How do solve this pde problem?

Question

EDIT: I know somehow, we end up with an equation relating the derivative of some coefficients to the rest of the stuff. I'm not sure where this equation, or even the constant that we use to get it, come from. Help with the derivation of this would be MUCH APPRECIATED I had these on an exam,and got zero credit for what I attempted, could someone suggest where to start? Please no full solutions.

Thank you.

You are given the Eigenvalues and eigenfunctions of the laplacian denoted $\lambda_{i,j},\ \psi_{i,j}(\vec{r}) $respectively ( these are obtained from $\nabla^2 u= \lambda u$). Explain how you would solve the following with the initial condition $u(\vec{r},t)=f_1(\vec{r})$ applied to the PdE. $$\frac{\partial u}{\partial t}=\nabla^2 u$$

$$\frac{\partial u}{\partial t}+ b_{2,12} \psi_{2,12}= \nabla^2 u$$

ADDENDUM

Original Form of Question: You are given a complete set of eigenfunctions $ \phi _{n,k}$ and eigenvalues$\lambda _{n,k}$ for the Laplacian on a domain R with homogeneous Dirichlet boundary conditions.The eigenfunctions are orthogonal with respect to a weight $\sigma$.

Explain how to solve the Heat Equation $u_t=\nabla^2 u$ on the domain R with the matching homogeneous boundary conditions with the intial condition $u(r,0)= f_1(r)$

.Explain how to solve $u_t+b_{2,12}\phi_{2,12}(r,\theta )=\nabla^2 u $ with $u(0,r,\theta )=0$ on the domain R with the matching homogeneous boundary conditions

Answer 1

I'll just remind you of some standard facts that I hope will lead you in the right direction, but you should refresh these in the book/references.

First the homogenous PDE, $ \partial_t u = \Delta u$. As mentioned, $u_{i,j}(r,t) := \psi_{i,j}(r) e^{\lambda_{i,j} t} $ solves the heat equation, with initial condition at time zero $ \psi_{i,j}(r) $. Since everything here is linear, a linear combination $ \sum_{i,j} c_{i,j} u_{i,j} $ is also a solution. So the problem reduces to solving for the $ c_{i,j} $ so that $ f_1 = \sum c_{i,j} \psi_{i,j} $. Without any further knowledge of $ f_1 $, I am not sure what more can be said.

Now for the next one, try to find a particular solution to: $$ \frac{\partial u}{\partial t}+ b_{2,12} \psi_{2,12}= \nabla^2 u $$ Hint: guess something of the form $ \psi_{i,j}(r) \; T(t) $, which upon substitution into the PDE will become an ODE for $ T(t) $. Then you can find a solution to the initial value problem by adding on the solution to the homogenous PDE.