Waching daily Jan 29 2019

Here's an overview of Euler's method

for solving differential equations of this form.

The derivative of some unknown function X is just a function of X,

and we assume that we know the starting value of this function X,

- so this could be temperature, this could be position, could be anything.

So, we start - we need to choose a step size,

- that's a choice that we get to make, and we'll start at time t equal 0,

where we know the X value.

So, then using the current value of X,

the equation - the differential equation - this function,

tells us the rate of change.

So, we know how fast X is changing.

We then use this rate of change

to determine the next value for X,

This is the current value for X,

this is how much X changes, in the time interval delta t.

This is a little bit of make-believe

- we're pretending that the rate of change is constant

for this time interval delta t, and we can use that

to figure out the next value of X,

then, we increased t by delta t

and we go back to this step, and we figure out the derivative again.

The derivative tells us how to go forward a little bit in time

to figure out the next X, then we update time, figure out the derivative again,

move forward to figure out X.

So 2 and 3 - these are the key steps here,

- wer'e constantly shuffling back and forth,

the function - the differential equation - tells us the derivative,

- here we use the derivative to figure out the value of X,

and then we go back, X gives us the derivative from the differential equation,

- we use a derivative to figure out X, and so on.

So, one repeats through these processes

until you have enough of a solution.

So, in practice, one would choose smaller and smaller delta t

until the solution curve stops changing.

So, if you chose a delta t of 2, and then 1, and then .01, and .001,

doing this on a computer, or a spreadsheet,

and eventually one would see that your table of values, if you plotted them,

your X's would stop changing, and that would be an indication

that you're delta t was small enough.

So, this is Euler's method in a nutshell.

It gets at the heart of what the differential equation means,

- the rate of change is given by the quantity X,

we use that rate of change to figure out X,

so, again we're thinking of this as a dynamical system,

- it's a rule that specifies how X changes,

the rule is written in terms of the derivative,

- the rate of change of X, rather than X directly,

but, not a problem, Euler's method, or things like it,

let us convert from rates of change in to the function itself.

Subtitles by the Amara.org community

For more infomation >> Dynamical Systems And Chaos: Computational Solutions Part 4 - Duration: 2:56.

-------------------------------------------

Dynamical Systems And Chaos: Computational Solutions Part 2 - Duration: 4:23.

So, let's continue with this example.

We just found the T(2) was 11, or approximately 11

because we had to do some make-believe to get this,

but now let's see if we can figure out T(4).

I can figure out how fast the temperature is changing

at time 2, assuming that the temperature is 11.

What's the rate of change? Well I just ask the equation

- that's what the differential equation does - it's a rule that tells me how fast

the temperature is changing, if we know the temperature.

So let's do that.

So we use the equation - we ask the equation:

When the temperature is 11, what's the rate of change? what's the derivative?

So, when time is 2, we plug in 11, so capital T is 11, 20 -11 is 9,

times .2 is 1.8

So now we know that when the temperature is 11,

it is warming up at 1.8 degrees per minute.

So now suppose we want to know T(4), 4 minutes in,

again, we have the same problem

- this rate isn't constant - it's changing all the time,

as soon as a temperature changes we get a new rate,

but as before, we'll ignore the problem

and pretend that it's constant.

So, again the problem is: the rate is not constant

- our solution is to ignore the problem

- not always a good way to go about things

but for Euler's method, it turns out to work okay

- we'll ignore the problem - pretend it is constant

and then we can figure out the temperature at time 4, 4 minutes in,

in these 2 minutes, that we're pretending:

how much temperature increase do we have,

well at 1.8 degrees per minute for 2 minutes, that's 3.6,

3.6 +11, where we started, gives us 14.6

So now, I know the temperature at T equals 4 minutes.

We can keep doing this,

continue along with this process, and we'll get

a series of temperature values for a series of times.

So, we continue this process,

and we can put our results in a table.

So these first 3 entries we've already figured out

- the initial temperature is 5, then at time 2 it was 11,

at 4, it was 14.6, and at 6,

if when one follow this process along, one would get 16.76,

and we could keep on going.

So, let's make a graph - let's make a plot of these numbers

and see what it looks like, and compare it to the exact solution.

So, for this equation, it turns out one can use calculus to figure out

an exact solution for this differential equation,

and that shown as this solid line here.

Towards the end of this sub unit, I'll talk a little bit about

how one would get this solid line.

The Euler solution - that's what we're doing here

- are these squares - so we start at

the initial condition, and then here at 11,

a little bit less than 15, almost 17, and so on.

So we can see that the Euler solution

- the squares connected by the dotted line

is not that close to the exact solution.

