Y.-S. Fok, One-dimensional infiltration into layered soils, J. Irrig. Drain. Div. 96 (1970), 121–129.Google Scholar
 D. E. Aylor and J.-Y. Parlange, Vertical infiltration into a layered soil, Soil Sci. Soc. Am. J. 37 (1973), 673–676.Google Scholar
 A. Y. Hachum and J. F. Alfaro, Rain infiltration into layered soils: Prediction, J. Irrig. Drain. Div. Am. Soc. Civil Eng. 106 (1980), 311–319.
 Z. Samani, A. Cheraghi, and L. Willardson, Water movement in horizontally layered soils, J. Irrig. Drain. Eng. 115 (1989), 449–456.Google Scholar
 C.-Y. Ku and Y.-H. Tsai, Solving nonlinear problems with singular initial conditions using a perturbed scalar homotopy method, Int. J. Nonlinear Sci. Numer. Simul. 14 (2013), 367–375.Web of ScienceGoogle Scholar
 R. Hanks and S. Bowers, Numerical solution of the moisture flow equation for infiltration into layered soils, Soil Sci. Soc. Am. J. 26 (1962), 530–534.Google Scholar
 F. Whisler and A. Klute, Analysis of infiltration into stratified soil columns Proc Wageningen Symp, Proc., IAHS Symposium on Water in the Unsaturated Flow, Wageningen, The Netherlands, 451–470, 1966.
 P. Moldrup, D. E. Rolston, and L. A. Hansen, Rapid and numerically stable simulation of one dimensional, transient water flow in unsaturated, layered soils, Soil Sci. 1483 (1989), 219–226.Google Scholar
 R. Srivastava and J. T.-C. Yeh, Analytical solutions for one-dimensional, transient infiltration toward the water table in homogeneous and layered soils, Water Resour. Res. 27 (1991), 753–762.Google Scholar
 N. Romano, B. Brunoneb, and A. Santini, Numerical analysis of one-dimensional unsaturated flow in layered soils, Adv. Water Resour. 21 (1998), 315–324.Google Scholar
 C. Corradini, F. Melone, and R. E. Smith, Modeling local infiltration for a two-layered soil under complex rainfall patterns, J. Hydrol. 237 (2000), 58–73.Google Scholar
 R. Leconte and F. P. Brissette, Soil moisture profile model for two-layered soil based on sharp wetting front approach, J. Hydrologic Eng. 62 (2001), 141–149.Google Scholar
 L. A. Richards, Capillary conduction of liquids through porous mediums, J. Appl. Phys. 1 (1931), 318–333.Google Scholar
 B. M. Das, Principles of geotechnical engineering, 7th edn, Cengage Learning, 200 First Stamford Place, Suite 400, Stamford, CT 06902, USA, 2010.
 W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes, the art of scientific computing, Cambridge University Press, Cambridge, 2007.Google Scholar
 C.-S. Liu and C.-W. Chang, Novel methods for ill-conditioned linear equations, J. Mar. Sci. and Technol. 17 (2009), 216–227.Google Scholar
 C. Vuik, A. Segal, and J. A. Meijerinky, An efficient preconditioned CG method for the solution of a class of layered problems with extreme contrasts in the coefficients, J. Computat. Phys. 152 (1999), 385–403.Google Scholar
 C.-S. Liu, Modifications of steepest descent method and conjugate gradient method against noise for ill-posed linear systems, Commun. Numer. Anal. 2012 (2012), 1–24.
 C.-S. Liu, H.-Ki. Hong, and S. N. Atluri, Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems, CMES: Comput. Model. Eng. Sci. 60 (2010), 279–308.Google Scholar
 C.-S. Liu, An optimal multi-vector iterative algorithm in a Krylov subspace for solving the ill-posed linear inverse problems, CMC: Comput. Mater. Continua. 33 (2013a), 175–198.Google Scholar
 C.-S. Liu, A two-side equilibration method to reduce the condition number of an ill-posed linear system, CMES: Comput. Model. Eng. Sci. 91 (2013b), 17–42.Google Scholar
 C.-Y. Ku, W. Yeih, and C.-S. Liu, Dynamical Newton-like methods for solving ill- conditioned systems of nonlinear equations with applications to boundary value problems, CMES: Comput. Model. Eng. Sci. 76 (2011), 83–108.Google Scholar
 M. T. Van Genuchten, A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J. 44 (1980), 892–898.Google Scholar
 W. Gardner, Some steady-state solutions of the unsaturated moisture flow equation with application to evaporation from a water table, Soil Sci. 85 (1958), 228–232.Google Scholar
 A. Warrick, Soil water dynamics, Oxford University Press, New York, 2003.Google Scholar
 C.-Y. Ku, A novel method for solving ill-conditioned systems of linear equations with extreme physical property contrasts, CMES: Comput. Model. Eng. Sci. 96 (2013), 409–434.Google Scholar
 W. H. Green and G. A. Ampt, Studies on soil physics I. The flow of air and water through soils, J. Agric. Sci. 4 (1911), 1–24.Google Scholar
 F. T. Tracy, Analytical and numerical solutions of Richards’ equation with discussions on relative hydraulic conductivity, INTECH Open Access Publisher, Janeza Trdine 9, 51000 Rijeka, Croatia, 2011.
About the article
Published Online: 2015-11-17
Published in Print: 2015-12-01