Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.15, No.2, pp. 69-86, 2006, DOI:10.3970/cmes.2006.015.069

Abstract In this paper we will study the numerical integrations of second order boundary value problems under the imposed conditions at t=0 and t=T in a general setting. We can construct a compact space shooting method for finding the unknown initial conditions. The key point is based on the construction of a one-step Lie group element G(u₀,u_T) and the establishment of a mid-point Lie group element G(r). Then, by imposing G(u₀,u_T) = G(r) we can search the missing initial conditions through an iterative solution of the weighting factor r ∈ (0,1). Numerical examples were examined to convince that the new More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.2, pp. 83-108, 2006, DOI:10.3970/cmes.2006.012.083

Abstract The system we consider consists of two parts: a purely algebraic system describing the manifold of constraints and a differential part describing the dynamics on this manifold. For the constrained dynamical problem in its engineering application, it is utmost important to developing numerical methods that can preserve the constraints. We embed the nonlinear dynamical system with dimensions n and with k constraints into a mathematically equivalent n + k-dimensional nonlinear system, which including k integrating factors. Each subsystem of the k independent sets constitutes a Lie type system of X^˙_i = A_iX_i with A_i ∈ so(n_i,1) and n₁ +···+n_k = n.… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.9, No.3, pp. 255-272, 2005, DOI:10.3970/cmes.2005.009.255

Abstract The group-preserving scheme developed by Liu (2001) for calculating the solutions of k-dimensional differential equations system adopted the Cayley transform to formulate the Lie group from its Lie algebra A ∈ so(k,1). In this paper we consider a more effective exponential mapping to derive exp(hA). In order to overcome the difficulty of numerical instabilities encountered by employing group-preserving schemes on stiff differential equations, we further combine the nonstandard finite difference method into the group-preserving schemes to obtain unconditional stable numerical methods. They provide single-step explicit time integrators for stiff differential equations. Several numerical examples are examined, some More >

Hokwon A. Cho¹, M. A. Golberg², A. S. Muleshkov¹, Xin Li¹

CMC-Computers, Materials & Continua, Vol.1, No.1, pp. 1-38, 2004, DOI:10.3970/cmc.2004.001.001

Abstract In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These… More >

N. Mai-Duy, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.4, No.1, pp. 85-102, 2003, DOI:10.3970/cmes.2003.004.085

Abstract This paper reports a mesh-free Indirect Radial Basis Function Network method (IRBFN) using Thin Plate Splines (TPSs) for numerical solution of Differential Equations (DEs) in rectangular and curvilinear coordinates. The adjustable parameters required by the method are the number of centres, their positions and possibly the order of the TPS. The first and second order TPSs which are widely applied in numerical schemes for numerical solution of DEs are employed in this study. The advantage of the TPS over the multiquadric basis function is that the former, with a given order, does not contain the… More >