#### Volume-8 ~ Issue-4

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**Abstract:** The purpose of this paper is to develop the space fractional order explicit finite difference scheme for
fractional order soil moisture diffusion equation with the initial and boundary conditions. We prove that the
solution of the space fractional order finite difference scheme is conditionally stable and the convergence of the
scheme is discussed at the length. Also as an application of this scheme, numerical solution for space fractional
soil moisture diffusion equation is obtained and it is represented graphically by the software 'Mathematica'.

**Keywards:** finite difference, explicit, diffusion equation, soil moisture.

[1] A. P. Bhadane, K. C. Takale, Basic Developments of Fractional Calculus and its Applications, Bulletin of Marathwada
Mathematical Society, Vol. 12, No. 2, p. 1-17 (2011).

[2] Florica MATEL and BUDIU , Differential equations and their applications to the Soil Moisture Study, Bulletin UASVM,
Horticulture 65(2)(2008) .

[3] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).

[4] S. M. Jogdand, K. C. Takale, V. C. Borkar, Fractional Order Finite Difference Scheme For Soil Moisture Diffusion Equation
and its Applications, IOSR Journal of Mathematics(IOSR-JM) , Volume 5, pp 12-18, 4 (2013).

[5] Daniel Hillel, Introduction to Soil Physics, Academic Press (1982).

[6] Don Kirkham and W.L. Powers, Advanced Soil Physics, Wiley-Interscience (1971).

[7] F.Liu, P. Zhuang, V. Anh, I, Turner, A Fractional Order Implicit Difference Approximation for the Space-Time Fractional
Diffusion equation , ANZIAM J.47 (EMAC2005), pp. C48-C68:(2006).

[8] Pater A.C., Raats and Martinus TH. Ven Genuchten, Milestones in Soil Physics, J. Soil Science 171,1(2006). 10

[9] I. Podlubny, Fractional Differential equations, Academic Press, San Diago (1999).

[10] S. Shen, F.Liu, Error Analysis of an explicit Finite Difference Approximation for the Space Fractional Diffusion equation with
insulated ends, ANZIAM J.46 (E), pp. C871-C887:(2005).

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Paper Type |
: | Research Paper |

Title |
: | Common Fixed Point Theorems for Generalisation of R-Weak Commutativity |

Country |
: | India |

Authors |
: | T. R. Vijayan |

: | 10.9790/5728-0840913 |

**Abstract:** The main purpose of this paper is to obtain fixed point theorems for R-weak commutativity which
generalizes theorem 1 of R.P.Pant [2].

**Key Words: **and Phrases. Fixed point, coincidence point, compatible maps, non-compatible,
R-weak commuting maps.

[1]. R.P.Pant, R-weak commutativity and common fixed points, Soochow journal of mathematics 25 (1999) 37-42.

[2]. R. P. Pant, Common fixed points of weakly commuting mappings, Math. Student, 62(1993), 97-102.

[3]. J. Jachymski, Common fixed point theorems for some families of maps, Indian J. Pure Appl. Math., 25(1994), 925-937.

[4]. G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 9(1986), 771-779.

[5]. R. P. Pant, Common fixed points of noncommuting mappings, J. Math. Anal. Appl.,

[6]. 188 (1994), 436-440.

[7]. R. P. Pant, Common fixed points of sequences of mappings, Ganita, 47(1996),43-49.

[8]. G. Jungck, Compatible mappings and common fixed points, Internat. J. Math. Math. Sci., 9 (1986), 771-779.

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Paper Type |
: | Research Paper |

Title |
: | Solitons Solutions to Nonlinear Partial Differential Equations by the Tanh Method |

Country |
: | Iraq |

Authors |
: | YusurSuhail Ali |

: | 10.9790/5728-0841419 |

**Abstract:** The Tanh method is implemented for the exact solutions of some different kinds of nonlinear partial differential equations. New solutions for nonlinear equations such as Benjamin-Bona-Mahony (BBM) equation, Gardner equation ,Cassama-Holm equation, and two component Kdv evolutionary system are obtained.

