Energy Form of a Parabolic PDE

The strong form of a parabolic PDE, that is, the governing equation (for example, the classical heat conduction equation with constant heat c, variable thermal conductivity k and heat production / see, e.g., Carslaw and Jaeger 1959 Selvadurai 2000a), the boundary conditions (BC) and the initial conditions (IC) are given as 

A convolution integral between (4.47a) and the step function h t) is given by 

Therefore the weak form and the variational principle (i.e., the principle of the energy minimization) of the parabolic PDE are obtained as follows  

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