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M257-316Notes_Lecture10

# M257-316Notes_Lecture10 - Chapter 8 Separation of Variables...

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Chapter 8 Separation of Variables Lecture 10 8.1 Types of Boundary Value Problems: Dirichlet Boundary Conditions 1. Heat Equation: α 2 = Thermal Conductivity. Heat Flow in a Bar Heat Flow on a Disk 2. Wave Equation: c = Wave Speed. Vibration of a String 3. Laplace’s Equation: 49

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Separation of Variables Neuman Boundary Conditions: What do you expect the solution to look like as t → ∞ ? Mixed Boundary Conditions: Ice Heat Bath u (0 , t ) = A u ( L, t ) = B Heat Bath 2. Ice 50
8.2. SEPARATION OF VARIABLES: 8.2 Separation of Variables: Consider the heat conduction in an insulated rod whose endpoints are held at zero degree for all time and within which the initial temperature is given by f ( x ). Fourier’s Guess: u ( x, t ) = X ( x ) T ( t ) (8.1) u t = X ( x ) ˙ T ( t ) = α 2 u xx = α 2 X ( x ) T ( t ) ÷ α 2 XT : X ( x ) X ( x ) = ˙ T ( t ) α 2 T ( t ) = Constant = α 2 . (8.2) > ˙ T ( t ) = α 2 λ 2 T ( t ) dT T = α 2 λ 2 dt ln | T | = α 2 λ 2 t + c T ( t ) = D e α 2 λ 2 t . (8.3) x > X ( x ) + λ 2 X ( x ) = 0 Guess X ( x ) = e rx ( r 2 + λ 2 )e rx = 0 r = ± λi (8.4) X = c 1 e iλx + c 2? e iλx = A sin λx + B cos λx. (8.5) Impose the boundary conditions: 0 = u (0 , t ) = X (0) T ( t ) = BT ( t ) B = 0 0 = u ( L, t ) = X ( L ) T ( t ) = ( A sin λL ) T ( t ) . (8.6) 51

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Separation of Variables Now we do not want the trivial solution so
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M257-316Notes_Lecture10 - Chapter 8 Separation of Variables...

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