Assignment 2: SEP291 Engineering Modelling (Submission due: 1st June 2015) 1. Find the Fourier series of the function “”##, of period ” = “# . “”##= !$ !% < # < & $ & < # <% ‘ $ = ( ” # $ %$ 2. Find the Fourier series of the function “”##, of period ” = “# . “”##= $%! #$ “!&< # <&#’ $ = $ 3. Solve the initial value problem by the Laplace transform “! +”” = #$ “%#&='() 4. Solve the initial value problem by the Laplace transform. If necessary, use partial fractions expansion. “!! ” ” #$ ” = %& “‘%(= )& “!’%(= % 5. Using the formulas L ” ” #$ = #L %”&’ “%(& or L ” “!!# = # $ L %”&’#”%(&” “!%(&, find L ” “# if “”## equals ( ” is a constant): “#$” . 6. Verify that the function “”##$$= %%& #’ + $’ ( )+& satisfies Laplace’s equation !” ” !#” + !” ” !$” = # and determine ” and ” so that ” satisfies the boundary conditions ” =””# on the circle “” + #” =# and ” = ” on the circle “” + #” = #$ . 7. Find “”##$$ for the string of length ” =” and “” = # when the initial velocity is zero and the initial deflection with small ” (say, 0.01) is as follows: “”##= $ $%&”‘!##. 8. Find “”##$$ for the string of length ” =” and “” =# when the initial velocity is zero and the initial deflection is as follows: “”##= $%$&#”‘! ##.

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