Nondimensionalising and solving nonhomogeneous heat equation

[ctrl.s]

I am given an equation of the form : dT/dt* = -lambda*(T-T_inf) + K.d^2T/dx*^2 over 0<x*<inf , t*>0
With initial condition: T(x*,0) = T_inf + T0[1-(x*/L)^2] , 0<= x* <= L
and T(x*,0) = T_inf , x*>L

I need to nondimensionalise the equation using the scaled variables:
t = Kt*/L^2, x = x*/L, theta = (T-T_inf)/T0

My attempt at nondimensionalising is as follows:

d^2/dx*^2 = 1/L^2 * d^2/dx^2
d/dt* = K/L^2 * d/dt

and

d^2T/dx*^2 = 1/L^2 * d^2T/dx^2 = 1/L^2 * T0 * dtheta/dx
dT/dt* = K/L^2 * dT/dt = K/L^2 * T0 * dtheta/dt

which I then substitute into the original equation to get:

K/L^2 * T0 * dtheta/dta = -lambda*(T-T_inf) + KT0/L^2 * d^2theta/dx^2

which leaves:

dtheta/dt = -lambda*L^2/K * theta + d^2theta/dx^2

and we are told to set y = lambda*L^2/K so

dtheta/dt = -y*theta + d^2theta/dx^2

Is this correct? I have had trouble in the past nondimensionalising so I am not sure. Additionally, I am unsure how to solve this equation. We were only taught how to solve the nonhomogeneous heat eqn with a source term in either x or t but not x,t which I am assuming this is (theta(x,t)) which makes me think I am wrong. I'd also appreciate help solving this regardless, source terms make it hard! Thanks!

edit: I'd like to note that I may have misinterpreted the question in terms of having to solve it. The subsequent question says that the internal heat of the system Q(t) = int[0,inf.] theta(x,t) dx and show that the heat is given by Q = Q0*e^(-yt)

 

[HallsofIvy]

To solve the partial differential equation \(\displaystyle \frac{\partial \theta}{\partial t}= -y\theta+ \frac{\partial^2\theta}{\partial x^2}\), I would "separate the variables". That is, look for a solution or the form \(\displaystyle \theta(x, t)= X(x)T(t)\). Putting that into the equation, \(\displaystyle X\frac{dT}{dt}= -yXT+ T\frac{d^2X}{dx^2}\). Divide both sides by XT to get \(\displaystyle \frac{1}{T}\frac{dT}{dt}= -y+ \frac{1}{X}\frac{d^2X}{dx^2}\). Now the left hand side depends only on t while the right hand side depends only on x. For those to be equal for all x and t, the two sides must be constants:

\(\displaystyle \frac{1}{T}\frac{dT}{dt}= \lambda\) or \(\displaystyle \frac{dT}{dt}= \lambda T\) and
\(\displaystyle -y+ \frac{1}{X}\frac{d^2X}{dx^2}=\lambda\) or \(\displaystyle \frac{d^2X}{dx^2}- yX= \lambda X\).

Those are ordinary differential equations with constant coefficients so should be easy to solve for given \(\displaystyle \lambda\). Now, in your post you give initial conditions, conditions on t, but no boundary conditions, on x. I see that you give x in an interval, [0, L], but what is to be true of the function at x= 0 and x= L? They are typically important in determining possible values for \(\displaystyle \lambda\).

 

[ctrl.s]

Thank you for the response. I didn't use the method of separation of variables since we were taught that this couldnt be used when a source/forcing term is present. However since that source term is in x and t, I see why we can use the method here.
The boundary conditions are stated to be 'insulating boundary conditions' which I take to mean: dtheta/dx (0,t) = 0 and dtheta/dx (L,t) = 0