How Can I Determine the New Initial State in a Homogenous Dirichlet Equation?

[onie mti]

I have this equation

View attachment 2365
given:

where the boundary conditions on p are homogenous dirichlet.

my work:
i know that q(0,t)=0 q(l,t)=0 for all τ>0.
and the initial conditon p(x,0)= p_0(x) also transltates to an initial condition on q.now how can i verify and indicate what the new initial state q_0 should be where i have to

 

[gtoguy97]

use the initial condition p_0(x)?The new initial condition for q can be determined by using the boundary conditions and the initial condition for p. At t = 0, q is equal to 0 at both x = 0 and x = l, so the initial condition for q must be 0. However, since q is related to p through the equation q = -dp/dx, the initial condition for q can also be determined from the initial condition for p. Taking the derivative of p_0(x) with respect to x gives us the initial condition for q:q_0(x) = -dp_0(x)/dx