iosr-Jm   Volume-12 ~ Issue-2 ~ Mar-Apr 2016

Abstract:In this papera new type of neural networks for the derivation of fourth order Runge-Kutta Method which involves tedious computation of many unknowns and its details.Its analysis can hardly be found in many literatures due to the vital role played by the method in the field of computation and applied science.

Keywords: Fourth order Rungekutta Method, Neural Network, Derivation, Analysis.

[1]. M.K. Jain,S .R. K.Iyengar, R. K. JainNumerical Methods for Scientific and Engineering computing,(2007).
[2]. J.D.Lambert , NumericalMethods for Ordinary Differential Systems, the initial value problem, John Wiley & Sons Ltd. , (1991).
[3]. J.C. Butcher, Numerical methods for Ordinary Differential Equations, John Wiley & Sons Ltd., (2003)
[4]. John R. Dorman , Numerical Methods for differential Equations, a Computational Approach, CRC Press, Inc. , (1996).
[5]. J .D. Lambert, Computational Methods in Ordinary Differential Equations, John Wiley & Sons Ltd., (1973).

Abstract:In this paper, we present a reliable iterative method proposed by Daftardar-Gejji and Hossein Jafari namely (DJM) for solving linear and nonlinear two-point boundary value problems. In comparing this method with the other methods proved it is easy in accessing to approximate solution and fast convergent series with easily computable components. Numerical examples are given to demonstrate the efficiency of the proposed method. The calculations in these examples are solved using the software MATHEMATICA® 10.0.

Keywords:Iterative method, Two-point boundary value problem, Numerical solution.

[1] C.Chun, R. Sakthivel, Homotopy pertubation for solving two-point boundary value problem-comparison with other method, Computer physics communications,181 (2010) 1021-1024.
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[3] A. G. Deacon, S Osher, Finite –element method for a boundary-value problem of mixed type, SIAM J. Numer. Anal. 16 (1979) 756-778.
[4] S. M. Roberts, J. S. Shipman, Two Point Boundary Value Problems: Shooting Methods, American Elsevier, New York, 1972.
[5] T. Y. Na, Computatinnonal Methods in Engineering Boundary Value Problems, Academic, New York, 1979.

Abstract:Approximate solutions of stationary exterior fields of Einstein's equations are obtained by expanding the metric in powers of a certain parameter and solving explicitly the first few orders in terms of two harmonic functions. Earlier approximate solutions up to third order were found. In the present paper we obtain the new fourth and fifth order equations and find their approximate solutions for the particular choice of the harmonic functions. There is a physical interpretation of the approximate solutions at the end of the paper.
Key word:Einstein's Field Equations, Approximate Solutions, Asymptotically Flat Solutions.

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Abstract:By improving the classical Lotka-Volterra model, a reasonable dynamic model is established for the unilateral fish which cannot survive independently. We combine the established model with the stability theory of differential equations to obtain the equilibrium point of the dynamic model about fish mutualism, and analyze the locally stability of the equilibrium point.By constructing Lyapunov function further, we try to analyze the global asymptotical stability of the equilibrium point, and give the corresponding explanations in the view of the evolution of the shoals of fish.

Keywords: Global asymptotical stability, dynamic model, the unilateral fish cannot survive independently

[1]. Gaoxiong Wang et al, Ordinary differential equations (Beijing China: Higher Education Press, 2006).
[2]. Zhenshan Lin, Population dynamics (Beijing China: Science Press, 2006).
[3]. Shuang Liu, Study on population dynamics model of biological system (Chongqing China, Chongqing University, MA, 2012).
[4]. Qiyuan Jiang, Jinxing Xie, ye Jun, Mathematical model (Beijing China: Higher Education Press, 2003).
[5]. Shengqiang Liu, Lansun Chen, Population biology model with stage structure (Beijing China: Science Press, 2010).
[6]. Fengde Chen, Xiangdong Xie, Study on dynamics of cooperative population model (Beijing China: Science Press, 2014).