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05b_3-Clickers-PDES and Separation of Variables

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

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1 PDEs Partial Differential Equations
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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):
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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, ...?)
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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, ...?)
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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!
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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!
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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
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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
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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!
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