Lecture 7.2: Different boundary conditions Matthew Macauley - PowerPoint PPT Presentation

Lecture 7.2: Different boundary conditions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 2080, Differential Equations M. Macauley (Clemson) Lecture 7.2: Different boundary conditions Differential Equations 1 / 6

Last time: Example 1a The solution to the following IVP/BVP for the heat equation: u t = c 2 u xx , u (0 , t ) = u ( π, t ) = 0 , u ( x , 0) = x ( π − x ) . ∞ 4(1 − ( − 1) n ) sin nx e − c 2 n 2 t . � is u ( x , t ) = π n 3 n =1 This time: Example 1b Solve the following IVP/BVP for the heat equation: u t = c 2 u xx , u (0 , t ) = u ( π, t ) = 32 , u ( x , 0) = x ( π − x ) + 32 . M. Macauley (Clemson) Lecture 7.2: Different boundary conditions Differential Equations 2 / 6

Last time: Example 1a The solution to the following IVP/BVP for the heat equation: u t = c 2 u xx , u (0 , t ) = u ( π, t ) = 0 , u ( x , 0) = x ( π − x ) . ∞ 4(1 − ( − 1) n ) sin nx e − c 2 n 2 t . � is u ( x , t ) = π n 3 n =1 This time: Example 1c Solve the following IVP/BVP for the heat equation: u t = c 2 u xx , u ( x , 0) = x ( π − x ) + 32 + 10 u (0 , t ) = 32 , u ( π, t ) = 42 , π x . M. Macauley (Clemson) Lecture 7.2: Different boundary conditions Differential Equations 3 / 6

A familiar theme Summary To solve the initial / boundary value problem u t = c 2 u xx , u (0 , t ) = a , u ( π, t ) = b , u ( x , 0) = h ( x ) , first solve the related homogeneous problem, then add this to the steady-state solution u ss ( x , t ) = a + b − a π x . M. Macauley (Clemson) Lecture 7.2: Different boundary conditions Differential Equations 4 / 6

von Neumann boundary conditions (type 2) Example 2 Solve the following IVP/BVP for the heat equation: u t = c 2 u xx , u x (0 , t ) = u x ( π, t ) = 0 , u ( x , 0) = x ( π − x ) . M. Macauley (Clemson) Lecture 7.2: Different boundary conditions Differential Equations 5 / 6

von Neumann boundary conditions (type 2) Example 2 (cont.) The general solution to the following BVP for the heat equation: u t = c 2 u xx , u x (0 , t ) = u x ( π, t ) = 0 , u ( x , 0) = x ( π − x ) . ∞ is u ( x , t ) = a 0 a n cos nx e − c 2 n 2 t . Now, we’ll solve the remaining IVP. � 2 + n =1 M. Macauley (Clemson) Lecture 7.2: Different boundary conditions Differential Equations 6 / 6