%0 Journal Article
%T A Nonlinear Shooting Method and Its Application to Nonlinear Rayleigh-B¨¦nard Convection
%A Jitender Singh
%J ISRN Mathematical Physics
%D 2013
%R 10.1155/2013/650208
%X The simple shooting method is revisited in order to solve nonlinear two-point BVP numerically. The BVP of the type is considered where components of are known at one of the boundaries and components of are specified at the other boundary. The map is assumed to be smooth and satisfies the Lipschitz condition. The two-point BVP is transformed into a system of nonlinear algebraic equations in several variables which, is solved numerically using the Newton method. Unlike the one-dimensional case, the Newton method does not always have quadratic convergence in general. However, we prove that the rate of convergence of the Newton iterative scheme associated with the BVPs of present type is at least quadratic. This indeed justifies and generalizes the shooting method of Ha (2001) to the BVPs arising in the higher order nonlinear ODEs. With at least quadratic convergence of Newton's method, an explicit application in solving nonlinear Rayleigh-B¨¦nard convection in a horizontal fluid layer heated from the below is discussed where rapid convergence in nonlinear shooting essentially plays an important role. 1. Introduction Let ,£¿£¿ , be a vector valued function defined by , where each map , , is smooth over the interval . We consider the following vector differential equation satisfied by : Throughout, the map is assumed to be smooth and satisfying the Lipschitz condition on a closed rectangle with a Lipschitz constant s.t. for all£¿£¿ and ; furthermore, the components of are assumed to satisfy initial conditions at first boundary given by and -conditions at the other boundary which are given by where is a permutation of the symmetric group . Equations (1)¨C(4) lead to a two-point boundary value problem whose solution is not known a priori at either of the boundaries. Such BVPs are a common object of study in mathematics, physics, engineering, stochastic analysis, and optimization. In general, it is not possible to solve these BVPs analytically, and one needs to look for their numerical solutions in order to unfold the inherent dynamics. To do so, the vector differential equation may be reduced to a set of nonlinear algebraic equations by approximating the solution with a (finite) Galerkin expansion in terms of a suitably chosen set of orthogonal functions already satisfying the conditions of the BVP in hand. The resulting set of the nonlinear algebraic equations is solved numerically for the unknown Galerkin coefficients using iterative methods such as the Newton-Raphson scheme. Despite its general applicability, the Galerkin method becomes handy as the number of
%U http://www.hindawi.com/journals/isrn.mathematical.physics/2013/650208/