**Keywords: **Tanh method, Benjamin-Bona-Mahony (BBM) equation, Gardner equation , Cassama-Holm equation

[1] W. Malfliet, The tanh method: a tool for solving certain classes of nonlinear evolution and wave equations, Journal of Computational and Applied Mathematics 164–165, 2004, 529–541.

[2] W. Malfliet, Solitary wave solutions of nonlinear wave equations, American Journal of Physics 60, 1992, 650–654.

[3] Abdul-Majid Wazwaz, Partial Differential Equations and Solitary Waves Theory, Springer Verlag, New York, NY. USA, 2009.

[4] Abdul-Majid Wazwaz, The Cole-Hopf transformation and multiple soliton solutions for the integrable sixth-order Drinfeld–Sokolov–Satsuma–Hirota equation, Applied Mathematics and Computation. 207, (1), 2009, 248–255. [5] Anwar Ja'afar Mohamad Jawad , Marko D. Petkovic, and Anjan Biswas, Soliton solutions of Burgers equations and perturbed Burgers equation, Applied Mathematics and Computation 216, (11), 2010, 3370–3377.

[6] D.D. Ganji, M. Nourollahi, M. Rostamian, A comparison of variational iteration method with Adomian's decomposition method in some highly nonlinear equations, International Journal of Science & Technology 2, (2), 2007, 179–188.

[7] Anwar Ja'afar Mohamad Jawad , Marko D. Petkovic, and Anjan Biswas , Soliton solutions of a few nonlinear wave equations, Applied Mathematics and Computation 216, (9), 2010, 2649–2658.

[8] Y.C. Hon, E.G. Fan, A series of new solutions for a complex coupled KdV system, Chaos Solitons Fractals 19, 2004, 515–525.

[9] Y. Ugurlu, D. Kaya, Exact and numerical solutions of generalized Drinfeld–Sokolov equations, Physics Letters A 372, (16) , 2008, 2867–2873.

[10] L. Wu, S. Chen, C. Pang, Traveling wave solutions for generalized Drinfeld–Sokolov equations, Applied Mathematical Modelling33, (11), 2009, 4126–4130.

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Paper Type |
: | Research Paper |

Title |
: | Common Fixed Point Theorems for Sequence of Mappings in Generalisation of Partial Metric Spaces |

Country |
: | India |

Authors |
: | T. R. Vijayan |

: | 10.9790/5728-0842023 |

**Abstract:**The main purpose of this paper is to obtain fixed point theorems for sequence of mappings under
partial metric spaces which generalizes theorem of four authors [5].

**Key Words: **Common fixed point, coincidence point, weakly compatible pair of mappings, partial metric space.

[1] S.G. Matthews, Partial metric topology, in: Proceedings Eighth Summer Conference on General Topology and Applications, in:

Ann. New York Acad. Sci. 728 (1994) 183–197.

[2] S. Oltra, O. Valero, Banach's fixed point theorem for partial metric spaces, Rend. Istit. Mat. Univ. Trieste. 36 (2004) 17 –26.

[3] G. Jungck, B.E. Rhoades, Fixed points for set valued functions without continuity, Indian. J. Pur. Appl. Math. 29 (1998) 227–238.

[4] O. Valero, On Banach fixed point theorems for partial metric spaces, Appl. Gen. Topol. 6 (2) (2005) 229–240.

[5] Ljubomir C´ iric´,Bessem Samet,HassenAydi,Calogero Vetro, Common fixed points of generalized contractions on partial metric

spaces and an application, Applied Mathematics and Computation 218 (2011) 2398-2406.

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Paper Type |
: | Research Paper |

Title |
: | Integral points on the homogeneous cone |

Country |
: | India |

Authors |
: | M. A. Gopalan, B. Sivakami |

: | 10.9790/5728-0842429 |

**Abstract:**The homogeneous cone represented by the ternary quadratic equation 2 2 2 z 3x 6y is analysed for
its non-zero integral solutions. Five different patterns of solutions are illustrated. In each pattern, interesting
relations among the solutions and some special polygonal and pyramidal numbers are exhibited.

