{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

s11_mthsc208_ws7c-2DHeatEqn

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

This preview shows pages 1–4. Sign up to view the full content.

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 π 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 ( π, y, t ) = u ( x, 0 , t ) = u ( x, π, t ) = 0 , u ( x, y, 0) = 2 sin x sin 2 y + 3 sin 4 x sin 5 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 λ , and write down two equations: an ODE for g ( t ), and a PDE f ( x, y ) (called the Helmholtz equation ), with four boundary conditions. (d) Solve the ODE for g ( t ). (e) To solve the PDE for f , assume that f ( x, y ) = X ( x ) Y ( y ). Plug this back in and separate variables. [For consistency, put the X 00 /X term on one side of the equation, and set equal to a constant μ .] Written by

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 4

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online