iosr-Jm   Volume-5 ~ Issue-3

Abstract: IIn this paper we study the method of solution of some stochastic differential equations of first order by using the ito integral and ito formula.
Keywords: stochastic differential equations, ito integral ito formula

1 Bernt Qksendal (2006) :Stochastic differential equations an introduction with application ,sixth edition Springer –Verlag Berlin
Heidelberg New York
2 Bernt Qksendal (1998): Stochastic differential equations an introduction with application, fifth edition Springer –Verlage Berlin
Heidelberg Italy
3 John Kerl (2008): Problems in stochastic differential equations, Springer
4 Protter, p.(1990):stochastic integration and differential equations ,Springer -Verlag
5 Gihman, 1.1., Skorohod, A.V.(1974a):stochastic differential equations ,Springer -Veralg
6 Hui- Hsiung Kuo (2006): introduction to stochastic integration -Springer
7 William, D.(1981)(edior):stochastic integrals –lecture notes in mathematics ,vol.851-Springer –Verlag
8 Karatzas, I., Shereve, S.E. (1991): Brownian motion and stochastic calculus. Second edition.Springer-Verlag

Abstract:In this paper, we established a traveling wave solution by using the proposed Tan-Cot function algorithm for nonlinear partial differential equations. The method is used to obtain new solitary wave solutions for various type of nonlinear partial differential equations such as, the (1+1)-dimensional Ito equation, Pochhammer-Chree (PC) equation, MIKP equation, Konopelchenko and Dubrovsky (KD) system of equations which are the important Soliton equations. Proposed method has been successfully implemented to establish new solitary wave solutions for the nonlinear PDEs.

Keywords: Nonlinear PDEs, Exact Solutions, Tan-Cot function method.

[1] Marwan Alquran, Kamel Al-Khaled, Hasan Ananbeh (2011). New Soliton Solutions for Systems of Nonlinear Evolution Equations
by the Rational Sine-Cosine Method. Studies in Mathematical Sciences, Vol. 3, No. 1, pp. 1-9.
[2] Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations. Am. J. Phys, Vol. 60, No. 7, pp. 650-654.
[3] El-Wakil, S.A, Abdou, M.A. (2007). New exact travelling wave solutions using modified extended tanh-function method, Chaos
Solitons Fractals, Vol. 31, No. 4, pp. 840-852.
[4] Xia, T.C., Li, B. and Zhang, H.Q. (2001). New explicit and exact solutions for the Nizhnik- Novikov-Vesselov equation, Appl. Math. E-Notes, Vol. 1, pp. 139-142.
[5] Inc, M., Ergut, M. (2005). Periodic wave solutions for the generalized shallow water wave equation by the improved Jacobi ell iptic
function method, Appl. Math. E-Notes, Vol. 5, pp. 89-96.
[6] Zhang, Sheng. (2006). The periodic wave solutions for the (2+1) -dimensional Konopelchenko Dubrovsky equations, Chaos Solitons
Fractals, Vol. 30, pp. 1213-1220.
[7] Feng, Z.S. (2002). The first integer method to study the Burgers-Korteweg-de Vries equation, J Phys. A. Math. Gen, Vol. 35, No. 2,
pp. 343-349.
[8] Mitchell A. R. and D. F. Griffiths (1980), The Finite Difference Method in Partial Differential Equations, John Wiley & Sons.
[9] Anwar J. M. Jawad, (2012), New Exact Solutions of Nonlinear Partial Differential Equations Using Tan-Cot Function Method,
Studies in Mathematical Sciences Vol. 5, No. 2, pp. 13-25.
[10] A.M. Wazwaz, Multiple-soliton solutions for the generalized (1+1)-dimensional and the generalized (2+1)-dimensional Ito equations, Appl. Math. Comput. 202 (2008): 840–849.

