# 4112017.pdf - MATH20411 TWO hours UNIVERSITY OF MANCHESTER...

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MATH20411 TWO hours UNIVERSITY OF MANCHESTER PARTIAL DIFFERENTIAL EQUATIONS AND VECTOR CALCULUS B 24th January 2017 14:00–16:00 2 Hours Answer THREE out of FOUR questions. If more than THREE questions are attempted, then credit will be given for the best THREE answers. Electronic calculators may be used, provided that they cannot store text. 1 of 5 P.T.O.
MATH20411 1. Consider Laplace’s equation 2 u ∂x 2 + 2 u ∂y 2 = 0 , 0 < x < π, 0 < y < 1 . a) State whether Laplace’s equation is (i) elliptic, parabolic or hyperbolic, (ii) linear or nonlinear, and (iii) homogeneous or non-homogeneous. [3 marks] b) Using the method of separation of variables, find all the non-trivial solutions to Laplace’s equation in the rectangular domain [0 , π ] × [0 , 1] of the form u n ( x, y ) = X n ( x ) Y n ( y ) , n = 0 , 1 , . . . , that satisfy the boundary conditions u (0 , y ) = 0 , ∂u ∂x ( π, y ) = 0 , ∂u ∂y ( x, 0) = 0 . [12 marks] c) Using your answer to part b), construct a series solution of the form u ( x, y ) = X n =0 A n X n ( x ) Y n ( y ) that also satisfies the additional boundary condition u ( x, 1) = f ( x ) , for a suitable function f ( x ). Define the coefficients A n . (You do not need