Replace this equation with an explicit finitedifference equa

Replace this equation with an explicit finite-difference equation what are the truncation errors ?

du/dt= v(d^2u/dy^2 + d^2u/dx^2)

Solution

du/dt = A*(d^{2u/dy^}2 +d^2u/dx2) u(x,0) = f(x) u(0,t) = 0 u(L,t) = V

Let w(x,t) = u(x,t) - (V/L)*x

Now, w(x,0) = u(x,t) - (V/L)x = f(x)-[(V/L)x] We can just call this g(x) so that\'s fine

w(0,t) = u(0,t) - (V/L)0 = 0-0 = 0 w(L,t) = u(L,t) - (V/L)L = V-V = 0

dw/dt = du/dt - 0 = du/dt dw/dx = du/dx - (V/L) (d^2w/dt2 + d^2w/dt2) = 2d^2u/dt2 - 0 = 2d^2u/dt2

du/dt = V*( d^2u/dy2 +d^2u/dx2) is given in the problem, and from above, this means that

dw/dt = V*( d^2w/dy2 + d^2w/dx2)

So I solve dw/dt = V*( d^2w/dy2 + d^2w/dx2)

the same way they do for the heat equation on wikipedia, then use the relation

u(x,t) = w(x/t) + (V/L)*x

To find u(x,t)