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s5_95c - ACM 95/100c May 7 2004 Problem Set V 0(2 pts Write...

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ACM 95/100c May 7, 2004 Problem Set V 0.- (2 pts) Write down your grading-section number. 1.- (55 pts) Solve the heat equation for w ( x, t ) in 0 < x < L ∂w ∂t - κ 2 w ∂x 2 = 0 with homogeneous boundary conditions w (0 , t ) = 0, w ( L, t ) = 0 and initial condition w ( x, 0) = f ( x ) using three different methods, and show that the various answers obtained coincide—as they should! (Method I) (5 pts) Separation of variables. (Method II) (30 pts) (a) (5 pts) Show that the Laplace transform in time transforms this equation into the inhomoge- neous ODE d 2 ˆ w dx 2 + λ ˆ w = - 1 κ f where λ = - s/κ and ˆ w ( x, s ) = 0 w ( x, t ) e - s t dt (b) (10 pts) Obtain the Green’s function for the equation d 2 ˆ w dx 2 + λ ˆ w = δ ( x - ξ ) with w (0 , t ) = 0, w ( L, t ) = 0, and use it to solve the inhomogeneous ODE for ˆ w ( x, s ) obtained in part (a) above. (c) (5 pts) Applying the inverse Laplace transform, show that w ( x, t ) can be written as w ( x, t ) = 1 2 πi n =0 A n c + i c - i sin nπ x L s + κn 2 π 2 L 2 e s t ds where A n = 2 L L 0 sin nπ ξ L f ( ξ )
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