On direct, implicit reductions of a nonlinear diffusion equation with an arbitrary function - generalizations of Clarkson's and Kruskal's method

Abstract

In this paper we determine exact, analytical solutions of the parabolic, nonlinear diffusion equation θ t = (f 2 (θ)θ x ) x + f 1 (θ), in which we insist that at least one of f 1 and f 2 is arbitrary. To do this we find reductions of the equation using a method based on the Clarkson and Kruskal direct method (Clarkson & Kruskal, 1989, J. Math. Phys., 30 2201-2213). A modification to the 'algorithm' used and a generalization of the ansatz with which one begins are necessary. Clarkson (1995) states that 'it is not clear how the direct method... may be applied to equations which contain arbitrary functions [of the dependent variable]'. Here we show how this can be accomplished by a modification of the algorithm used. We also consider a more general ansatz in which the new independent variable depends not only on the given independent variables but also on the given dependent variable. Whilst this principle is now well known, to the author's knowledge, how this is done in practice has not been explicitly described, nor have any results determined from this generalized ansatz previously been reported. Finally we consider an ansatz which is equivalent to a combination of a hodograph transformation and the usual reduction. In each of the three cases we find new reductions in which f 2 remains arbitrary and f 1 is given in terms of f 2 . The first and third cases are of particular interest: for each, a large class of new reductions and solutions are found.