An Exact Solution of Stikker's Nonlinear Heat Equation

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].. Abstract. Exact solutions are derived for a nonlinear heat equation where the conductivity is a linear fractional function of (i) the temperature gradient or (ii) the product of the radial distance and the radial component of the temperature gradient for problems expressed in cylindrical coordinates. It is shown that equations of this form satisfy the same maximum principle as the linear heat equation, and a uniqueness theorem for an associated boundary value problem is given. The exact solutions are additively separable, isolating the nonlinear component from the remaining independent variables. The asymptotic behaviour of these solutions is studied, and a boundary value problem that is satisfied by these solutions is presented. 1. Introduction. Considerable work has been done in the analysis of certain types of nonlinear heat conduction (diffusion) equations where the conductivity (dif-fusivity) is a function of the temperature (concentration) [2], [6]. The porous media equation, a nonlinear diffusion equation of this type, is an example [1], [9]. A review of some nonlinear equations that admit exact solutions has been completed recently by Rogers and Ames [10]. Some analysis has also been done for equations where the conductivity is a function of the temperature gradient or its magnitude [3]. In this paper we shall develop exact solutions for the following two nonlinear heat conduction equations in 1R3: