Dirichlet problem for unbounded domain

Consider a semi-infinite strip as shown in Fig.  like previous example $u(0,y)=0,~u(a,y)=0$ and $u(x,0)=f(x).$ But this time, there is no fixed boundary on the upper side. What can be the boundary condition on unbounded side? From physical arguments, we know that $u(x,y)$ is bounded i.e. if $u-f(x)$ is heat source then we know that $u\to 0~$as$~y\to \infty.$

Figure:

Now let us solve this problem

$$u_{xx}- u_{yy} = 0,~ ~R$$ (3.87)

$$u(0,y) =u(a,y) = 0,$$ (3.88)

$$u(x,0) = f(x)$$ (3.89)

$$u(x,y) ~ ~y \to \infty.$$ (3.90)

Solution: Assume that the solution can be expressed in separation form as

u(x,y)=X(x)T(y)
$\frac{X''}{X} = \frac{Y''}{Y} = \lambda$

With the same procedure as above, we will get for this problem

$$\lambda_n=\frac{n^2 \pi^2}{a^2}$$

are the eigenvalues and corresponding eigen-functions are

$$X_n(x)=\alpha_n \sin(\frac{n\pi}{a}x)$$

In the previous example, the solution of equation for $Y(y)$ was expressed in terms of $\sinh$ and $\cosh$ functions as given by Eq. (). This particular form is not useful here. Here we express the solution in terms of exponential form as

$$Y_n(y)=\alpha_n\exp(\frac{n\pi}{a}y)+\beta_n\exp(-\frac{n\pi}{a}y)$$

For boundedness of solution as $y\to \infty,$ we get

$$Y_n(y)=\beta_n\exp(-\frac{n\pi}{a}y)$$

$$u_n(x,y) =X_n(x)Y_n(y)$$

$$= \beta_n\sin(\frac{n\pi}{a}x)\exp(-\frac{n\pi}{a}y)$$

By the principle of superposition of solutions,we get the general solution

$$u(x,t) = \sum_{n=0}^{\infty}u_n(x,t)$$ (3.91)

$$= \sum_{n=0}^{\infty}\beta_n \sin(\frac{n\pi}{a}x)\exp(-\frac{n\pi}{a}y)$$ (3.92)

The unknown coefficients $\{\beta_n\}$ are found by using the remaining boundary condition $u(x,0)=f(x)$

$$u(x,0)=f(x)=\sum_{n=1}^{\infty}\beta_n \sin(\frac{n\pi}{a}x).$$ (3.93)

Multiply both sides by $\sin(\frac{m\pi}{a})x$ and integrate both sides $[0,a]$ and using orthogonality, one gets

$$\beta_n=\frac{2}{a}\int_{0}^{a}f(x)\sin\sin(\frac{n\pi}{a}x).$$ (3.94)

Thus the required solution is given by Eq. () and Eq. () .

Example 3.2.2  

$$f(x) = \begin{cases} T_0=100 &0\leq x \leq a\\ 0& ~\text{otherwise} \end{cases}$$ (3.95)

Show that

$$u(x,y)=4\frac{T_0}{\pi}\sum_{n=\text{odd}}^{} \sin(\frac{n\pi}{a}x)\sinh(\frac{n\pi}{a}(b-y))n\sinh(\frac{n\pi b}{a}).$$

Explore further: Plot level curves of $u(x,y).$ i.e. $u(x,y)=$constant contour of equi-temp points.

Example 3.2.3   Solve the above problem for semi-infinite case and show that

$$u(x,y)=4\frac{T_0}{\pi}\sum_{n=\text{odd}}^{} \frac{1}{n}\exp(-\frac{n\pi}{a}y)\sin(\frac{n\pi }{b})x.$$

Explore further: Compare the contour plots of this case with that of previous (on $[0,10]\times[0,10]$) interpret the results.

Dirichlet problem for a circular disk