s11_mthsc208_ws7c-2DHeatEqn

s11_mthsc208_ws7c-2DHeatEqn - MthSc 208 (Spring 2011)...

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Unformatted text preview: MthSc 208 (Spring 2011) Worksheet 7c MthSc 208: Differential Equations (Spring 2011) In-class Worksheet 7c: The 2D Heat Equation NAME: We will solve for the function u ( x,y,t ) defined for 0 x,y and t 0 which satisfies the following initial value problem of the heat equation: u t = c 2 ( u xx + u yy ) u (0 ,y,t ) = u ( ,y,t ) = u ( x, ,t ) = u ( x,,t ) = 0 , u ( x,y, 0) = 2sin x sin2 y + 3sin4 x sin5 y . (a) Carefully descsribe (and sketch) a physical situation that this models. (b) Assume that u ( x,y,t ) = f ( x,y ) g ( t ). Compute u xx , u yy , and u t , find boundary conditions for f ( x,y ). Written by M. Macauley 1 MthSc 208 (Spring 2011) Worksheet 7c (c) Plug u = fg back into the PDE and separate variables by dividing both sides of the equation by c 2 fg . Set this equal to a constant , and write down two equations: an ODE for g ( t ), and a PDE f ( x,y ) (called the Helmholtz equation ), with four boundary conditions....
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s11_mthsc208_ws7c-2DHeatEqn - MthSc 208 (Spring 2011)...

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