Strang doesn't explain what exactly we are modelling when using boundary conditions as opposed to initial conditions, and it had me completely confused the first time watching. This video (ua-cam.com/video/lsY7zYaezto/v-deo.html) really helped me get an intuition for the whole field of PDEs as modelling real things. 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).
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).
Mathematicians, generally speaking and rightfully so, prefer chalk. There are universities where the math faculty have agreed to refuse to teach in whiteboard rooms, I remember in my undergrad the university began replacing the chalk boards with whiteboards in the math department building, they only got to do some rooms in the basement before that was put to a stop.
Prof. Gilbert Strang is a legend
Thank you, sir so much for your amazing videos. I actually learned better in a much faster way than my regular course hours.
These videos are helpful for partial differential equations.
Strang doesn't explain what exactly we are modelling when using boundary conditions as opposed to initial conditions, and it had me completely confused the first time watching. This video (ua-cam.com/video/lsY7zYaezto/v-deo.html) really helped me get an intuition for the whole field of PDEs as modelling real things.
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).
Samuel Laferriere -Amazing explanation, i had the same doubt, thanks for sharing
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).
unison.bu@gmail.com
6:00 I have no idea what his drawing there...
I understand the math, but not the application.
Omg what a great teacher
I am still that why these universities are using board and chalk instead of whiteboard and Marker or smartboard (digital board)? Just a question?
Mathematicians, generally speaking and rightfully so, prefer chalk. There are universities where the math faculty have agreed to refuse to teach in whiteboard rooms, I remember in my undergrad the university began replacing the chalk boards with whiteboards in the math department building, they only got to do some rooms in the basement before that was put to a stop.
Can you please be my grandfather
“I should do one more example”, Professor Gilbert said.
Thank you, Professor!
This just confirms my belief that my professors are horrible and don't care about teaching well
Feels like he rushed through this one...
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the DE playlist is becoming boring