Yuliya Gorb

Nonhomogeneous Heat Equation

Heat Equation

Lecture 11

October 01, 2013

Lecture 11 Nonhomogeneous Heat Equation

Yuliya Gorb

Nonhomogeneous Heat Equation

Goal: solve inhomogeneous heat eq. with non-zero initial cond.:

“white elephant”

∂φ

∂t+△φ = f (t, x), t > 0, x ∈ R

n

φ(0, x) = g(x), x ∈ Rn

(1)

Then φ(t, x) = u(t, x) + v(t, x) where u and v solve the following problems:

“blue elephant”

∂u

∂t+△u = 0, t > 0, x ∈ R

n

u(0, x) = g(x), x ∈ Rn

(2)

“pink elephant”

∂v

∂t+△v = f (t, x), t > 0, x ∈ R

n

v(0, x) = 0, x ∈ Rn

(3)

Lecture 11 Nonhomogeneous Heat Equation

Yuliya Gorb

Duhamel’s Principle for Heat Equation

Recall, the solution to (2) is given by

u(t, x) =

∫

Rn

G(t, x − y)g(y) dy , t > 0, x ∈ Rn

with the heat kernel G(t, x) defined in last lecture

For problem (3) we apply the Duhamel’s principle similar to one we used forthe linear transport equation, namely, define

∂ψ

∂t+△ψ = 0, t ≥ s, x ∈ R

n

ψ(t = s, x ; s) = f (s, x), x ∈ Rn

whose solution is

ψ(t, x) =

∫

Rn

G(t − s, x − y)f (s, y) dy , t > 0, x ∈ Rn

hence, the solution v(t, x) of (3) is given by

v(t, x) =

∫

t

0

ψ(t, x ; s) ds, t > 0, x ∈ Rn

Lecture 11 Nonhomogeneous Heat Equation

Yuliya Gorb

References

Evans pp. 49–51

Lecture 11 Nonhomogeneous Heat Equation