Mixed convolved action variational principles in heat diffusion

A true variational formulation is developed for dissipative processes based upon the concept of mixed convolved action. Here the focus is on continuum problems associated with heat diffusion, as well as related second sound phenomena. The convolved action can overcome the shortcomings of typical action principles, such as Hamilton's principle, to address dissipative processes without the need for separate dissipation functionals and ad hoc variational operations. In addition, the mixed convolved action is compatible with the initial and boundary conditions of a well-posed heat problem. In fact, the stationarity of the mixed convolved action is shown to provide the governing partial differential equations, the initial conditions and the boundary conditions as its Euler-Lagrange equations. Thus, the mixed convolved action encapsulates the entire description of the initial/boundary value heat problem. In addition to the theoretical significance, this new formulation can establish the basis for effective numerical methods, for example, involving finite element representations over both space and time. One particular two-dimensional formulation is developed here and then applied to two example problems to illustrate the viability of the proposed approach.