{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

05b_3-Clickers-PDES and Separation of Variables

# 05b_3-Clickers-PDES and Separation of Variables - PDEs...

This preview shows pages 1–10. Sign up to view the full content.

1 PDEs Partial Differential Equations

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 How does the prop constant depend on the area , A? A) linearly B) ~ some other positive power of A C) inversely D) ~ some negative power of A E) It should be independent of area! H ( x , t ) T ( x , t ) x Heat flow (H = Joules passing by/sec):
3 Thermal heat flow H(x,t) has units (J passing)/sec If you have H(x,t) entering on the left, and H(x+dx,t) exiting on the right, what is the energy building up inside, in time dt? x dx A A) H(x)-H(x+dx) B) H(x+dx)-H(x) C) ( H(x)-H(x+dx) ) dt D) ( H(x+dx)-H(x) ) dt E) Something else?! (Signs, units, factor of A, ...?)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
4 Thermal heat flow H(x,t) has units (J passing)/sec If you have H(x,t) entering on the left, and H(x+dx,t) exiting on the right, what is the energy building up inside, in time dt? x dx A A) H(x,dt)-H(x+dx,dt) B) H(x+dx,t+dt)-H(x,t) C) ( H(x,t)-H(x+dx,t) ) dt D) ( H(x+dx,t)-H(x,t) ) dt E) Something else?! (Signs, units, factor of A, ...?)
5 What is the general solution to Y’’(y)-k2Y(y)=0 (where k is some real constant) A) Y(y)=A eky+Be-ky B) Y(y)=Ae-kycos(ky-δ) C) Y(y)=Acos(ky) D) Y(y)=Acos(ky)+Bsin(ky) E) None of these or MORE than one!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
6 What is the general solution to X’’(x)+k2X(x)=0 A) X(x)=A ekx+Be-kx B) X(x)=Ae-kxcos(kx-δ) C) X(x)=Acos(kx) D) X(x)=Acos(kx)+Bsin(kx) E) None of these or MORE than one!
7 When solving T(x,y)=0, separation of variables says: try T(x,y) = X(x) Y(y) i) Just for practice , invent some function T(x,y) that is manifestly of this form . (Don’t worry about whether it satisfies Laplace's equation, just make up some function!) What is your X(x) here? What is Y(y)? ii) Just to compare, invent some function T(x,y) that is definitely NOT of this form. Challenge questions: 1) Did your answer in i) satisfy Laplace’s eqn? 2) Could our method (separation of variables) ever FIND your function in part ii above? 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
8 When solving T(x,y)=0, separation of variables says try T(x,y) = X(x) Y(y). We arrived at the equation f(x) + g(y) = 0 for some complicated f(x) and g(y) Invent some function f(x) and some other function g(y) that satisfies this equation. Challenge question: In 3-D, the method of separation of variables would have gotten you to f(x)+g(y)+h(z)=0. Generalize your “invented solution” to this case. 2
9 When solving T(x,y)=0, separation of variables says try T(x,y) = X(x) Y(y). We arrived at 2 d 2 X ( x ) dx 2 = cX ( x ) and d 2 Y ( y ) dy 2 = - cY ( y ) Write down the general solution to both of these ODEs!

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}