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