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?
Best Answer
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.