I have a system described an equation, and I want to find an (DE) equation for z(x,t), in the limit as l->0.
First some definitions to simplify it some:
$Z1=z(x,t)-z(x+l,t)$
$Z2=z(x,t)-z(x-l,t)$
These are not constants, just to simplify expr.
c however, is a constant.
EQ:
$c*l^2*d^2z(x,t)/dt^2=(1-l/sqrt(Z1^2+l^2))*Z1+(1-l/sqrt(Z2^2+l^2))*Z2$
So, I want to find the limit of this expression as l->0, from the positive side. (l is a physical distance, and this is a physical realizable system, hopefully in the limit too).
So z(x,t) initially only makes physical sense for x=0,l,2l,3l.
And I want to make it into a continuous function for all x.
I dont know if its relevant but z(x,t) and x is also a distance, and t is time.
Please tell me if I should clarify something, and whether the problem as stated is well defined mathematically.