Finding v0 in a 5-point grid using Laplace numerical solution

[purejoker]

hi there, I am a bit stuck can someone help?

i have a 5 point grid
--_______v1
--_______|
--_______|
v3-------v0-------v4
--_______|
--_______|
--_______v2

--_______ used to postion the | (y-axis) the grid is a plus sign.

they are seprated by $$\delta$$x and $$\delta$$y

i've worked out upto the second differential:
(v3-v0/$$\delta$$x - v0-v4/$$\delta$$x) /$$\delta$$x + (v1-v0/$$\delta$$y - v0-v2/$$\delta$$y) /$$\delta$$y = 0

the problem is i don't know how to get to this stage:

v0 = 1/4 (v1+v2+v3+v4)


thanks for your help

pure!

 

[lippyka]

To get to that stage, you need to expand the derivatives. Using the definition of the derivative, you can expand the above equation as: (v3 - (2v0 - v4)/2)/\deltax + (v1 - (2v0 - v2)/2)/\deltay = 0Now, solve for v0 by adding the two sides and rearranging: v0 = (v1+v2+v3+v4)/4