I follow this course for solving the heat equation. However, I cannot understand a possible case of semi-infinite.
The general equation obtained from the separation of variables is
$$\phi(x) = c_1\cos (\sqrt{\lambda}x) + c_2 \sin(\sqrt{\lambda} x)$$
as we have to obtain $\lambda$ by applying the boundary conditions. All example are for $x=0$ and $x=L$. I am trying to figure out how to solve it for a simple set of boundary conditions:
$$u(x,0)=C_0, u(0,t)=C_1, u(\infty,t)=C_2$$
How do we apply $\phi(\infty) = 0$ to obtain $\lambda$?
In fact, I don't know how to apply the condition of $\infty$ to $\phi(x)$
I know that the common solution for the semi-infinite boundary conditions is Laplace transform, but I am curious what is the mathematical limitations for applying the semi-infinite boundary conditions to this solution by the approach of separation of variables.
Best Answer
The heat equation doesn't make good physical sense if the total amount of heat is infinite. The total heat at time $t$ is a constant multiple of $\int_{0}^{\infty}u(x,t)dx$. Assuming positive temperatures also makes sense, which makes $L^1[0,\infty)$ a natural space in $x$. These types of conditions replace explicit values of the solution at $\infty$. Then if you think of a limit of problems for $0 \le x \le L$ as $L\rightarrow\infty$, it might not surprise you do think in terms of integral combinations of solutions instead of sums. For example, a general type of solution of $u_t = c^2u_{xx}$ might look like $$ u(x,t)=\int_{0}^{\infty}e^{-c^2\lambda^2 t}\{C(\lambda)\sin(\lambda x)+D(\lambda)\cos(\lambda x)\}d\lambda. $$
The coefficients $C(\lambda),D(\lambda)$ in such a solution would then be determined from initial data through Fourier sine and cosine transforms: $$ u(x,0)=\int_{0}^{\infty}C(\lambda)\sin(\lambda x)+D(\lambda)\cos(\lambda x)dx. $$