Energy Form of a Hyperbolic PDE

The strong form of a hyperbolic PDE, that is, the governing equation, the boundary conditions (BC), the displacement-strain relation (DS) and the initial conditions (IC) are given as follows (for example, the equation of classical elasto-dynamics for a Hookean solid with mass density p and elasticity constant tensor D see, e.g., Gurtin 1972 Selvadurai 2000b)  

A convolution integral between (4.51a) and the step function i(t) is given by 

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

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