Abstract: Our main purpose in this project is to help reader find a clear and glaring relationship between linear algebra and differential equations, such that the applications of the former may solve the system of the latter using exponential of a matrix. Applications to linear differential equations on account of eigen values and eigenvectors, diagonalization of n-square matrix using computation of an exponential of a matrix using results and ideas from elementary studies form the core study of our project.
Keywords: matrix,fundamental matrix, ordinary differential equations, systems of ordinary differential equations, eigenvalues and eigenvectors of a matrix, diagonalisation of a matrix, nilpotent matrix, exponential of a matrix

[1] Cleve Moler, Charles Van Loan, Nineteen dubious ways to compute Exponential of a matrix, Twenty five years later, Siam Reviews
Vol 45 No 1 pp. 3-000 (2003 Society for industrial and Applied Mathematics
[2] J. Gallier and D. Xu, Computing Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices (International
journal of robotics and automation Vol 17 no. 4 2002)
[3] Su-Jing Wang,Cheng, Cheng jia, Hui-ling Chen, Chun Guang Zhou, college of computer science and technology, Jilin university,
Chang Chun 130012 P.R china.
[4] M. Arioli, B. Cobenotti and C. Fassino, The Pade method for computing metrix exponential , Lin. Alg. Application , 240 (1996) pp.
111-130
[5] R.B Sidje, explosive, software package for computing matrix exponential. ACM Trans Math software 24 (1998) pp. 130-156.
[6] G.F. Simmons, Differential equations with applications and historical notes (Tata McGraw hill publ ishing company Ltd)
[7] E.A. Coddington, A textbook on ordinary differential equations (Prentice Hall Publications)
[8] Krishnamurthy and others, An introduction to linear algebra (Affiliated East-West Press Pvt. Ltd)
[9] Shantinarayan, A textbook on matrices (S. Chand and company Ltd.)
[10] K.B. Datta, Matrix and linear algebra (Prentice Hall of India Pvt. Ltd.)
[11] M. Rama MohanaRao, Ordinary differential equations, Theory and applications (Affiliated East-West Press Pvt. Ltd)

Abstract: The problem of two-phase MHD flow in an inclined channel contain porous medium has been considered. The differential equations governing the flow and heat transfer are solved analytically in both fluid regions of the channel. Effects of physical parameter such as Pressure parameter, ratio of heights, Hartmann number and ratio of viscosities on the velocity and temperature fields are shown graphically. It was found that the pressure parameter on both the velocity and temperature drop as result of magnetic field and porous material. Results have been presented for a wide range of governing parameters..
Keywords: incline channel, porous medium, electromagnetic force,

[1] Cheng W, Cheng W (1996) mixed convective in a vertical parallel-plate channel partially filled with porous media of high
permeability. Int. J. Heat Mass Transfer 39:1331
[2] Al-Nimr MA, Haddad OM (1999) Fully developed free convection in open-ended vertical channels partially filled with porous
materials J. Porous Media 2:179
[3] Jain R.K and Mahta KN (1962) Laminar hydromagnetic flow in an annulus with porous walls pysics of fluids 5(10) 1207-1211
[4] Bathaiah D. and Venugopal R. (1982) Effects of porous lining of the MHD flow between two concentric rotating cylinders under
the influence of a uniform magnetic field, ActaMechanica, 44, 3-4 141-158
[5] Prasad VR, Takhar HS, Zucco J, Ghosh SK, and Beg OA (2008) Numerical study of hydromagnetic viscous plasma flow with Hall
current effects in rotating porous media 53rd congress Indian society Theoretical and Applied Mechanics, University college of
Engineering, Osmania University, Hyderabad, India December, 147-157
[6] Beg OA Zucco J. and Takhar HS (2009) Unsteadymagnetohydrodynamic Hartmann-Coutte flow with Hall current, ionslip, Viscous
and Joule heating effects: Network numerical solutions communications in Nonlinear science and Numerical Simulation, 14 1082-
1097
[7] J.C Hsieh, TS Chen, BF Armoly, (1993)Mixed convection along a nonisothermal vertical flat plate embedded in a porous medium,
intl. J. Heat Mass Transfer 36(7) 1819-1825
[8] C-H Chen, TS Chen, C-K, Chen, (1996) Non-Darcy mixed convection along nonisothermal vertical surface in porous media, Intl. J.
Heat Mass Transfer 36(6), 1157-1164
[9] Jat, R.N, and SantoshChaudhary (2010) Hydromagnetic flow and Heat transfer on a continuously flow and Heat transfer on a
continuously moving surface. Applied Mathematical Sciences, 4; 2; 65-78

Abstract: The object of this paper is to evaluate an improper integral involving generalized Kampé de Fériet function and then make its application to solve a boundary value problem on heat conduction. Expansion formula for generalized Kampé de Fériet function has also been obtained. A few interesting special cases along with the application of one of them in the radial wave function for the hydrogen like atoms have also been discussed.
Keywords: Generalized Kampé de Fériet function, Multiplication formula for Gamma-function, Heat conduction, Radial wave function.

