iosr-Jm   Volume-8 ~ Issue-4

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).

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.

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.

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.