# Conventions in PDEs

Suppose the function $u(x,y,z,t)$ satisfies $$\frac{\partial u}{\partial t} = \Delta u +5$$

on a sphere of radius 2, with $\Delta$ as the Laplace operator and $u(x,y,z,0)=f(x,y,z)$. On the surface of the sphere it is given that $\nabla u \cdot \hat{n} = 6$, where $\hat{n}$ is a unit outward normal vector. Calculate the total thermal energy for this sphere as a function of time.

Solving the problem is straightforward enough (write down the integrated conservation law, use divergence theorem, etc.), but what do I do about the other physical quantities such as specific heat, density and thermal conductivity ($c,\rho ,K_0$), which are present in the original heat equation? Do I assume $c\rho =1$ and $K_0 = 1$? The entire heat equation is given by: $$c \rho \frac{\partial u}{\partial t} =K_0 \frac{\partial ^2 u}{\partial x^2} + Q(x,t)$$

Edit (this is what my instructor had to say): The only thing that I want to mention, is that if the equation is given to you, it means that specific values have already been assigned to all the problem parameters (e.g. $c, \rho , K_0$ are known).
• Compare the parameters to the coefficients of the given equation. You indeed have $c\rho = 1$, $K_0=1$ and $Q(x,t) = 5$ Commented May 19, 2018 at 14:10
I would prefer to think that the problem has been expressed in dimensionless units. That is to say, the units of measurement have been chosen in such a way that the numerical value of $c\rho/K_0$ is unity.