midterm2

# midterm2 - u xx-10 xu x = u t t> 0 0 ≤ x ≤ 1 u(0,t =...

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Name Instructor: Stefano Guerra No Calculators. Closed book and notes. Write your ﬁnal answers in the boxes provided. You may use the back of the pages. Each question is worth 15 points. Second Midterm Exam Problem 1 Problem 2 Problem 3 Problem 4 1

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1. (a) Consider the function: f ( x ) = x - 2 x 0 0 0 x 1 2 x 1 x < 2 (1) exteded by periodicity outside the interval - 2 x 2. Compute the Fourier series of f ( x ) (Note: the integrals you have to compute are in the formala sheet. You might have to performe a change of variable ﬁrst!). 2
(b) Sketch the graph of f ( x ) and of the function which is the limit of the Fourier series in the interval - 6 x 6. 3

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2. Solve the heat equation: u xx = u t t > 0 0 x 1 u (0 ,t ) = - 1 t > 0 u (1 ,t ) = 1 t > 0 u ( x, 0) = 1 - 2 x - 4 sin (3 πx ) + 11 sin (10 πx ) 0 x 1 (2) 4
3. Consider the boundary problem for partial diﬀerential equation:

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Unformatted text preview: u xx-10 xu + x = u t t > 0 0 ≤ x ≤ 1 u (0 ,t ) = u (1 ,t ) t > u x (0 ,t ) = u x (1 ,t ) t > u ( x, 0) = x 2 (3) (a) Use separation of variables to ﬁnd the boundary value problem that the function X ( x ) has to solve in order for u ( x,t ) = X ( x ) T ( t ) to be a nonzero solution of the homogenous boundary problem associated to the equation above. 5 (b) Is the boundary problem you found self-adjoint? Justify yout answer. 6 4. Find the solution of the Laplace equation: u xx + u yy = 0 ≤ x ≤ a ≤ y ≤ b u ( x, 0) = 0 ≤ x ≤ a u ( x,b ) = f ( x ) 0 ≤ x ≤ a u (0 ,y ) = 0 ≤ y ≤ b u ( a,y ) = g ( y ) 0 ≤ y ≤ b (4) 7...
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midterm2 - u xx-10 xu x = u t t> 0 0 ≤ x ≤ 1 u(0,t =...

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