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 )= Au ( L, t B Heat Bath 2. Ice 50
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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 0 ( x ) T ( t ) ÷ α 2 XT : X 0 ( 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 0 ( 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 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 A 6 = 0. Thus we look for values of λ such that
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This note was uploaded on 05/04/2011 for the course MATH 25 taught by Professor Lo during the Spring '11 term at BC.

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

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