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Math · Advanced Math
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Question 4 (50 marks) A thin rod of unit length has its lateral surface insulated against the flow of heat. The material comprising the го1 has thermal conductivity 5.2, specific heat 4. 0, and density 1.3. The left end of the rod is insulated: 4(0, t) = 0 for t > 0 14(1,t) + 0. 511(1,t) = 0 for 2 0, u(x, 0) 100 for 0sxs 1. its right end radiates fi-eely into air of constant temperature zero: and the initial temperature distribution in the rod is uniform: (a) Use separation of variables and deduce the corresponding eigenvalue problem. (b) Find the condition(s) satisfied by the eigenvalues of this problem. Please show the details for the calculation in each case (c) Show that there exists an infinite sequence of eigenvalues for this problem and numerically approximate to eight decimal place accuracy the first three eigenvalues d) Write the corresponding eigenfunctions and the asymptotic behavior of the eigenvalues(e) Find a series expression for the temperature u - u(x,t) at any point x in the rod at any subsequent time t. Although you do not have to evaluate them, be sure to give explicit formulas for the coefficients in your series expression for the temperature function ) Approximate to eight decimal place accuracy the first three coefficients of your series expression. g Truncate your series expression for the temperature to the first three terms and use this to estimate the temperature of the rod at positionx0.5 and time t1.0. How accurate is your answer? How do you know this?

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