An exact methodology for solving nonlinear diffusion equations based on integral transforms

This paper develops a new methodology for the solution of nonlinear diffusion equations. The solution technique is based on integral transforms and leads to exact numerical results. We apply the formal methodology to the problem of one-dimensional transient heat conduction. A new form of the heat equation is developed using a generalized expression for temperature-dependent thermal conductivity, based on a power-series expansion, for the three standard orthogonal coordinate systems. The resulting form of the heat equation suggests that the finite integral transform technique may reduce the dimensionality of the heat equation prior to the initiation of any numerical procedure. An example in a slab with linearly varying thermal conductivity is shown to produce exact results for the temperature distribution.