[1] Jaeger, J.C.: A continuation formula for Appell's function 𝐹1 , J. Lond. Math. Soc., 13, (1938).
[2] Nagel, B., Olsson, P. and Weissglas, P.: Ark. Fys., 23, (1963), 137-143.
[3] Olsson, P.O.M.: A hypergeometric function of two variables of importance in perturbation theory, I,II, Ark. Fys., 30, (1965), 187-
191, 29, 459-465.
[4] Singh, F.: Expansion formulae for Kampé de Fériet function and radial wave function and heat conduction, Def. Sci. Jr., 21, (1970),
p. 265-272.
[5] Bhonsle, B.R.: Heat conduction and Hermite Polynomials, Proc. Nat. Acad. Sci., India, 36, (1966), 356-360
[6] Bajpai, S.D.: Applications of Chebyshev-Hermite polynomials and Fox's 𝐻 -function in heat conduction, Vijnana
ParishaAnusandhan Patrika, 37, (1994), 29-36.
[7] Shrivastava, R. & Mishra, P.P.: Expansion Formulae for Fox's 𝐻- function, radial wave functions, and heat conduction (accepted
for publication in the Journal Reflection des- ERA)
[8] Yang, J.: 𝑈(𝑛 + 1) extensions of some basic hypergeometric series identities, Adv. Stud. Contem. Math., 18, (2009), no. 2, 201-218.
[9] Kampé de Fériet, J.: Heat conduction & Hermite Polynomials, Bull. Calcutta Math. Soc. The Golden Jubilee Commemoration
Volume, (1958), 193-204.
[10] Srivastava, H.M. and Karlsson, P.W.: Multiple Gaussian hypergeometric series, Halsted Press Wiley, New York (1985).

Abstract: Closed-form analytical expressions for the displacement and stresses at an arbitrary point caused by strike-slip line source buried in a homogeneous, isotropic, perfectly elastic half-space with rigid boundary are obtained. These expressions are used, further, to find the expressions for the displacement and stresses caused by vertical as well as horizontal strike-slip line source. The variation of the displacement and stress fields due to vertical strike-slip line source and horizontal strike-slip line source with distance from the fault and depth from the fault is studied numerically.

Keywords – Half-space, rigid boundary, static deformation, strike-slip faulting

[1] Chinnery, M. A., The Deformation of the Ground Around Surface Faults, Bull. Seismol. Soc. Am., vol. 61, 1961, 355-372.
[2] Chugh, S., Madan, D. K. and Singh, K., Static Deformation Due To Non-Uniform Slip Along a Very Long Vertical Strike-Slip
Geological Fault In Two- Phase Medium, Int. J. of Ecological Economics and Statistics, vol. 19, 2010, 47-65.
[3] Kumar, A., Singh, S. and Singh, J., Static Deformation of Two Welded Monoclinic Elastic Half-spaces due to a Long Inclined Strikeslip
Fault, Proc.of Indian Acad. Of Sci-Earth and Planet. Sci., vol. 111, 2002, 125-131.
[4] Madan, D.K. and Garg, N.R., Static Deformation of an Orthotropic Horizontal Elastic Layer Coupling in Different ways to a base
due to a very Long Inclined Strike-slip Fault Embedded in a Layer, Indian J. pure. Appl. Math., vol. 28, 1997, 697-712.
[5] Mahrer, K. D. and Nur, A., Strike-Slip Faulting in a Downward Varying Crust, J. Geophys. Res., vol. 84,1979a, 2296-2302.
[6] Mahrer, K. D. and Nur, A., Static Strike-Slip Faulting in a Horizontally Varying Crust, Bull. Seismol. Soc. Am., vol. 69, 1979b, 975-
1009.
[7] Malik, M., Singh, M. and Singh, J., Static Deformation of a Uniform Half-space with a Rigid Boundary Due to a Vertical Dip- Slip
Line Source, IOSR-J. Math., vol. 4, 2013, 26-37.
[8] Maruyama, T., On Two-dimensional Elastic Dislocations in an Infinite and Semi-infinite Medium, Bull. Earthq. Res. Inst., vol. 44,
1966, 811-871.
[9] McHugh, S. and Johnstone, M., Surface Shear Stress, Strain and Shear Displacement for Screw Dislocations in a Vertical Slab With
Shear Modulus Contrast, Geophys. J. R. astr. Soc., vol. 49, 1977, 715-722.
[10] Mindlin, R.D. and Cheng, D. H., Nuclei of Strain in the Semi-Infinite Solid, J. Appl. Phys., vol. 21, 1950, 926-930

