A The boundary conditions in reference to Laplace's equation

chwala
Gold Member
Messages
2,825
Reaction score
413
TL;DR Summary
I am looking at Laplace equation. The beginning steps are pretty clear...kindly see attached... I need some insight on the given boundary conditions
1669895975385.png


We have inhomogenous dirichlet boundary conditions (well understood)....the laplace equation is a steady state equation and we can clearly see that in 2D..it will be defined by 4 boundary conditions and NO initial condition...having said that; kindly have a look at the continuation below...

1669896171570.png


I can follow the steps; In the attempt to solve; they made ##3## boundary conditions homogenous thereby leaving one inhomogenous boundary condition to work with to realize steps to solution...
Now my question is; Why the choice of ##f_1(x)## ? or it does not matter...could we as well have worked with ##u(0,y)=g_1(y)## and made the other ##3## conditions homogenous? The other steps are not difficult to follow....am on that now cheers!
 

Attachments

  • 1669896106517.png
    1669896106517.png
    21.9 KB · Views: 148
Physics news on Phys.org
chwala said:
TL;DR Summary: I am looking at Laplace equation. The beginning steps are pretty clear...kindly see attached... I need some insight on the given boundary conditions

Now my question is; Why the choice of f1(x) ?
The example simply chooses one of the edges to be a nonzero function. The choice is arbitrary and could be any edge (although a particular choice may make the notation simpler)
 
Ok I will explore this...looks like if we were to use ##g_1(y)## then I guess that we shall be dealing with solutions (both general and particular) in terms of ##y##...
 
Hmmmmm...getting to understand this isn't a walk in the park...spent a whole 25 minutes trying to go through the problem...Will post my area of doubts/clarity later...
 
Now my question is; Why the choice of ##f_1(x)## ? or it does not matter...could we as well have worked with ##u(0,y)=g_1(y)## and made the other ##3## conditions homogenous? The other steps are not difficult to follow....am on that now cheers!

The problem is linear; you can decompose it as the sun of four problems, each of which is homogenous on three sides and inhomogenous on the fourth.
 
I am still on this problem...i have a query on the highlighted part of the notes...can we use the alternative approach below?

We have,

...

##Y=C \cosh (λy) + D \sinh (λy)## now on using the boundary condition ##u(x,b)=0##

We can also get;

##0=C\cosh (λb)+0##

##0=C+ 0##

##⇒C=0##

Instead of the ##C=-D\tanh (λb)## given on the text. Would that also be correct?

It then follows that,

##⇒Y=D \sinh (λy)##

then the other steps may follow i.e
...
##U_n( x,y) = \sin \dfrac{nπx}{a} ⋅ \sinh \dfrac{nπy}{a} ##

Just another question on the given general solution; why did they ignore/leave out the Constants B and D ?

We ought to have;

##U_n( x,y) = B\sin \dfrac{nπx}{a} ⋅ D \sinh \dfrac{nπy}{a} ##
 

Attachments

Last edited:
chwala said:
I am still on this problem...i have a query on the highlighted part of the notes...can we use the alternative approach below?

We have,

...

##Y=C \cosh (λy) + D \sinh (λy)## now on using the boundary condition ##u(x,b)=0##

We can also get;

##0=C\cosh (λb)+0##

Why is D\sinh (\lambda b) zero?

##0=C+ 0##

##⇒C=0##

Instead of the ##C=-D\tanh (λb)## given on the text. Would that also be correct?

No. You need Y(0) \neq 0 in order to apply the boundary condition on y = 0. That means you need a non-zero coefficient of \cosh (\lambda y). What the text is saying is that <br /> \begin{split} <br /> \sinh(\lambda(b-y)) &amp;= \sinh(\lambda b)\cosh(\lambda y) - \cosh(\lambda b)\sinh(\lambda y) \\<br /> &amp;= \cosh(\lambda b)\left(\tanh(\lambda b)\cosh(\lambda y) - \sinh(\lambda y)\right) \end{split} is a solution of Y&#039;&#039; = \lambda^2 Y which satisfies Y(b) = 0 and Y(0) \neq 0 - and every non-zero multiple of that solution is also a solution.

It then follows that,

##⇒Y=D \sinh (λy)##

then the other steps may follow i.e
...
##U_n( x,y) = \sin \dfrac{nπx}{a} ⋅ \sinh \dfrac{nπy}{a} ##

Just another question on the given general solution; why did they ignore/leave out the Constants B and D ?

