Question

The heat equation is: ∂u/∂t = α∇^2u

where α is the thermal diffusivity of a substance, and u(x, y, z, t) describes the temperature of a substance as a function of space and time. Show that the separation of variables procedure can be successfully applied to this partial differential equation to separate the space and time variables. You should obtain, as your final answer, a series of ordinary differential equations.

Answer 1

let u=f(x)*g(y)*h(z)*T(t)

then LHS of the equation=f*g*h*T'

RHS of the equation becomes:

hence the resultant equation becomes:

dividing both sides by f*g*h*T,

we get

now each term is independently function of their own variable only.

hence assigning

we get 4 ODEs and one constraint i.e. k1=k2+k3+k4