The attached image is from "Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations" by J.W. Thomas. In the beginning the set of test functions, [itex]\phi[/itex] is defined. They arrive at equation (9.2.11) by using [itex]\phi(x,T) = \phi(a,t) = \phi(b,t) = 0[/itex]. Where does this condition come from? Thanks.