We ought to have;

##U_n( x,y) = B\sin \dfrac{nπx}{a} ⋅ D \sinh \dfrac{nπy}{a} ##

How do you determine the coefficients a_n in u(x,0) = \sum_{n} a_n U_n(x,0), and how does the value of U_n(x,0) \neq 0 affect the value of a_n?
 
pasmith said:
Why is D\sinh (\lambda b) zero?
No. You need Y(0) \neq 0 in order to apply the boundary condition on y = 0. That means you need a non-zero coefficient of \cosh (\lambda y). What the text is saying is that <br /> \begin{split}<br /> \sinh(\lambda(b-y)) &amp;= \sinh(\lambda b)\cosh(\lambda y) - \cosh(\lambda b)\sinh(\lambda y) \\<br /> &amp;= \cosh(\lambda b)\left(\tanh(\lambda b)\cosh(\lambda y) - \sinh(\lambda y)\right) \end{split} is a solution of Y&#039;&#039; = \lambda^2 Y which satisfies Y(b) = 0 and Y(0) \neq 0 - and every non-zero multiple of that solution is also a solution.
How do you determine the coefficients a_n in u(x,0) = \sum_{n} a_n U_n(x,0), and how does the value of U_n(x,0) \neq 0 affect the value of a_n?

'Why is D\sinh (\lambda b) zero?'

Response;

I used the boundary condition ##u(x,b)=0## on

##Y=C \cosh (λy) + D \sinh (λy)## to give me

##0=C \cosh (λb) + D \sinh (λb)##

##0=C\cosh (λb)+0##

##0=C+ 0##

##⇒C=0##

It then follows that,

##Y=D \sinh (λy)##

when ##b=0## then it follows that ##\sinh (λb)=0## and ##\cosh (λb)=1## .

Let me know if this reasoning is correct. I used that in arriving at my general solution as indicated above...

Kindly note that the difference in my approach and the text is on ##y## and the ##(y-b)##...

##U_n( x,y) = \sin \dfrac{nπx}{a} ⋅ \sinh \dfrac{nπ(y-b)}{a} ## [text approach]

##U_n( x,y) = \sin \dfrac{nπx}{a} ⋅ \sinh \dfrac{nπy}{a} ## [My approach]
 
Last edited:
pasmith said:
Why is D\sinh (\lambda b) zero?
No. You need Y(0) \neq 0 in order to apply the boundary condition on y = 0. That means you need a non-zero coefficient of \cosh (\lambda y). What the text is saying is that <br /> \begin{split}<br /> \sinh(\lambda(b-y)) &amp;= \sinh(\lambda b)\cosh(\lambda y) - \cosh(\lambda b)\sinh(\lambda y) \\<br /> &amp;= \cosh(\lambda b)\left(\tanh(\lambda b)\cosh(\lambda y) - \sinh(\lambda y)\right) \end{split} is a solution of Y&#039;&#039; = \lambda^2 Y which satisfies Y(b) = 0 and Y(0) \neq 0 - and every non-zero multiple of that solution is also a solution.
How do you determine the coefficients a_n in u(x,0) = \sum_{n} a_n U_n(x,0), and how does the value of U_n(x,0) \neq 0 affect the value of a_n?
If i am getting you right, we are going to use the inhomogenous boundary condition to determine the coefficient ##a_n##.

My question is in reference to the step before applying the inhomogenous boundary condition...i can follow the steps quite well...i just want to know if the reason behind ignoring the coefficents ##B## and ##D## in

##U_n( x,y) = B\sin \dfrac{nπx}{a} ⋅ D \sinh \dfrac{nπ(y-b)}{a} ##

is due to the fact that they are inconsequential...or they will be taken care when determining value of ##a_n##. Cheers.
 
  • #10
chwala said:
'Why is D\sinh (\lambda b) zero?'

Response;

I used the boundary condition ##u(x,b)=0## on

##Y=C \cosh (λy) + D \sinh (λy)## to give me

##0=C \cosh (λb) + D \sinh (λb)##

##0=C\cosh (λb)+0##

But you can't have D\sinh(\lambda b) = 0 withouh at least one of D, \lambda, and b being zero. We know that b \neq 0, and by assumption \lambda &gt; 0; therefore you must have D = 0. So when you subsequently conclude that C = 0 you in fact have Y \equiv 0 which is not allowed.
 
  • #11
pasmith said:
But you can't have D\sinh(\lambda b) = 0 withouh at least one of D, \lambda, and b being zero. We know that b \neq 0, and by assumption \lambda &gt; 0; therefore you must have D = 0. So when you subsequently conclude that C = 0 you in fact have Y \equiv 0 which is not allowed.
Thanks, let me look at this again slowly and in detail...fun going through this...though a bit confusing...

...i noted the use of the homogenous boundary conditions in this particular order ##u(0,y), u(a,y), ## that were applied on ##X## and ##u(x,b), u(a,y)## that were applied on ##Y##.

I will need to go all over again then finish with the last part that will make use of the inhomogenous boundary condition. Cheers mate.
 