It's not that bad, but it's not a perfect match

and we wouldn't expect a perfect match

because we had to do some pretending in order to get this.

So, as is often the case, ignoring the problem

- remember the problem was that:

the derivative - the rate of change wasn't constant.

Ignoring the problem actually wasn't a great solution

because we have these errors here.

For this example, I'd chose a step size of 2, a delta t of 2.

I said: let's figure out the temperature, capital T, every 2 minutes,

but it's this step size that got us into trouble

because I had to pretend that a constantly changing rate

was actually constant over this time of 2 minutes,

and that's clearly not true,

so, a way we could do better with this Euler method is to use a smaller delta t.

For more infomation >> Dynamical Systems And Chaos: Computational Solutions Part 2 - Duration: 4:23.

-------------------------------------------

Mùa Xuân Ơi remix 2019 [Video Gái Xinh] - Duration: 3:53.

For more infomation >> Mùa Xuân Ơi remix 2019 [Video Gái Xinh] - Duration: 3:53.

-------------------------------------------

Dynamical Systems And Chaos: Computational Solutions Part 5 - Duration: 5:49.

I'll conclude this sub unit on computational methods

for solving differential equations, with some more general remarks.

So, we've been discussing Euler's method,

- Euler's method is known as a numerical

or computational method for solving the differential equation.

It's computational because it involves doing computations

using, almost always, a computer,

and it's called numerical - that's its may be more traditional term,

because the result of Euler's method is not a formula for a function

but it's a list of numbers, so we call it a numerical method.

Euler's method is quite simple conceptually,

and it gets at the heart of what difference equations are:

- a rule for how something changes,

and that rule is written in terms of the derivative,

- the rate of change, not the function itself.

So, I think if you understand Euler's method,

and may be code it up yourself,

then you really have a good understanding

of what differential equations mean,

so, I definitely recommend if you have some programming skills,

- the language doesn't matter at all,

and you can use a spreadsheet,

to try coding up Euler's method yourself.

I'll list this in the homework for this unit.

So, Euler's method is very nice conceptually,

however it's not very efficient computationally,

so, it's not used much in practice.

Let me say a little bit about how one can improve on Euler's method.

So, to improve on Euler's method,

there are two things one might like to do:

first, there's a family of techniques,

a group of techniques, known as Runge-Kutta methods,

and here's the idea behind them:

I'm not going to go into this in details,

but It's worth mentioning.

So, in Euler's method,

we pretend that the rate of change is constant, over an interval delta t,

and then we have to choose what rate of change

we're going to use, and we just choose the rate of change

at the left part of that interval.

We just take the starting rate of change in that interval

and pretend that it's constant for all of delta t.

Runge-Kutta methods say,

well, rather than using the rate of change at the start of the interval,

what if we use the rate of change at the start of the interval,

and at the end of the interval, and average those two.

That would probably be a more fairer representation

of what's happening in that interval,

or better still, maybe we could sample the rate of change

at three different points along that interval,

and there are different schemes for doing that sampling

and different ways of averaging the different derivatives

that one estimates, but that's the general idea.

Rather than just use one derivative,

sample a couple derivatives, and average those.

So, that's one way to improve on Euler's method.

It's not immediately obvious, but this turns out to be

more efficient, in the sense that, with less computational effort

you can get an equally accurate answer.

The other thing that one typically does

is something called adaptive step size,

and this is: we have the program automatically adjust delta t

on the fly - as it's doing, as it's trying to find a solution.

So, delta t needs to be small

when the derivative is changing rapidly.

We get into trouble with these methods

when we pretend the derivative is

constant over delta t, but it actually changes a lot.

So, if we have a situation where, sometimes,

as time goes on, the derivative is changing rapidly,

and other times, it's not changing rapidly,

then we don't need to use the same delta t.

If the change in derivative is slow,

we can use a large delta t,

when it's fast we need to use a small delta t,

and so these adaptive step-size methods

figure that out on the fly.

I sort of think about it - adaptive step size - as this way,

and this is a rough analogy but it may be gives the right idea,

- imagine you're walking across a landscape,

and you're blindfolded - you can't see.

If the landscape is flat, you can take very big steps,

and you're not going to miss anything,

but if the landscape is very bumpy,

then you'll need to take small steps

to make sure you don't trip or miss something.

So, if you're blindfolded, you might adjust your step size

depending on what you sense the terrain to be.

OK, in any event, the standard way to improve on Euler's method

is to do these two things:

some type of Runge-Kutta method,

and some type of adaptive step size.

