Dual extremum principles for a nonlinear diffusion problem
Maximum and minimum principles for a nonlinear boundary value problem in diffusion with concentration-dependent coefficient D(c) are derived in a unified manner from the theory of dual extremum principles. The results are illustrated by a calculation in the case D(c) = exp c. The purpose of this note is to give a new variational formulation of the nonlinear boundary value problem described by the equation c( 0) = 1, c(oo) = o. (2) This problem arises in the study of diffusion into a semi-infinite medium, where the diffusion coefficient D is a function of the concentration c. The associated flux is given by /(z) = D(c) dc/dz, / < 0, (3) and we shall suppose that D(c) £ C'[0, 1] and that D(c) > 0 for c > 0. Shampine [1] has shown that this kind of problem possesses a unique solution and has discussed in [2] some aspects of an approximate solution due to Macey [3], This latter work formulates the solution c implicitly via an expression for z(c). That is, the dependent and independent variables are interchanged. If we adopt this approach, the problem in (1) to (3) may be written as [2] dz/dc = D(c)/f, z(I) = 0, (4) — df/dc = \z, 0 < c < 1, /(0) = 0. (5) Here z and / are regarded as functions of the independent variable c. To give a variational formulation of the problem we now observe that (4) and (5) are examples of the canonical equations dz/dc — dH/df, (6) -df/dc = dH/dz, (7) with the Hamiltonian H(/, z) given by H(j, z) = D{c) In |/| + i22. (8) We note that d'H/df = -D(e)/f, d2H/dz2 = i (9) Since D(c) > 0, H is concave in / and convex definite in z, which in turn means that we can obtain dual extremum principles [see 4, 5],