The energy method
The energy method [ edit ] The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems. [2] In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by ∂ � ∂ � + � ∂ � ∂ � = 0 , � ∈ [ � , � ] , � > 0 , where � ≠ 0 is a constant and � ( � , � ) is an unknown function with initial condition � ( � , 0 ) = � ( � ) . Multiplying with � and integrating over the domain gives ∫ � � � ∂ � ∂ � d � + � ∫ � � � ∂ � ∂ � d � = 0. Using that ∫ � � � ∂ � ∂ � d � = 1 2 ∂ ∂ � ‖ � ‖ 2 and ∫ � � � ∂ � ∂ � d � = 1 2 � ( � , � ) 2 − 1 2 � ( � , � ) 2 , where integration by parts has been used for the second relationship, we get ∂ ∂ � ‖ � ‖ 2 + � � ( � , � ) 2 − � � ( � , � ) 2 = 0. Here ‖ ⋅ ‖ denotes the standard...