Almost all numerical programming environments, that I'm familiar with:

MATLAB, Octave, Maple, Mathematica, Python

- have some sort of built-in Runge-Kutta, adaptive step size solver.

So, perhaps in the form, some of you who have experience

with these different things,

and maybe have solved differential equations before

can post some examples

for how to use these different built-in functions.

OK, lastly, may be just take a look ahead.

So, in the next units, I'll be frequently presenting

solutions to differential equations,

and almost always, the solutions that I present

and show you and discuss will be numerical solutions.

In order to do this course you don't need to

solve differential equations on your own,

you don't need to code up your own algorithms,

you don't need to use other peoples' algorithms,

however, I think it's important that you have

a sense of where these numerical solutions come from.

So, if I show you a solution to a differential equation

you have some idea where that solution came from,

- it's not magic, it's just from using something like

Euler's method - a very simple, but repetitive way

of solving differential equations.

Subtitles by the Amara.org community

For more infomation >> Dynamical Systems And Chaos: Computational Solutions Part 5 - Duration: 5:49.

-------------------------------------------

PULANGLAH UDA (OFFICIAL LIRIK VIDEO) Lely Tanjung - Duration: 6:35.

For more infomation >> PULANGLAH UDA (OFFICIAL LIRIK VIDEO) Lely Tanjung - Duration: 6:35.

-------------------------------------------

Dynamical Systems And Chaos: Computational Solutions Part 3 - Duration: 5:07.

A smaller delta t makes Euler's method

more accurate, and we can see why:

The reason Euler's method isn't accurate

is that we're pretending that a constantly changing rate

is actually constant over some time interval.

Over a time interval of 2 minutes, that rate might change quite a bit,

but the rate will change less if the time interval is smaller.

So, if the time interval is 1 minute, instead of 2,

the bit of make-believe, where we pretend that

the rate isn't changing, will be closer to reality,

and I can illustrate that on this next plot.

So, I won't go through all the numerics of this,

but here is Euler's method

for 2 different delta t's.

So, first, the squares, which we've already seen,

we calculated those before,

that's Euler's method with a delta t of 2,

where we're pretending that a constantly changing rate

is actually constant for a whole two minutes.

Delta T of 1, that's the triangles with the dash, and not dotted line.

a little hard to see, but the key thing is it's between these two

- it's closer to the exact solution which is the solid curve.

It's closer because the ignoring of the problem is a less bad thing to do,

- we're pretending now that a continually changing rate

is only constant for 1 minute, instead of 2, so it's not as much of a lie,

and by now you can probably guess

how we could make this better and better and better

- we would let delta t get smaller and smaller and smaller,

and then we would see that the Euler method

would be exactly on top of this line.

Now that we've seen part of a particular example

let me talk about Euler's method a bit more generally.

Euler's method applies to differential equations of this form.

A differential equation is a dynamical system,

a rule for how something changes in time.

What makes differential equations a little bit tricky

is that the rule is indirect.

This tells us how the derivative changes

and we're interested in how the quantity X itself changes.

Euler's method is just a way to go from this

indirect information derivative, to the direct information about X

So Euler's method converts this indirect rule,

the differential equation - indirect rule involving the derivative

- the rate of change, and it converts that in to values for X.

It does so, by pretending that this rate of change

is constant over a time interval.

So, Euler's method does this conversion by pretending that the derivative,

which is constantly changing, is actually constant

over some time interval delta t.

This bit of make-believe gets better

closer to the true value, as delta t gets smaller.

So, as delta t, our time interval over which were pretending the rate isn't changing,

as delta t gets closer and closer to 0,

this Euler pretending will get less and less wrong,

and in this way, Euler's method, a solution obtained

from Euler's method, will get closer and closer to the true answer.

So, as delta t gets closer and closer to 0,

a solution obtained by Euler's method will get

closer and closer to the exact solution.

So, Euler's method is a computational way of

finding a solution to a differential equation.

It requires doing a computation,

and you can see that as delta t gets smaller and smaller

the computational will get longer and longer.

- we'll need to do more and more steps to get anywhere,

so these are almost always done on a computer.

So, this is an algorithmic solution to differential equations.

It's a procedure, it's well-defined,

for well-defined differential equations, it's guaranteed

to converge to the exact solution.

So, Euler's method is very general,

it almost always works, and I think it gets at

the core idea of a differential equation:

a differential equation is a dynamical system,

a rule for how something changes.

The rule is a little bit indirect, because it is in terms of the derivative,

the rate of change of this quantity X, and not X itself

but Euler's method is a little bit of a trick

that converts this indirect information about the derivative

into direct information about values for X

Subtitles by the Amara.org community

For more infomation >> Dynamical Systems And Chaos: Computational Solutions Part 3 - Duration: 5:07.

