5.2: Parabolic Equation

Let us first study the heat equation in 1 space (and, of course, 1 time) dimension. This is the standard example of a parabolic equation.

\[\begin{aligned} \dfrac{\partial}{\partial t} u = k \dfrac{\partial^2}{\partial x^2} u,\;\;0<x<L,\;t>0.\end{aligned} \nonumber \]

with boundary conditions

\[u(0,t) = 0,\;u(L,t)=0,\;\;t>0, \nonumber \]

and initial condition \[u(x,0) = x,\;0<x<L. \nonumber \] We shall attack this problem by separation of variables, a technique always worth trying when attempting to solve a PDE, \[u(x,t) = X(x) T(t). \nonumber \] This leads to the differential equation

\[X(x) T'(t) = kX"(x) T(t). \label{eq5.5} \]

We find, by dividing both sides by \(XT\), that

\[\frac{1}{k}\frac{T'(t)}{T(t)} = \frac{X"(k)}{X(k)}. \nonumber \]

Thus the left-hand side, a function of \(t\), equals a function of \(x\) on the right-hand side. This is not possible unless both sides are independent of \(x\) and \(t\), i.e. constant. Let us call this constant \(-\lambda\).

We obtain two differential equations

\[\begin{aligned} T'(t) &= -\lambda k T(t) \\ X''(x) &= -\lambda X(x)\end{aligned} \nonumber \]

Exercise \(\PageIndex{1}\)

What happens if \(X(x)T(t)\) is zero at some point \((x=x_0, t=t_0)\)?

Answer

Nothing. We can still perform the same trick.

Note

This is not so trivial as I suggest. We either have \(X(x_0)=0\) or \(T(t_0)=0\). Let me just consider the first case, and assume \(T(t_0)\neq 0\). In that case we find (from \ref{eq5.5})), substituting \(t=t_0\), that \(X''(x_0)=0\).

We now have to distinguish the three cases \(\lambda>0\), \(\lambda=0\), and \(\lambda<0\).

\(\lambda>0\)

Write \(\alpha^2=\lambda\), so that the equation for \(X\) becomes \[X''(x)=-\alpha^2 X(x). \nonumber \] This has as solution \[X(x) = A\cos\alpha x +B\sin\alpha x. \nonumber \] \(X(0) = 0\) gives \(A\cdot 1 + B \cdot 0=0\), or \(A=0\). Using \(X(L)=0\) we find that \[B\sin\alpha L = 0 \nonumber \] which has a nontrivial (i.e., one that is not zero) solution when \(\alpha L = n\pi\), with \(n\) a positive integer. This leads to \(\lambda_n= \frac{n^2\pi^2}{L^2}\).

\(\lambda=0\)

We find that \(X = A+Bx\). The boundary conditions give \(A=B=0\), so there is only the trivial (zero) solution.

\(\lambda<0\)

We write \(\lambda=-\alpha^2\), so that the equation for \(X\) becomes \[X''(x)=-\alpha^2 X(x). \nonumber \] The solution is now in terms of exponential, or hyperbolic functions, \[X(x) = A \cosh x + B \sinh x. \nonumber \] The boundary condition at \(x=0\) gives \(A =0\), and the one at \(x=L\) gives \(B=0\). Again there is only a trivial solution.

We have thus only found a solution for a discrete set of “eigenvalues” \(\lambda_n>0\). Solving the equation for \(T\) we find an exponential solution, \(T=\exp(-\lambda k T)\). Combining all this information together, we have \[u_n(x,t) = \exp\left(-k \frac{n^2\pi^2}{L^2}t\right)\sin\left(\frac{n\pi}{L}x \right). \nonumber \] The equation we started from was linear and homogeneous, so we can superimpose the solutions for different values of \(n\), \[u(x,t) = \sum_{n=1}^\infty c_n \exp\left(-k \frac{n^2\pi^2}{L^2}t\right)\sin\left(\frac{n\pi}{L}x\right). \nonumber \] This is a Fourier sine series with time-dependent Fourier coefficients. The initial condition specifies the coefficients \(c_n\), which are the Fourier coefficients at time \(t=0\). Thus

\[\begin{aligned} c_n &=& \frac{2}{L} \int_0^L x \sin\frac{n\pi x}{L} dx \nonumber\\ &=& - \frac{2L}{n\pi}(-1)^n = (-1)^{n+1} \frac{2L}{n\pi}.\end{aligned} \nonumber \] The final solution to the PDE + BC’s + IC is \[u(x,t) = \sum_{n=1}^\infty (-1)^{n+1} \frac{2L}{n\pi} \exp\left(-k \frac{n^2\pi^2}{L^2}t\right)\sin\frac{n\pi}{L}x. \nonumber \]

This solution is transient: if time goes to infinity, it goes to zero.