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: ut = c2uxx, u(0, t) = u(π, t) = 0, u(x, 0) = x(π − x) . is u(x, t) =

∞

• n=1

4(1−(−1)n) πn3

sin nx e−c2n2t.

This time: Example 1b

Solve the following IVP/BVP for the heat equation: ut = c2uxx, 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: ut = c2uxx, u(0, t) = u(π, t) = 0, u(x, 0) = x(π − x) . is u(x, t) =

∞

• n=1

4(1−(−1)n) πn3

sin nx e−c2n2t.

This time: Example 1c

Solve the following IVP/BVP for the heat equation: ut = c2uxx, u(0, t) = 32, u(π, t) = 42, u(x, 0) = x(π − x) + 32 + 10

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 ut = c2uxx, 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 uss(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: ut = c2uxx, ux(0, t) = ux(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: ut = c2uxx, ux(0, t) = ux(π, t) = 0, u(x, 0) = x(π − x) . is u(x, t) = a0 2 +

∞

• n=1

an cos nx e−c2n2t. Now, we’ll solve the remaining IVP.

• M. Macauley (Clemson)

Lecture 7.2: Different boundary conditions Differential Equations 6 / 6