MATH36041.pdf - MATH36041 Two hours UNIVERSITY OF...

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MATH36041 Two hours UNIVERSITY OF MANCHESTER ESSENTIAL PARTIAL DIFFERENTIAL EQUATIONS 16 January 2017 14:00-16:00 Answer ALL five questions in Section A (40 marks in total) and TWO of the three questions in Section B (60 marks in total). If more than two questions from Section B are attempted, then credit will be given for the best two answers. University approved calculators may be used 1 of 5 P.T.O.
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MATH36041 SECTION A Answer ALL questions A1. Give definitions of the following (a) A norm on a vector space. [2 marks] (b) A complete normed vector space. [3 marks] (c) An inner product on a vector space. [2 marks] (d) The weak derivative of u L 2 (0 , 1). [3 marks] (e) The Sobolev space H 1 (0 , 1) [3 marks] (Total marks: 13) A2. State the Lax-Milgram theorem. [6 marks] The remaining problems in section A refer to the following classical problem. For f C 0 ([0 , 1]), find u C 2 (0 , 1) C 0 ([0 , 1]) such that (BVP1) - u 00 ( x ) = f ( x ) , x (0 , 1) u (0) = u (1) = 0 . A3. What is the weak formulation of the classical problem (BVP1)? [5 marks] A4. Apply the Lax-Milgram theorem to show that there is always a unique solution to the weak for- mulation of (BVP1). Fully explain why all hypotheses of the theorem are satisfied.
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