Homework1 - u n ’s that satisfies u ( x, 1) = sin(2 πx...

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Math 212 , Fall 2007 Instructor : Friedemann Brock Homework assignment, 3–10 October 2007 1. Show that u ( x, y, t ) = (1 /t )exp[ - ( x 2 + y 2 ) / (4 kt )] satisfies the heat equation u t = k ( u xx + u yy ). 2. Show that u ( x, y, z ) = ( x 2 + y 2 + z 2 ) - 1 / 2 satisfies Laplace’s equation u xx + u yy + u zz = 0 for ( x, y, z ) 6 = (0 , 0 , 0). 3. Suppose that u 1 and u 2 are both solutions of the linear PDE Lu = F , where F 6 = 0. Under what conditions is the linear combination c 1 u 1 + c 2 u 2 also a solution of the equation ? 4. What form must G have for the differential equation u tt - u xx = G ( x, t, u ) to be linear ? Linear and homogeneous ? 5. (a) Show that for n = 1 , 2 , . . . , u n ( x, y ) = sin( nπx ) sinh( nπy ) satisfies u xx + u yy = 0 , u (0 , y ) = u (1 , y ) = u ( x, 0) = 0 . (b) Find a linear combination of the
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Unformatted text preview: u n ’s that satisfies u ( x, 1) = sin(2 πx )-sin(3 πx ). (c) Show that for n = 1 , 2 , . . . , e u n ( x, y ) = sin( nπx ) sinh( nπ (1-y )) satisfies u xx + u yy = 0 , u (0 , y ) = u (1 , y ) = u ( x, 1) = 0 . (d) Find a linear combination of the e u n ’s that satisfies u ( x, 0) = 2 sin( πx ). (e) Let D = (0 , 1) × (0 , 1). Solve the problem u xx + u yy = 0 in D, u (0 , y ) = u (1 , y ) = 0 , (0 ≤ y ≤ 1) , u ( x, 0) = 2 sin( πx ) , u ( x, 1) = sin(2 πx )-sin(3 πx ) , (0 ≤ x ≤ 1) . 1...
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