General Method for Solving Systems of First-Order, Linear PDES

Question

When solving systems of first-order, linear, constant coefficient ODEs, the general method to arrive at an analytical solution is to make a matrix of the coefficients, find its eigenvalues and eigenvectors and use them to construct a general solution wherein the eigenvalues are the exponents of the exponential portion of the solution and the eigenvectors help determine the values of the arbitrary constants given a set of initial conditions. Is there a general method for obtaining analytical solutions of systems of first-order, linear, constant coefficient PDEs as well? If so, can someone direct me to where I can find more information about it?

Answer 1

Yes, there is. If you are looking at parabolic equations like \begin{align} \partial_t u(t)=A u \end{align} for some linear operator $A$ (e.g. $A=\partial_{xx}$), you can just as in ODEs solve the equation by writing \begin{align} u(t)=e^{At}u(0). \end{align} However, it is now a very nontrivial question to ask what $e^{At}$ means, and for what operators it is defined, and how to compute it. The theory on this subject is named semigroup theory, which you can find on wiki and in a lot of books. However, it is not an easy subject, and it is often mainly useful as technical tool, so if you actually want to solve an equation, this might not be the way to go.