A community for students.
Here's the question you clicked on:
 0 viewing
anonymous
 3 years ago
Find the steadystate temperature distribution for the semiinfinite plate problem if the temperature of the bottom edge is T = f(x) = x (in degrees; that is, the temperature at x cm is x degrees), the temperature of the other sides is 0o, and the width of the plate is 10 cm.
anonymous
 3 years ago
Find the steadystate temperature distribution for the semiinfinite plate problem if the temperature of the bottom edge is T = f(x) = x (in degrees; that is, the temperature at x cm is x degrees), the temperature of the other sides is 0o, and the width of the plate is 10 cm.

This Question is Closed

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0@UnkleRhaukus @oldrin.bataku @sirm3d @niksva @ParthKohli and others, would you kindly help me?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0would u give us a geometry of problem? :)

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0i would give a partial differential equation...,

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0ok, and boundary conditions

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0\[\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}=0\]???

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0Hint: Answer:\[T = \frac{ 20 }{ \pi } \sum_{n=1}^{\infty} \frac{ (1)^{n+1} }{ n } e^{n \pi y/10} \sin (n \pi x/10)\] How to get that.., ??

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0ok, u must use "separation of variables"

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0suppose that steady state solution can be written as\[T(x,y)=X(x).Y(y)\]plug in the original equation, u will get\[X''Y+XY''=0\]\[\frac{X''}{X}=\frac{Y''}{Y}=\color\red{\pm}\lambda^2\]now what do u think positive sign or positive?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0sorry positive or negative?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0lambda is a positive number, an arbitrary constant...but why we put it there? see this equation\[\frac{X''}{X}=\frac{Y''}{Y}\]LHS is a function of only \(x\) and RHS is a function of only \(y\) its not possible unless both sides are equal to a constant number

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0sorry i've no idea about positive and negative, but i just know that it's separation constant

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0what do u think? what are boundary conditions?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0there are several methods to verify the sign of constant of separation

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0the boundary conditions is l from 0 up to 10, right ?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0what math u are in? Advanced Engineering Mathematics?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0mathematical physics ..,

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0ok, boundary conditions\[T=f(x) \ \ @ \ \ y=0\]\[T=0 \ \ @ \ \ y \rightarrow \infty\]\[T=0 \ \ @ \ \ x=0\]\[T=0 \ \ @ \ \ x=10\]what do u think?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0i think dw:1366283650608:dw

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0dw:1366283758632:dw

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0exactly, and \(f(x)=x\) of course

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0so lets think what will be the sign of \(\lambda^2\)

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0because of solution may be either positive or negative ??

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0first suppose positive\[\frac{X''}{X}=\lambda^2\]\[X''\lambda^2 X=0\]\[X=A\sinh \lambda x+B\cosh \lambda x\]set boundary conditions\[T=0 \ \ @ \ \ x=0\]will result in \(X(0)=0\) and it gives \(B=0\)\[T=0 \ \ @ \ \ x=10\]will result in \(X(10)=0\) and it gives \(A=0\) so \(X\) will be equal to zero which is not acceptable so we must choose negative sign

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0ok, so we have\[\frac{X''}{X}=\frac{Y''}{Y}=\color\green{}\lambda^2\]\[X''+\lambda^2 X=0\]\[Y''\lambda^2 Y=0\]\[X=A\sin \lambda x+B\cos \lambda x\]\[Y=A\sinh \lambda y+B\cosh \lambda y\]and finally setting boundary conditions

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0\[T=0 \ \ @ \ \ x=0\]will result in \(X(0)=0\) and it gives \(B=0\)\[T=0 \ \ @ \ \ x=0\]will result in \(X(10)=0\) and it gives \(10\lambda=n\pi\) and\[\lambda=\frac{n\pi}{10} \ \ \ \color\green{\text{eigenvalues}}\]

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0or if u want get the exact form wich is given in solution u can write\[Y=Ce^{\lambda y} +De^{\lambda y} \]

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0general solution of\[Y''\lambda^2 Y=0\]can be written as\[Y=C\sinh \lambda y+D\cosh \lambda y\]or\[Y=Ce^{\lambda y} +De^{\lambda y}\]thats for u...find out why :)

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0now we know as \(y \rightarrow \infty\) temperature approches zero so \(C=0\)

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0its a simple limit calculation, right?

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0so summing the answers for \(n=1,2,3,...\) we will have\[T = \sum_{n=1}^{\infty} a_ne^{\frac{n \pi}{10} y} \sin (\frac{n \pi}{10} x)\]for calculating \(a_n\) final boundary condition will be useful\[T=f(x)=x \ \ @ \ \ y=0\]so\[x = \sum_{n=1}^{\infty} a_n\sin (\frac{n \pi}{10} x)\]from fourier series\[a_n=\frac{2}{10} \int_{0}^{10} x\sin (\frac{n \pi}{10} x) \ \text{d}x=\frac{20}{n \pi} (1)^{n+1}\]We are done,I hope its helpful

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0let me know if it is unclear.

anonymous
 3 years ago
Best ResponseYou've already chosen the best response.0ah thnak you so much :)
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.