Nonlinear heat diffusion simulation using Volterra series expansion

A method for solving a nonlinear heat diffusion problem based on the use of the Volterra series is presented. Application in a practical configuration shows that the method comes to solve a linear problem at time t with a source term that depends on solutions calculated at previous instants. Unknowns of this linear problem are the generalized transfer functions Hk(p1,…pk), k = 1,2,…, that are the k  th order Laplace transforms L(k)L(k) of the Volterra kernels hk(t). The method allows splitting naturally the solution as a linear contribution and a nonlinear one. Interest of such a method stands on the fact that a very small number of kernels is required to simulate accurately the nonlinear contribution (in practice 1 or 2). Furthermore, it is demonstrated that the Volterra method can be used efficiently to find the expression of a single simplified kernel that simulates well the entire nonlinear contribution.