Some variants of Newton's method with third order convergence, Applied Mathematics and Computation 140 (2003) 419-426.

[6] A.Y. Ozban, Some new variants of Newton's method, Applied Mathematics Letters 17 (2004) 677-682.

[7] H.H.H. Homeier, On Newton type methods with cubic convergence, Journal of Computational and Applied Mathematics 176 (2005) 425-432.

[8] D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton's method with third order convergence, Applied Mathematics and Computation 183 (2006) 659–684.

[9] K. Jisheng, L.Yitian, W. Xiuhua, Third-order modification of Newton's method, Journal of Computational and Applied Mathematics 205 (2007) 1 – 5.

[10] T. Lukic, N. M. Ralevic, Geometric mean Newton's method for simple and multiple roots, Applied Mathematics Letters 21 (2008) 30–36. [11] O. Y. Ababneh, New Newton's method with