ExamFinalSolutions

ExamFinalSolutions - CHEE 3334 Fall '97 FINAL EXAM Michael...

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CHEE 3334 Fall ’97 Michael Nikolaou FINAL EXAM Open-book, open-notes. Total points 100. Please, budget your time ! Do NOT try to read the book during the exam! Write neatly! Do NOT use your programmable calculator to find results you are asked to compute in a certain way--after all, the calculator may find the wrong answer or not all answers! Show all work. Return this sheet with your exam . GOOD LUCK and HAPPY HOLIDAYS! The concentration u ( x , t ) of a species diffused through a membrane is described by the partial differential equation (PDE) 2 2 2 + = x u x u t u 0 , 1 0 < < t x (1) with initial condition ) 1 ( ) 0 , ( x x x u - = , for all x in the interval (0,1) (2) and boundary conditions 0 ) , 0 ( = t u , for all 0 t (3) t e t u - - = 1 ) , 1 ( , for all 0 t (4) (10 pts.) 1. Is the above PDE (equation (1)) linear or nonlinear? Justify your answer. An equation of the form L(u) = f, (where u is the unknown function and f is a forcing function that does not depend on u) is linear if the mapping L is linear. A mapping is linear if L(u 1 + u 2 ) = L(u 1 ) + L(u 2 ) for any u 1 , u 2 and L(cu) = c L(u) for any u and any number c. The above equations can be written as 2 2 2 0 u u u t x x - - = . It is straightforward that L(u 1 + u 2 ) L(u 1 ) + L(u 2 ), i.e. 2 2 2 2 2 2 1 2 1 2 1 2 1 1 1 2 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) u u u u u u u u u u u u t x t x t x x x x + + + - - ³ - - + - - ± ς ³ ÷♠ , and L(cu) c L(u), i.e. 2
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ExamFinalSolutions - CHEE 3334 Fall '97 FINAL EXAM Michael...

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