**Keywords**: Homogeneous cone, Integral solutions, Polygonal numbers, Pyramidal numbers, Ternary quadratic.

[1] L.E.Dickson, History of theory of numbers, Vol 2, Chelsea publishing company, New York, 1952.

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

[3] M.A.Gopalan and D.Geetha, Lattice points on the hyperboloid of one sheet x2 6xy y2 6x 2y 5 z2 4 , Impact

J.Sci.Tech. Vol..4, No. 1, 2010,23-32.

[4] M.A.Gopalan and V.Pandichelvi, Integral solutions of Ternary quadratic Equation Z(X Y) 4XY , Impact J.Sci.Tech., Vol.

5(1),2011, pp. 1-6.

[5] M.A.Gopalan and J.Kaliga Rani, On Ternary Quadratic Equation X2 Y2 Z2 8 , Impact J.Sci.Tech., Vol. 5(1), 2011, pp. 39-

43.

[6] M.A.Gopalan, S.Vidhyalakshmi and T.R.Usharani, Integral points on the homogeneous cone 2z2 4xy 8x 4z 2 0 , Global
Journal of Mathematics and Mathematical Sciences, Vol. 2(1), 2012, pp. 61-67.

[7] M.A.Gopalan, S.Vidhyalakshmi, T.R.Usharani and S.Mallika, Integral points on the Homogeneous cone 6z2 3y2 2x2 0 ,

Impact J.Sci.Tech. Vol.6(1), 2012,pp. 7-13.

[8] M.A.Gopalan, S.Vidhyalakshmi and K.Lakshmi, Integral points on the hyperboloid of two sheets 3y2 7x2 z2 21 , Diophantus
J. Math., 1(2), 2012, 99-107.

[9] M.A.Gopalan, S.Vidhyalakshmi and E.Premalatha, On Ternary Quadratic Diophantine Equation x2 3y2 7z2 , Diophantus J.

Math., 1(1), 2012, 51-57.

[10] M.A.Gopalan, S.Vidhyalakshmi and A.Kavitha, Integral points on the homogeneous cone z2 2x2 7y2 , Diophantus J. Math.,

1(2), 2012, 127-136.

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Paper Type |
: | Research Paper |

Title |
: | Fuzzy Semi-Pre-Generalized Super Closed Sets |

Country |
: | India |

Authors |
: | M. K. Mishra, M. Shukla, R. Deepa, B. Ambiga |

: | 10.9790/5728-0843032 |

**Abstract:**In this paper, a new class of sets called fuzzy semi-pre-generalized super closed sets is introduced and its properties are studied and explore some of its properties.

**Keywords : **Fuzzy topology, fuzzy super closure, fuzzy super interior ,fuzzy super closed set, fuzzy super open set, fuzzy super closed set, fuzzy super generalized closed set .

[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 July –December2012.

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Paper Type |
: | Research Paper |

Title |
: | The conditional neighborhood for graph and its algorithm. |

Country |
: | Egypt |

Authors |
: | Mabrok EL- Ghoul, Habiba El-Zohny, Hend El- Morsy |

: | 10.9790/5728-0843341 |

**Abstract:**In this paper we will define the conditional neighborhood for graph and classified the conditions into many types. In each type we will compute the algorithm for graph . We will prove that the neighborhood will be give different neighborhood by different algorithm.

**Keywords:** Neighborhood , Graph , Algorithm. AMS Subject Classification:05C85, 68R10

[1] Casey J,. Exploring Curvature,Wiesbaden, Germany, 1996.

[2] Gray, A. "Drawing Space Curves with Assigned Curvature." in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 222-224, 1997.

[3] Fournier,Jean-Claude,Graph Theory and applications with Exercises and problems,ISTE Ltd,2009.

[4] Susanna S.Epp, Discrete Mathematics With Application, Third Edition,Thomson Learning,Inc,2004.

[5] http://en.wikipedia.org/wiki/Algorithm .

