An exact methodology for solving nonlinear diffusion equations based on integral transforms (Q1093338)

Let T and k(T) denote the temperature and the thermal conductivity, respectively. Assuming a truncated Taylor-series expansion of the form \(k(T)=k_ 0\sum^{N}_{n=0}\beta_ n(T-T_ 0)^ n,\) where \(\beta_ 0\equiv 1\), \(T_ 0\) is the reference temperature, the authors derive the nonlinar temperature field equation at once for a slab, a cylinder and a sphere. Suitable initial and boundary conditions are derived as well. The case of the slab for \(N=1\) is examined in details. Applying an integral transform enables to obtain the solution in the form of an infinite series. To get terms of this series one has to solve a system of nonlinear Volterra integral equations of the second kind. To do this one a numerical procedure is proposed. Numerical calculations are performed and results are compared to those ones obtained by a standard finite difference method.