Abstract: This study primarily attempts to investigate if there is any relationship between selected factors namely gender, school factors and academic achievement in Statistics of high school students in Tripoli, Libya. The study specifically answered the following questions 1-Is there any relationship between students' achievement in Statistics and school factors in terms of the Availability and use of school library?

[1] Anderson, L.W. (1987) . "The Classroom Environment Study- Teaching for Learning". Comparative Education Review, 31(i) • 69- 87
[2] Benbow, C.P. and Arjmand, O. (1990) Predictors of High Academic Achievement in Mathematics and Science by Mathematically
Talented Students- A Longitudinal Study. Journal of Educational Psychology. 82 (3) • 430-441.
[3] Bourke, S. (1986) "How Smaller is Better- Some Relationships Between Class Size, Teaching Practices, and Student
Achievement", American Educational Research Journal, .v. 23, no. (4) ,1986, 558-71.
[4] Boyle, G.J. and Start, K.B. (1989) . "Sex differences in the prediction of academic achievement using the children's Motivation
Analysis Test". British Journal of Educational Psychology, 59 • 245-252.
[5] Coleman, J.S., et al (1966) Equality of Educational Opportunity. Washington, D.C. • U.S. Government Printing Office.
[6] Drew, D. and Gray, J. (1990) . "The fifth year examination achievement of black young people in England and Wales". Educational
Research, 32 (2) • 107-117.
[7] Eccleston, G., Borkin, I. and Burraos, A. (1990). "Sex-related differences in an academic performance at G.C.E. (A) level.
Educational Research, 32 (3) • 229-232.
[8] Erickson, G.L. and Erickson, L.J. (1984) . "Females and science achievement- achievement, evidence, explanations and
implications". Science Education, 68(2) • 63-89.
[9] Fennema E. and Sherman J. (1977). Sex-related differences in Mathematics achievement, spatial visualisation, and socio cultural
factors, American Education Research Journal, 14(1) • 51-71.
[10] Gallagher, J.J. (1987). A summary of research in science education. Science Education , 71" 277-284.

Abstract: The purpose of this paper is to give a new type of open function called quasi -open function. Also, we obtain its characterizations and its basic properties. 2000 AMS Subject Classification 54C10;54C08;54C05

Keywords Topological spaces, -open set, -closed set, -interior, -closure, quasi -open function and quasi -closed functions

[1] J. Antony Rex Rodrigo and A. Hana Selvi Jansi,Topological Mappings via h* - sets,International Journal of Science,Engineering and
Technology Research (IJSETR) Volume 2,Issue 1, January 2013.
[2] J. Antony Rex Rodrigo and A. Hana Selvi Jansi, On h* - closed sets in topological spaces.(communicated)
[3] J. Antony Rex Rodrigo, A. Hana Selvi Jansi, and Jessie Theodore - On h* - open sets in topological spaces.(communicated)
[4] K. Balachandran, P. Sundaram and H. Maki. On generalized continuous maps in topological spaces . Mem, Fac. Sci. Kochi Univ.
Ser. A. Math. (1991), 5 – 13.
[5] P. Bhattacharya and B. K. Lahiri, Semi – generalized closed sets in topology, Indian J . Pure. Appl, Math., closed sets in topology,
(1987), 375 – 382.
[6] S.G. Crossley and S. K. Hildebrand Semi – topological properties, Fund. Math., (1972). 233 – 254.
[7] R. Devi, H. Maki and K. Balachandran Semi – generalized homeomorphisms and generalized Semi - homeomorphisms in
topological spaces, Indian J . Pure. Appl, Math., (1995), 271 – 284.
[8] N. Levine.Generalized closed sets in topology, Rend. Circ. Math.. Palermo, (1970), 89 – 96.
[9] N. Levine, Semi – open sets and semi – continuity in topological spaces, Amer.Math.Monthly, (1963), 36-41.
[10] N. Rajesh and E. Ekici , On Quasi - open and Quasi - closed Functions , Bull. Malays.
Math.Sci.Soc.(2)31(20(2008),217-221.