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Paper Type |
: | Research Paper |

Title |
: | Literary Analysis using CUSUM Technique on Bharathiar Writings |

Country |
: | India |

Authors |
: | Arbind Kumar Singh, G. Manimannan |

: | 10.9790/5728-0844250 |

**Abstract:** Stylometry is an attempt to capture the essence of the style of a particular author by reference to a variety of quantitative criteria, usually lexical in the nature, called discriminators, or more succinctly the statistical analysis of literary style. A written passage of any kind can be analysed by the method called CUSUM analysis. This analysis reveals whether an article is written by one person or more than one person. In this paper, an attempt is made to authorship attribution on the basis of CUSUM technique to certain articles written on Indian freedom movement published in the magazine called India. Seven articles written by renowned Tamil poet Bharathiar and another six articles not attributed to any author, but belonging to the same period, are considered for authorship identity in the present study. The three features of writings used in this analysis are (i) the use of the 2, 3 and 4 letter words, (ii) words starting with a vowel and (iii) the third combination of these two together. Among the six unattributed articles, CUSUM analysis establishes that all of the writings are very close to Bharathiar's style. This result supported the claims made by many scholars that these six articles could have been written by Mahakavi Bharathiar (MB).

**Keywords: **Authorship, Stylometry, CUSUM Analysis and Test of divergence.

[1] A. F. Bissell (1995), Weighted Cumulative Sums for Text Analysis using Word Counts, Journal of Royal Statistical Society, Series A, 158, part 3, pp. 525-545.

[2] J. Tweedie, and R. H. Baayen (1996), Lexical constant in stylometry and authorship studies www.cs.queensu.ca/achallc97/papers/s004.html

[3] J. Farrington (1996), How to be literary detective: Authorship Attribution, members.aol.com/qsum/QsumIntroduction.html. [4] Manimannan G. and Bagavandas. M (2001), The authorship attribution: the case Bharathiar, presented at National conference on Mathematical and Applied Statistics, Nagpur University, Nagpur.

[5] Mathew Stephen Herse (2001), Automatic Detection of Plagiarism: An Approach Using the Qsum Method, University of Sheffield. [7] T. McEnery and M, Oakes, Lancaster University, Authorship Identification and Computational Stylometry.

[8] Tom Ashford (2001), Computerised Determination of Disputed Authorship: The CUSUM Method. www.dcs.shef.ac.uk/intranet/teaching/public/projects/archive/ug2001/pdf/u8tja.pdf .

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**Abstract:**In this investigation, we intend to present the influence of the prominent Soret effect on natural
convection boundary-layer flow of a non-Newtonian nanofluid over a vertical cone embedded in a porous
medium. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The
governing partial differential equations are transformed into a set of non-linear equations and solved
numerically using an efficient numerical shooting technique with a fourth-order Runge–Kutta scheme (MATLAB
package). The numerical results for the temperature, volume fraction, and concentration profiles, as well as the
reduced local Nusselt number
1/ 2
x x Nu Ra , the nanoparticle Sherwood number
1/ 2
x x NSh Ra and the
regular Sherwood number
1/ 2
x x Sh Ra are presented through plots which reveal interesting features.
Comparisons with previously published work are performed and excellent agreement is obtained.

**Keywords :** natural convection, non-Newtonian nanofluid, porous medium ,Soret effect.

[1] J.A. Eastman, S.U.S. Choi, S. Li, W. Yu, L.J. Thompson, Anomalously Increased Effective Thermal Conductivities Containing

Copper Nanoparticles, Applied Physics Letters 78 (2001) 718-720.

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Suspensions, Applied Physics Letters 79 (2001) 2252-2254.

[3] J. Buongiorno, Convective transport in nanofluids, ASME J Heat Transfer 128 (2006) 240–250.

[4] S. Kakac, A. Pramuanjaroenkij, Review of convective heat transfer enhancement with nanofluids, Int. J. Heat Mass Transfer 52

(2009) 3187–3196.

[5] A.V. Kuznetsov, D.A. Nield, Natural convective boundary-layer flow of a nanofluid past a vertical plate, Int. J. of Thermal Sciences

49 (2010) 243–247.

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