Simple Heat Transfer Problem
Partial
Differential Equations (PDEs)
an equation containing derivatives of a function of
two or more variables
a second order PDE contains no derivatives higher than
2
Linear second-order partial differential equations
form:
Auxx + 2Buxy + Cuyy + Eux + Fuy + Gu = H
examples second order PDEs
4uxx + 6xyuxy = 0
πxxy + x2uyy = sin (xy)
not-linear second order PDEs
u2xx + uyy = 0
Difference
Equations
f’(x)
≈ f (x + h/2) – f (x - h/2)
h
f (x + h/2 – h/2) – f (x + h/2 – h/2) f (x – h/2 + h/2) – f (x – h/2 – h/2)
f”(x) ≈ h
h
_
h
≈ f (x +h) – f (x) – ((f (x) – f (x – h))
h2
≈ f (x + h) – 2f (x) + f (x – h )
h2
Steady State Heat (voltage)
Distribution
uxx + uyy = f (x,y), 0 ≤ x ≤ a, 0 ≤ y ≤ b
u (x,0) = G1 (x) and u (x,b) = G2 (x) 0 ≤ x ≤ a
u (o,y) = G3 (y) and u (a,y) = G4 (x) 0 ≤ y ≤ b
Deriving Finite Difference Equations
uxx (xi,yj) ≈
u (xi + h,yj) – 2u (xi + yj) + u
(xi - h,yj)
h2
≈ ui + 1,j – 2ui,j + ui
– 1,j
h2
uyy (xi,yj) ≈ ui,j + 1 – 2ui,j
+ ui,j – 1
k2
ui + 1,j – 2ui,j + ui – 1,j + ui,j + 1 – 2ui,j + ui,j – 1 = f (xi, yj)
h2
k2
wi,j = λ2
(wi + 1,j +
2 (1 + λ2)
wi,j = (wi + 1,j + wi
- 1,j + wi,j +
1 + w i,j – 1
4