  • #12
pasmith said:
But you can't have D\sinh(\lambda b) = 0 withouh at least one of D, \lambda, and b being zero. We know that b \neq 0, and by assumption \lambda &gt; 0; therefore you must have D = 0. So when you subsequently conclude that C = 0 you in fact have Y \equiv 0 which is not allowed.
...just confirm i thought we have ##Y(b)=0## thus ##b## can indeed be equal to ##0##... or am i not getting it right?
 
  • #13
chwala said:
...just confirm i thought we have ##Y(b)=0## thus ##b## can indeed be equal to ##0##... or am i not getting it right?

b is fixed by the problem: it's part of the definition of the domain of u. You don't get to choose it to your advantage.
 
  • #14
pasmith said:
b is fixed by the problem: it's part of the definition of the domain of u. You don't get to choose it to your advantage.
If you look at the text; they have indicated that

##Y=C \cosh (λy) + D \sinh (λy)## can match ##Y(b)=0##. I just followed the text and did not choose it...not unless my interpretation is wrong.
 
  • #15
chwala said:
If you look at the text; they have indicated that

##Y=C \cosh (λy) + D \sinh (λy)## can match ##Y(b)=0##. I just followed the text and did not choose it...not unless my interpretation is wrong.

I don't see that in the text. Do you have an equation number reference for it?
 
  • #16
pasmith said:
I don't see that in the text. Do you have an equation number reference for it?
Check on page ##4## of the attached pdf.
 
  • #17
I see
Y&#039;&#039; - \lambda^2 Y = 0 \Rightarrow Y(y) = C\cosh \lambda y + D\sinh \lambda y can only match Y(0) = 0 or Y(b) = 0
and
Y&#039;&#039; + \lambda^2 Y = 0 \Rightarrow Y(y) = C\cos \lambda y + D\sin \lambda y can match Y(0) = 0 = Y(b)
We are looking at the first case.
 
  • #18
chwala said:
I am still on this problem...i have a query on the highlighted part of the notes...can we use the alternative approach below?

We have,

...

##Y=C \cosh (λy) + D \sinh (λy)## now on using the boundary condition ##u(x,b)=0##

We can also get;

##0=C\cosh (λb)+0##

##0=C+ 0##

##⇒C=0##

Instead of the ##C=-D\tanh (λb)## given on the text. Would that also be correct?

It then follows that,

##⇒Y=D \sinh (λy)##

then the other steps may follow i.e
...
##U_n( x,y) = \sin \dfrac{nπx}{a} ⋅ \sinh \dfrac{nπy}{a} ##

Just another question on the given general solution; why did they ignore/leave out the Constants B and D ?

We ought to have;

##U_n( x,y) = B\sin \dfrac{nπx}{a} ⋅ D \sinh \dfrac{nπy}{a} ##
This was wrong thinking on my part! I am on it now... actually before this step we have;

...

##u(x,b)=X(x)⋅Y(b)=0##, that is from the given boundary condition; ##u(x,b)=0##, therefore, it follows that ## Y(b)=0##.

Thus we shall have;

##y(b)=C \cosh(λb)+D \sinh(λb)##

##0=C \cosh(λb)+D \sinh(λb)##

##⇒C=-D\tanh (λb)## ...

My point is that i was wrong by stating that ##C=0##.

Cheers
 
  • #19
I am conversant now with solving this problem at ease; just need some clarification on the highlighted part;

1671799724268.png


1671799754756.png


on the integral part... looks like something to do with Fourier?...or something else...cheers mate!
 
  • #20
Yes, it is a fourier series.
 
  • #21
Thanks @pasmith happy festivities mate!:cool:
 
  • #22
hutchphd said:
The example simply chooses one of the edges to be a nonzero function. The choice is arbitrary and could be any edge (although a particular choice may make the notation simpler)
Just to be absolutely clear on this; You said we can choose to make any of the boundary conditions: ##u(x,0), u(x,b), u(a,y) ## and ##u(0,y)## to be inhomogenous?
 
  • #23
chwala said:
I am conversant now with solving this problem at ease; just need some clarification on the highlighted part;

View attachment 319260

View attachment 319261

on the integral part... looks like something to do with Fourier?...or something else...cheers mate!

Yeah now i understand this; from fourier sine series;

$$B_n=\dfrac{1}{a}\int_{-a}^a f_1(x) \sin (\dfrac{nπx}{a})dx=\dfrac{2}{a}\int_0^a f_1(x) \sin (\dfrac{nπx}{a})dx$$ with ##n=1,2.3,...##

Am looking at the Fourier series now (for both ##\sin## and ##\cos## ) -am refreshing actually...they are pretty easy! only thing challenging is the integration of parts-sometimes 2 to 3 times- phew :wink: and dealing with the constants ...here and there... otherwise, they're quite easy!
 
Last edited:
Back
Top