Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.47, No.1, pp. 1-22, 2009, DOI:10.3970/cmes.2009.047.001

Abstract In this paper an analytical method for approximating the solution of backward heat conduction problem is presented. The Fourier series expansion technique is used to formulate a first-kind Fredholm integral equation for the temperature field u(x,t) at any time t < T, when the data are specified at a final time T. Then we consider a direct regularization, instead of the Tikhonov regularization, by adding the term αu(x,t) to obtain a second-kind Fredholm integral equation. The termwise separable property of kernel function allows us by transforming it to a two-point boundary value problem, and thus a closed-form solution More >

Chih-Wen Chang¹, Chein-Shan Liu², Jiang-Ren Chang¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.2, pp. 69-80, 2007, DOI:10.3970/icces.2007.003.069

Abstract By using a quasi-boundary regularization we can formulate a two-point boundary value problem of the backward heat conduction equation. The ill-posed problem is analyzed by using the semi-discretization numerical schemes. Then, the resulting ordinary differential equations in the discretized space are numerically integrated towards the time direction by the Lie-group shooting method to find the unknown initial conditions. The key point is based on the erection of a one-step Lie group element G(T) and the formation of a generalized mid-point Lie group element G(r). Then, by imposing G(T) = G(r) we can seek the missing More >

C.-S. Liu¹, C.-W. Chang², J.-R. Chang²

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.1, pp. 67-82, 2006, DOI:10.3970/cmes.2006.012.067

Abstract In this paper we are concerned with the backward problems governed by differential equations. It is a first time that we can construct a backward time dynamics on the past cone, such that an augmented dynamical system of the Lie type X^˙ = B(X,t)X with t ∈ R⁻, X ∈ Mⁿ⁺¹ lying on the past cone and B ∈ so(n,1), was derived for the backward differential equations system x^· =f(x,t), t ∈ R⁻, x ∈ Rⁿ. These two differential equations systems are mathematically equivalent. Then we apply the backward group preserving scheme (BGPS), which is an explicit single-step algorithm… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.1, pp. 55-66, 2006, DOI:10.3970/cmes.2006.012.055

Abstract In this paper we are concerned with the numerical integration of Burgers equation backward in time. We construct a one-step backward group preserving scheme (BGPS) for the semi-discretization of Burgers equation. The one-step BGPS is very effectively to calculate the solution at an initial time t = 0 from a given final data at t = T, which with a time stepsize equal to T and with a suitable grid length produces a highly accurate solution never seen before. Under noisy final data the BGPS is also robust to against the disturbance. When the solution appears steep gradient, More >

C.-W. Chang¹, C.-S. Liu², J.-R. Chang^1,3

CMES-Computer Modeling in Engineering & Sciences, Vol.10, No.1, pp. 13-38, 2005, DOI:10.3970/cmes.2005.010.013

Abstract In this paper, the inverse heat conduction problem governed by sideways heat equation is investigated numerically. The problem is ill-posed because the solution, if it exists, does not depend continuously on the data. To begin with, this ill-posed problem is analyzed by considering the stability of the semi-discretization numerical schemes. Then the resulting ordinary differential equations at the discretized times are numerically integrated towards the spatial direction by the group preserving scheme, and the stable range of the index r = 1/2ν Δt is investigated. When the numerical results are compared with exact solutions, it More >