MATH 3150: PDE FOR ENGINEERS FINAL EXAM (VERSION D) 1. Consider the heat equation in a wire whose diffusivity varies over time: u k(t) 2 x 2

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1 MATH 35: PDE FOR ENGINEERS FINAL EXAM (VERSION D). Consider the heat equation in a wire whose diffusivity varies over time: u t = u k(t) x where k(t) is some positive function of time. Assume the wire is infinitely long in both directions. (a) Find the general solution to the heat equation in the form of convolution u(x, t) = G(x, t) u(x, ) and (b) write out explicitly the function G(x, t) in terms of t k(t) dt and x. Hint: the ordinary differential equation dz dt = A(t)Z(t) has solution Z(t) = e t A(t) dt Z(). (c) Calculate G(x, t) for k(t) = e t an exponentially decaying diffusivity. (a) Let a(t) = t k(t) dt. Then a(t) G(x, t) = e a(t)x /4. (b) a(t) = t e t dt so that = e t G(x, t) = ( e t ) e x /4( e t).. f(x) = { x x < π x π x < and f(x) has period π. (a) Draw the graph of f(x), including at least two periods in your graph. Date: May 9,.

2 MATH 35: PDE FOR ENGINEERS FINAL EXAM (VERSION D) (b) Calculate the real Fourier amplitudes a m and b m of f(x). Hint: x cos x dx = cos x + x sin x. (c) Find the energy of f(x). (d) In the form of a fraction, find the total percentage of energy contained among the amplitudes a, a, b, a, b. Do not use any decimal approximations. (e) Show that over 95% of the energy is stored among the amplitudes a, a, b, a, b. Hint: π (a) See figure a on the facing page. (b) Because the function is even, b m =. a = fracl = π = π. L x dx f(x) dx (c) a m = L = so a k+ = while Thus f(x) = π 4 π L f(x) cos ( πmx ) dx L = x cos (mx) dx π = πm (( )m ) { 4 πm if m is odd if m is even 4 a k+ = π(k + ). (cos(x) + 3 cos(3x) + 5 cos(5x) +... ). f = = T/ T/ = π3 3. f(x) dx x dx

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4 4 MATH 35: PDE FOR ENGINEERS FINAL EXAM (VERSION D) (d) ( T a + T a + b + a + b) f = π (π/) + π 6 π ( ) π 3 /3 (e) = π3 / + 6 π π 3 /3 = π π > = = 99 = 99%. 3. True or false: the Fourier transform of the function { e x if x f(x) = otherwise is False. ˆf(ω) = + ω. 4. Calculate the kernel K(x, t) for the heat equation with convection u t = k u x + c u x where c and k are constants, so that the solution of the heat equation with convection will be u(x, t) = u(s, )K(x s, t) ds. π It might help to know the integral: π e ap e ipx dp = a e x /4a where a is any constant. û t = ( kip c p ) û so that û(p, t) = e t(kip c p )û(p, ) and u(x, t) = F ( e t(kip c p ) ) u(x, ).

5 MATH 35: PDE FOR ENGINEERS FINAL EXAM (VERSION D) 5 So K(x, t) = F ( e t(kip c p ) ) = π = π e tc p e tkip e ipx dp e tc p e ip(x+tk) dp = tc e (x+tk) /4tc. 5. Expand the function f(x) = arctan x in a Taylor expansion to four terms about x =. Hint: f (x) = + x = x + x 4 x (a geometric series), and the Taylor expansion is f(x) = f() + f ()! x + f () x +! f(x) = x x3 3 + x5 5 x Take a disk of unit radius and heat the right half of the edge of the disk to o and the left half of the edge to o. Let the disk sit with these temperatures on its edges for a long time, until it reaches a steady state. Center the disk at origin of coordinates. Draw a graph of u as a function of radius r at θ =, from r = to r =, clearly labelling the maximum and minimum values on the vertical axis. Hint: recall that if we write the temperature of the edge of the disk as f(θ), and write the Fourier amplitudes of f(θ) as a = π a m = π b m = π π π π f(θ) dθ f(θ) cos(mθ) dθ f(θ) sin(mθ) dθ then the steady state temperature inside the plate, in polar coordinates, is u(r, θ) = a + r m (a m cos (mθ) + b m sin (mθ)) m= Hint: now compare with the last problem. The function is { if π/ < θ < π/ f(θ) = otherwise.

6 6 MATH 35: PDE FOR ENGINEERS FINAL EXAM (VERSION D) It Fourier coefficients are u(r, θ) = + π a = a m = π = πm b m = / π/ (since f(θ) is even). Thus k= Plugging in θ = gives cos(mθ) dθ if m = + 4k, k =,,,... if m = 3 + 4k, k =,,,... if m is even. ( ) r +4k + 4k cos (( + 4k) θ) r3+4k cos ((3 + 4k) θ) k u(r, θ) = + π ) (r r3 3 + r5 5 r = + π arctan(r). The graph is drawn in figure 6 on the next page. 7. Take a function f(x) with period T and form the function g(x) = f(ax) for some constant a. The period of g(x) is (a) at (b) T/a (c) π (d) πa/t (e) none of the above. The energy of g(x) is related to the energy of f(x) by (a) E(g) = E(f) (b) E(g) = a E(f) (c) E(g) = ae(f) (d) E(g) = a E(f) (e) E(g) = a E(f) (f) none of the above. (b) and (d). 8. (a) Suppose that u(x, t) satisfies the wave equation u t = c u x with u = at x = and x = L at all times t. Why is it true that ( ) u u = u u x x t x t + u u x x t?

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8 8 MATH 35: PDE FOR ENGINEERS FINAL EXAM (VERSION D) (b) Show that u(x, t) has constant (in time) energy, where the energy of u(x, t) is defined to be E = ( L ( u ) ( ) ) u + c dx. t x Hint: differentiate in t, use the wave equation to get rid of u t. Do not use any Fourier series or transforms. Do not separate variables. Warning: this notion of energy is different from the one we studied for Fourier series. (a) The chain rule for partial derivatives. (b) de dt = d ( L ( u ) ( ) ) u + c dx dt t x = L ( u u u ) u + c dx t t x t x L ( u = t c u u ) u + c dx x x t x L ( ) = c u u dx x x t = c u x=l u x t x= = because u = at x = and x = L all the time, and so its rate of change is u t = there too.