Solutions to the diffusion equation

Question

When it comes to solving the heat diffusion equation u_t=u_xx the two most important solutions are a) a combination (sum) of sin-terms to resemble the function of the initial condition (that is essentially a fourier series) b) a convolution-integral of the function of the initial condition with the Gauss-curve

In most books you only find a) or b).

My question: How does it come that you get to such different solutions? Why is it that some books end up with a) and others with b)? What is the essential difference of deriving a) or b) in the end?

Answer 1

The two solutions solve different problems for the same equation.

The Fourier series solution solves the heat equation u_t=u_xx on a bounded interval [a,b] with an initial condition at t=0 of the form u(x,0)=f(x), a <=x<=b, and boundary conditions at both ends of the interval. These conditions can be of different types, leading to different series expansions. The most general (homogeneous) conditions are of the form

\alpha u(a)+\beta u_x(a)=\alpha u(b)+\beta u_x(b)=0, \alpha^2+\beta^2!=0.

If \beta=0 they are called Dirichlet conditions; if \alpha=0 Neumann conditions, if both \alpha and \beta are non zero, Robin conditions. Conditions can also be mixed: of one type on one end, of another type on the other end.

The convolution solutions solves the pure initial value problem, or Cauchy problem, on the whole real line, with initial value u(x,0)=f(x), x\in R.

Answer 2

The only real difference is what "basis" you choose to work in, roughly speaking.

The diffusion equation is du/dt = L(u), where L(u) = u_xx is a linear operator. If this were a finite-dimensional system, then you would probably want to find a basis in which the matrix of L were diagonal; then the solution is easy, because your system is now decoupled, and you can solve for each coordinate independently. If you want, you can then change back to standard coordinates.

This system isn't finite-dimensional, but a lot of the same intuition applies: there is a "basis" in which the operator L is effectively diagonal, namely the Fourier basis. (This is because differentiation can be understood as a convolution in a somewhat generalized sense, and all convolution operators are diagaonalized by the Fourier transform.) This means the Fourier transform decouples your system, making it easy to solve. If Fu is the Fourier transform of u in the spatial variable, then the solution is given by (Fu)(t) = T_t((Fu)(0)), where T_t is a diagonal operator that depends on t.

On the other hand, any operator that is diagonal in Fourier space is represented by a convolution in ordinary space. If you want to look at everything in ordinary space, then you thus obtain a representation for your solution as a convolution with a kernel that depends on t. In other words, you have u(t) = g_t*u(0), for some g_t.

Here is some general discussion of the Fourier transform; my reply there goes into convolutions and why they should all be diagonal in an appropriate basis.

Edit: I think a pretty good reference for this stuff is Fourier Analysis: An Introduction by Stein & Shakarchi (Princeton Lectures in Analaysis I). Diffusion equations (a. k. a. heat equations) are a sort of running example throughout much of the book; I think you would understand the connection between your two methods of solution pretty thoroughly if you worked your way through this book.