midterm1a

# midterm1a - OCTOBER 2009 MIDTERM TEST 1 APM346H1 Partial...

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Unformatted text preview: OCTOBER 2009 MIDTERM TEST 1 APM346H1 Partial Diﬀerential Equations (closed book test, no calculators) Instructors: M. Chugunova, V. Ivrii Marker: Ioannis Anapolitanos Duration: 1.5 hours Total Marks: 100 Question 1 (5 marks ). Write the general form of the linear second order homogenous PDE with the unknown function u(x, y ). Question 2 (5 marks ). Write the general form of the ﬁrst order PDE with the unknown function u(x, y, z ). Question 3 (5 marks ). Write any example of the non-homogeneous hyperbolic PDE with the unknown function u(x, y ). Question 4 (5 marks ). Find the general solution u(x, y ) of the equation uxy = x + cos(y ). Question 5 (5 marks ). Find the general solution u(x, y ) of the equation ux − 2 uy = sin(x). Question 6 (5 marks ). Find the general solution u(x, y ) of the equation uxx − 4 uyy = x. Question 7 (5 marks ). Find the solution u(x, y ) of the initial value problem ux = ey , u(1, y ) = y. Question 8 (5 marks ). Find the characteristic curves for the equation ux + y uy = 0. Question 9 (5 marks ). Find any change of variables ξ = ξ (x, y ), the equation uxx − uxy − 6 uyy = 0 to uξη = 0. η = η (x, y ) that will transform Question 10 (5 marks ). Find the characteristic lines for the equation uxx − 9 uyy = 0. Problem 1 (20 marks ). Find the solution u(x, y ) of the initial value problem uxx + 2 uxy − 3 uyy = x + y, u(x, 0) = sin(x), uy (x, 0) = 1. Problem 2 (20 marks ). Solve the initial value problem 2 ux + (u + 1) uy = u, Problem 3 (10 marks ). Compute bolic equation uxx − 4 uyy = ex−2 y . u(x, 2 x) = ex . is a characteristic triangle of the hyper- ex−2 y dxdy where ...
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