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Another PDE and boundary conditions

Markov

Member
Feb 1, 2012
149
1) Solve

$\begin{aligned}
{{u}_{t}}&=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\
{{u}_{x}}(0,t)&=0,\text{ }{{u}_{x}}(L,t)=0,\text{ for }t>0, \\
u(x,0)&=6+\sin \frac{3\pi x}{L}
\end{aligned}$

2) Transform the problem so that the boundary conditions get homogeneous:

$\begin{aligned}
{{u}_{t}}&=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\
{{u}_{x}}(0,t)&=Ae^{-at},\text{ }{{u}_{x}}(L,t)=B,\text{ for }t>0, \\
u(x,0)&=0
\end{aligned}$

Attempts:

1) No ideas for this one, I don't know how to proceed when the initial conditions have the first derivative.

2) I think I need to define a new function right? But how?
 

dwsmith

Well-known member
Feb 1, 2012
1,673
1) Solve

$\begin{aligned}
{{u}_{t}}&=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\
{{u}_{x}}(0,t)&=0,\text{ }{{u}_{x}}(L,t)=0,\text{ for }t>0, \\
u(x,0)&=6+\sin \frac{3\pi x}{L}
\end{aligned}$

2) Transform the problem so that the boundary conditions get homogeneous:

$\begin{aligned}
{{u}_{t}}&=K{{u}_{xx}},\text{ }0<x<L,\text{ }t>0, \\
{{u}_{x}}(0,t)&=Ae^{-at},\text{ }{{u}_{x}}(L,t)=B,\text{ for }t>0, \\
u(x,0)&=0
\end{aligned}$

Attempts:

1) No ideas for this one, I don't know how to proceed when the initial conditions have the first derivative.

2) I think I need to define a new function right? But how?
What book are you using? If it isn't helpful, you should get Elementary Partial Differential Equations by Berg and McGregor.
 

Markov

Member
Feb 1, 2012
149
I have no chances to get books, the only source I have left is the forum. :(

Think you could help me please? :(
 

dwsmith

Well-known member
Feb 1, 2012
1,673
I have no chances to get books, the only source I have left is the forum. :(

Think you could help me please? :(
The book I suggested is relatively cheap and it is a good book. I suggest you think about picking a book up whether it is that one or another one.
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
For 1, because the boundary conditions are homogeneous, try a solution of the form $\sum_{n=0}^\infty A_n(t)cos(n\pi x/L)$.
Use cosine because the derivative of cosine is sine which is 0 at 0 and L.
For 2, yes, you need to change the function. And you want to make the boundary conditions 0 so find a function, f(x,t), that satisfies the boundary conditions and subtract f from u.
 
Last edited:

Markov

Member
Feb 1, 2012
149
Okay but on (1) why can we adopt that kind of solution?

As for second question, I have $v(x,t)=A{{e}^{-at}}-\left( A{{e}^{-at}}-B \right)\dfrac{x}{L}$ so I need $f(x,t)=v(x,t)+u(x,t),$ does this work? (I had a typo, the boundary conditions don't have the first derivative.)
 
Last edited: