Harinakshi Karkera^a , Nagaraj N. Katagi^a,† , Ramesh B. Kudenatti^b

Frontiers in Heat and Mass Transfer, Vol.18, pp. 1-10, 2022, DOI:10.5098/hmt.18.41

Abstract In this paper, we study the characteristics of laminar boundary-layer flow of a viscous incompressible fluid over a moving wedge. The transformed boundary-layer equation given by the Falkner-Skan equation is solved by an efficient easy-to-use approximate method based on uniform Haar wavelets in conjunction with quasilinearization and collocation approach. The residual and error estimates are computed to confirm the validity of the obtained results. A meaningful comparison between the present solutions with existing numerical results in the literature is carried out to highlight the benefits and efficiency of proposed method. Furthermore, the influence of variable pressure gradient and the wedge… More >

Chein-Shan Liu¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.5, No.2, pp. 55-84, 2008, DOI:10.3970/icces.2008.005.055

Abstract The present paper provides a Lie-group shooting method for the numerical solutions of second-order nonlinear boundary value problems exhibiting multiple solutions. It aims to find all solutions as easy as possible. The boundary conditions considered are classified into four types, namely the Dirichlet, the first Robin, the second Robin and the Neumann. The two Robin type problems are transformed into a canonical one by using the technique of symmetric extension of the governing equations. The Lie-group shooting method is very effective to search unknown initial condition through a weighting factor r(0,1). Furthermore, the closed-form solutions are derived to calculate the… More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.13, No.2, pp. 149-164, 2006, DOI:10.3970/cmes.2006.013.149

Abstract The present paper provides a Lie-group shooting method for the numerical solutions of second order nonlinear boundary value problems exhibiting multiple solutions. It aims to find all solutions as easy as possible. The boundary conditions considered are classified into four types, namely the Dirichlet, the first Robin, the second Robin and the Neumann. The two Robin type problems are transformed into a canonical one by using the technique of symmetric extension of the governing equations. The Lie-group shooting method is very effective to search unknown initial condition through a weighting factor r ∈ (0,1) Furthermore, the closed-form solutions are derived… More >

Chein-Shan Liu¹, Weichung Yeih², Satya N. Atluri³

CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.3, pp. 301-324, 2010, DOI:10.3970/cmes.2010.059.301

Abstract When problems in engineering and science are discretized, algebraic equations appear naturally. In a recent paper by Liu and Atluri, non-linear algebraic equations (NAEs) were transformed into a nonlinear system of ODEs, which were then integrated by a method labelled as the Fictitious Time Integration Method (FTIM). In this paper, the FTIM is enhanced, by using the concept of arepellorin the theory ofnonlinear dynamical systems, to situations where multiple-solutions exist. We label this enhanced method as MSFTIM. MSFTIM is applied and illustrated in this paper through solving boundary-layer problems, boundary-value problems, and eigenvalue problems with multiple solutions. More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.30, No.1, pp. 1-16, 2008, DOI:10.3970/cmes.2008.030.001

Abstract In this paper we propose a new numerical integration method of second-order boundary value problems (BVPs) resulting from the elastica of slender rods under different loading conditions and boundary conditions. We construct a compact space shooting method for finding unknown initial conditions. The key point is based on the construction of a one-step Lie group element G(T) and the establishment of a generalized mid-point Lie group element G(r) by using the mean value theorem. Then, by imposing G(T) = G(r) we can search the missing initial condition through a closed-form solution in terms of the weighting factor r ∈ (0,1).… More >