Abstract: In this paper we introduce two spaces of Boehmians each of which contains the dual of a certain space of entire functions. Both these spaces of Boehmians are shown to be isomorphic to each other under the Fractional Fourier transform. We extend the theory of the Fractional Fourier transform on this new Boehmians space. Mathematics Subject Classification: (2000) 44A15, 44A40, 46F12, 44-99.

Key Words and Phrases: Boehmians, Convolution, entire function, Fractional Fourier transform, Tempered distributions

[1] Alieva, T. and Barbe, A.M, Fractional Fourier analysis of objects with scaling symmetry in the book: Fractals in engineering,
ed.J. Lavyvehel, E.luton and C. Tricot, Springer Verlag, (1997), 252-265. Allain, C. and Cloitre, M.: Physics Rew E 33,
(1986), 3566-3569.
[2] Bhosale, B.N., Topics in the Theory of integral Transformations of generalized Functions, Doctoral Thesis, University of
Kolhapur, India (2001).
[3] Bhosale, B.N.and Chaudhary, M.S., Fractional Fourier Transform of Distributions of Compact support, Bull. Cul. Math. Soc.
94, No. 5, (2002), 349-358.
[4] Howell, K.B., A new theory of Fourier analysis, part I, The space of test functions, J. Math. Anal. Appl. 168, (1992), 342-350.
[5] Howell, K.B., A new theory of Fourier analysis, part II, Further analysis on space of test functions, J. Math. Anal. Appl. 173,
(1993), 419-429.
[6] Howell, K.B., A new theory of Fourier analysis, part III, Basic analysis on the dual space, J. Math. Anal. Appl. 175, (1993),
257-267.
[7] Howell, K.B., A new theory of Fourier analysis, part IV, Basic multiplication and Convolution on dual spaces J. Math. Anal.
Appl. 168, (1998), 342-350.
[8] V. Karunakaran, Generalized Integral Transforms, International workshop & seminar on Trans. Techniques & their Appl.,
Kerala, (2001), 13-16.
[9] V. Karunakaran & R. Roopkumar, Ultra Boehmians & their Fourier Transforms, Fractional Calculus & Applied Ana., An
International J. for theory & Appl., Vol. 5, No. 2, (2002), 181-194.
[10] V. Karunakaran & N.V. Kalpakam, Boehmians & Fourier Transform, Integral Transforms & Special Functions, Vol. 9, No.3,
(2000), 197-216.

Abstract: In this paper we are going to define splitting on bigraphs. The purpose of this paper is that however investigation of created graph by splitting is bipartite graph (Bigraph) or not? We show that after of operation of splitting on the one of the vertices of bigraph, the graph is also bigraph. Another rule is proved in this paper, is that if the graph is connected then the resulting bi graph is also connected.

Keywords: bipartite graph, bigraph, splitting, connected graph, matrix, incident matrix, adjacency matrix

[1] H. Azanchiler, Extension of Line-Splitting from graphs to Binary Matroid, Lobachevskii, Journal of Mathematics, volum 24, 2006
3-12
[2] T. T.Raghunathan, Splitting in binary matroids, 1998
[3] K. Ruohonen, Graph Theory, Translation by J.Tamminen, K.G.lee, and R.Piche, 2008
[4] M. M.Shikare, Gh, Azadi, Generalized splitting operation for binary matroids and applications, 1998
[5] M. M. Shikare, Gh, Azadi, Generalization of splitting of operation to binary matroids, 2003
[6] D. B. West, Introduction To Graph Theory, New Delhi-110001, 2009