A tentative solution to Assignment 5, Problem3(g)

[Shentao YANG]

The problem is here: http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter3/S3.2.P.html
Here is my tentative solution, hope everyone for correction:
consider:
$$w(x,t) = \left\{ {\matrix{
   {u(x,t) - 1} & {0 < x < \infty } & {t < 1}  \cr
   {u(x,t)} & {0 < x < \infty } & {t > 1}  \cr

 } } \right.$$
Then we have:
$${w_t} = {u_t}, {w_{xx}} = {u_{xx}}$$
Therefore:

$$\matrix{
   {{w_t} = k{w_{xx}}} & {t > 0} & {x > 0}  \cr
   {w\left| {_{t = 0} = u(x,0) - 1 =  - 1} \right.} & {} & {x > 0}  \cr
   {w\left| {_{x = 0} = \matrix{
   {\left\{ {\matrix{
   {u(0,t) - 1 = 1 - 1 = 0} & {t < 1}  \cr
   {u(0,t) = 0} & {t > 1}  \cr

 } } \right\}} & { = 0}  \cr

 } } \right.} & {t > 0} & {}  \cr

 } $$
which is the Dirichlet boundary condition.
Solving this problem wrt $w(x,t)$, I get (The calculation may be wrong):
$$w = 1 - erf({x \over {\sqrt {4kt} }})$$
Therefore:
$$u = \left\{ {\matrix{
   {2 - erf({x \over {\sqrt {4kt} }})} & {x > 0} & {0 < t < 1}  \cr
   {1 - erf({x \over {\sqrt {4kt} }})} & {x > 0} & {t > 1}  \cr

 } } \right.$$
However, in this case $u(x,t)$ is not continuous at $t = 1$.
Hope everyone for correction / verification.

[Shentao YANG]

By the way, I guess there are typos in problem (4):
The $g(t)$ in equation (10) should be $g(x)$ and the r.h.s. of equation (11) should be $h(t)$.

[XinYu Zheng]

Your function $w(x,t)$ is not continuous, so if you solve for $u$ in terms of $w$, it wouldn't be continuous either. There is a section in the textbook (http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter3/S3.2.html#sect-3.2.2) regarding inhomogeneous boundary conditions, but this has neither been covered in lectures nor tutorials.

[Shentao YANG]

Your function $w(x,t)$ is not continuous, so if you solve for $u$ in terms of $w$, it wouldn't be continuous either.

Yes, on each subinterval will be continuous, but not the whole. However, when I applied this same method on Problem3(f), the solution is really continuous (hopefully I am correct, you may check it yourself), so I am not sure whether this discontinuousness is the "nature" of this particular problem or just because my method is wrong...

There is a section in the textbook (http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter3/S3.2.html#sect-3.2.2) regarding inhomogeneous boundary conditions, but this has neither been covered in lectures nor tutorials.

Yes, I am thinking of a way to do it alternatively, that's why I titled it as "a tentative solution"...

By the way, can any one tell me will this kind of inhomogeneous boundary conditions problems be tested? (since it is not covered in both lectures and tutorials)

[Victor Ivrii]

Use formula (14) from Section 3.2

Similarly, for Neumann b.c. use (15)

Consider as bonus problems