NONLINEAR EIGENFUNCTION EXPANSIONS FOR THE SOLUTION OF NONLINEAR DIFFUSION PROBLEMS
A general nonlinear eigenvalue problem approach is proposed for the hybrid numerical-analytical solution of nonlinear diffusion and convection-diffusion problems via the Generalized Integral Transform Technique (G.I.T.T.). While the GITT has been extensively employed in the solution of nonlinear problems, the preferred approach has always been that of rewriting the partial differential system with linear characteristic coefficients, in both the equation and corresponding boundary conditions, and moving the nonlinearities to the respective source terms. Then, the eigenvalue problem is generally chosen in terms of the linear equation and boundary conditions coefficients, and the nonlinear source terms appear only in the transformed ordinary differential system. Here, a different solution path is considered by following the original nonlinear problem formulation, and adopting the eigenvalue problem that carries along the information on the nonlinear operators. The eigenvalue problem then needs to be solved simultaneously with the transformed ODE system. A significant gain is expected to result from this more complete eigenfunction expansion basis, to be observed on the improved convergence rates and reduced computational costs. The approach is illustrated for a test-case related to a diffusion problem with nonlinear boundary conditions, in the form of a natural convection heat transfer coefficient. An implicit filtering solution is also implemented for comparison purposes.