Chem Differential Eq HW Solutions Fall 2011 46

Chem Differential Eq HW Solutions Fall 2011 46 - 46 Chapter...

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Unformatted text preview: 46 Chapter 3 Partial Differential Equations in Rectangular Coordinates Solving the equation for T , we find T (t) = et ; thus we have the product solution c 0 e− x et , where, for convenience, we have used c0 as an arbitrary constant. (d) If k = −α2 < 0, then X + µ2 X = 0 ⇒ X = c1 cos µx + c2 sin µx; X (0) = −X (0) ⇒ µc2 = −c1 X (1) = −X (1) ⇒ −µc1 sin µ + µc2 cos µ = −c1 cos µ − c2 sin µ ⇒ −µc1 sin µ − c1 cos µ = −c1 cos µ − c2 sin µ ⇒ −µc1 sin µ = −c2 sin µ c1 ⇒ −µc1 sin µ = sin µ. µ Since µ = 0, take c1 = 0 (otherwise you will get a trivial solution) and divide by c1 and get µ2 sin µ = − sin µ ⇒ sin µ = 0 ⇒ µ = nπ, where n is an integer. So X = c1 cos nπx + c2 sin nπx. But c1 = −c2µ, so X = −c1 nπ cos nπx − sin nπx . Call Xn = nπ cos nπx − sin nπx. (e) To establish the orthogonality of the Xn ’s, treat the case k = 1 separately. For k = −µ2 , we refer to the boundary value problem X + µ2 X = 0, X (0) = −X (0), X (1) = −X (1), n that is satisfied by the Xn ’s, where µn = nπ . We establish orthogonality using a trick from Sturm-Liouville theory (Chapter 6, Section 6.2). Since Xm = µ2 Xm and Xn = µ2 Xn , m n multiplying the first equation by Xn and the second by Xm and then subtracting the resulting equations, we obtain Xn Xm = µ2 Xm Xn and Xm Xn = µ2 Xn Xm m n Xn Xm − Xm Xn = (µ2 − µ2 )XmXn n m X n Xm − X m Xn = (µ2 − µ2 )XmXn n m where the last equation follows by simply checking the validity of the identity Xn Xm − Xm Xn = Xn Xm − Xm Xn . So 1 (µ2 − µ2 ) n m 1 Xn (x)Xm (x) − Xm (x)Xn (x) dx Xm (x)Xn (x) dx = 0 0 1 = Xn (x)Xm(x) − Xm (x)Xn(x) , 0 because the integral of the derivative of a function is the function itself. Now we use the boundary conditions to conclude that 1 Xn (x)Xm(x) − Xm (x)Xn(x) 0 = Xn (1)Xm(1) − Xm (1)Xn(1) − Xn (0)Xm(0) + Xm (0)Xn(0) = −Xn (1)Xm(1) + Xm (1)Xn(1) + Xn (0)Xm(0) − Xm (0)Xn(0) = 0. ...
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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