RECENT PROGRESS IN THE SOLUTION OF NONLINEAR DIFFUSION EQUATIONS

Important contributions of the late Dr. E. C. Childs included early recognition of the nonlinear diffusion form of the nonhysteretic flow equation for unsaturated nonswelling soils, and of the importance of a predictive system based on the flow equation. The full equation is a nonlinear Fokker-Planck equation, and reduces to a diffusion equation when gravity may be neglected. Equations of the same form describe water movement and volume change in swelling soils. This paper reviews recent work on quasianalytical and analytical methods of solving these equations. Quasilinear solutions of the steady equation in two and three dimensions have received much attention. The extension to heterogeneous soils can be taken further to include sloping stratification. Theorems enable solutions for surface sources, and sources at arbitrary finite depth, to be deduced from mathematically simpler solutions in a region extending infinitely in all directions. A generalization to the case of sloping soil surface is given. In many real-world problems where similarity methods do not apply, integral methods offer the same possibility of effectively reducing the number of independent variables. Green and Ampt introduced a simple integral method; and more sophisticated methods have entered soil-water studies through the work of Parlange. Recent studies reveal the importance of preserving integral continuity in integral methods involving iteration. For diffusivity D = a(b - 0)−2, all problems of one-dimensional nonlinear diffusion subject to arbitrary initial conditions plus a flux boundary condition can be linearized. Exact solutions are therefore available for various problems, some of them in explicit form.