Heat equation

Question

For a constant c > 0 find the solution to the heat equation

u_t = c^2u_(xx)

with the initial condition T(x, 0) = φ(x)

Consider a change of variables t'=c^2t or x'=cx

 

thank you to whoever can do it in steps for me :)

Answer 1

u_t = c²u_xx   with   u(x,0) = φ(x)

 

Using the transformation x′ = cx, the equation becomes

 

u_t = u_x′x′

 

 

The common strategy is to use separation of variables.

 

Let u_k(x,t) = T(t)X(x′).  Then,

 

dT/dt X = T d²X/dx'²

 

 

Separating variables,

 

(1/T) dT/dt = (1/X) d²X/dx′² = -k²  where -k² < 0 is a constant because the LHS depends only on t and the RHS depends only on x'.

 

 

Integrating,

 

ln(T) = -k²t + c_k  or T = C_ke^{(-k^2)t}   and  X = D_ksin(kx) + E_kcos(kx).

 

 

So u_k(x',t) = T(t)X(x') = C_ke^{(-k^2)t[}D_ksin(kx') + E_kcos(kx')]

 

 

The general solution would be a superposition for all possible values of A_k and B_k:

 

u(x',t) = Σe^{(-k^2)t}[A_ksin(kx') + B_kcos(kx')]   with

 

 

u(x',0) = Σ[A_ksin(kx') + B_kcos(kx')] = φ(x)