Boundary Conditions Replace Initial Conditions

MIT RES.18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015
Instructor: Gilbert Strang

A second order equation can change from two initial conditions to boundary conditions at two points.

License: Creative Commons BY-NC-SA

[eng-khalil]

Thank you, sir so much for your amazing videos. I actually learned better in a much faster way than my regular course hours.

[georgesadler]

These videos are helpful for partial differential equations.

[samlaf]

Although he implies he, he never states it explicitly, but the difference here is that ODEs with initial conditions usually model behavior with time (start a system at time 0 with initial conditions, and then solving the ODE will tell you how it behaves over time. Note that we only need initial values and then the solution can be used to simulate forever). ODEs with boundary conditions (as here) are different in that they DO NOT take time into account. Rather, they only take position into account. So for example, we could use them to model the curvature in a spoon after we bend the metal. Or to model how a chain will position itself if we hold on to its two ends and let it hang. Notice that here we need boundary conditions (the two ends of the chain). And time will not matter. As long as there are no external forces, the chain will reach equilibrium and stay there forever.

Both the time (with initial value) and position (with boundary conditions) models could be the same in the case when there is only one boundary condition (at position 0). Imagine for example holding onto an infinite string and letting it drop into an infinite empty space. Now take the model where we place mass on a table and model its behavior across time. Both of these two models will have the same mathematical solution (y = 0). Yet one of them describes that the string has the same position (0) at all of its x positions, and the other one describes that the mass has the same position (0) at all times t (stays put on the table).

[blissunison]

Please help me solve this
Consider a perfectly flexible or elastic string stretched to a length 𝜋 and fixed at both
ends with homogeneous Dirichlet boundary conditions. The string is plugged
initially at its midpoint and released so that it vibrates. The wave speed is given by
9 m/ s and has zero initial velocity. Use this information to answer questions 1-5.
1. Determine the initial condition for this string.
2. Setup an initial boundary value problem in this situation.
3. Describe each of the equations given by this problem.
4. Justify that this problem is well-posed.
5. Use the method of separation of variables to determine the deflection 𝑢(𝑥, 𝑡) at
any point 𝑥 and time 𝑡 > 0 of the wave problem given by (2).

[TheudosGauh]

“I should do one more example”, Professor Gilbert said.

[IRFANALI-zpon]

I am still that why these universities are using board and chalk instead of whiteboard and Marker or smartboard (digital board)? Just a question?

[justpaulo]

6:00 I have no idea what his drawing there...
I understand the math, but not the application.

[HeyImRod]

This just confirms my belief that my professors are horrible and don't care about teaching well

[seamus]

Feels like he rushed through this one...