-------------------------------------------

Dynamical Systems and Chaos: Computational Solutions Part 1 - Duration: 4:58.

In the last sub unit I introduced qualitative,

or geometric techniques, for solving differential equations.

We saw that, with the graph of the right hand side of the equation,

we could figure out the phase line

and get a general picture for what solutions look like.

In this sub unit, I'll present another way of solving differential equations

these are sometimes called numerical solutions

but I like to think of them as computational, or algorithmic solutions.

I like these methods a lot - they're very versatile,

they're very commonly used,

and they get at the heart of what differential equations are

from a dynamical systems point of view.

So, I'll present this material in three parts:

first, I'll give an overview of the computational methods,

then in the second part, which is optional,

I'll go in to those methods in some detail

so that you could code them up for yourself, if you wanted,

and then in the third part of this sub unit,

I'll summarize, take a step back,

and compare and contrast a number of different solution methods.

The computational method that I'd like to present

is known as Euler's method.

Let's return to the example that we started with:

Newton's Law of Cooling - this describes

the rate of change of the temperature

capital T is temperature, little t is time,

of an object that is originally at 5 degrees,

that's placed in a 20 degree room.

We'd like to know the temperature at later times

we would really like to have capital T as a function of time,

but in this approach

I'll try to estimate capital T, the temperature,

at 2 minutes, 4 minutes, 6 minutes, and so on.

So, we start - where else can we start - at the beginning.

We know that the temperature is 5,

and we have this differential equation - this rule,

so I can use these two facts to figure out

how fast the temperature is changing,

at the initial time, when T equals 5,

that's what this differential equation does,

it tells me - if I know the temperature,

it tells me how fast the temperature is changing,

so let's do that.

So, I use the equation, defined dT/dt,

the rate change of the temperature, at the initial time.

So, I just ask the equation - by plugging in 5,

so, capital T is 5, 20 minus 5 is 15,

times 0.2, gives me 3 degrees per minute.

So, this is how fast the object is initially warming up

so I can use this rate to figure out the temperature at a later time,

- uh oh - but there's a problem, which is that this rate is not constant,

the rate is always changing,

as soon as a temperature changes, a little bit,

the derivative changes, the rate changes a little bit,

so, we're in a bit of a dilemma.

So, we have a problem: the rate is not constant,

so, we have to cope with this problem somehow,

and the coping mechanism that we use in Euler's method

is just to ignore the problem - we'll just pretend that it is constant

for, let's say, a 2 minute interval.

So, we'll pretend that this rate is constant for 2 minutes,

it's not, but we're just going act as if it is,

and if this rate is constant for these 2 minutes,

then, I can figure out the temperature 2 minutes later.

OK, so I want to know T(2) - the temperature T, at time 2 minutes,

2 minutes after this object is placed in the room.

So, how much does it warm up in those 2 minutes?

Well, it's warming up at 3 degrees per minute,

and we pretend that that's constant

it's not really but we'll just ignore the problem and pretend

3 degrees per minute, for 2 minutes

that's a total increase of 6 minutes

I take those 6 minutes, add them to 5, I get 11,

and in this way I've figured out T(2)

- probably not accurate, or probably not exact

because we had to do a little bit of make-believe,

but it actually isn't that bad.

We can do a similar thing to get T(4), so let's get that a try.

Subtitles by the Amara.org community

For more infomation >> Dynamical Systems and Chaos: Computational Solutions Part 1 - Duration: 4:58.

-------------------------------------------

SLIME ASMR - Synthesize The Most - Relaxing Slime Video #017 😎😎😎 - Duration: 10:01.

PLEASE LIKE, SHARE, SUBCRIBE video! Thank you very much!

For more infomation >> SLIME ASMR - Synthesize The Most - Relaxing Slime Video #017 😎😎😎 - Duration: 10:01.

-------------------------------------------

Video Captures Fiery Crash In Dublin, NH - Duration: 0:25.

For more infomation >> Video Captures Fiery Crash In Dublin, NH - Duration: 0:25.

-------------------------------------------

Most Satisfying Relaxing Slime #54 - Duration: 10:01.

PLEASE LIKE, SHARE, SUBCRIBE video! Thank you very much!

For more infomation >> Most Satisfying Relaxing Slime #54 - Duration: 10:01.

-------------------------------------------

Evening Video Forecast - Duration: 0:54.

For more infomation >> Evening Video Forecast - Duration: 0:54.

-------------------------------------------

Angel's Dance Video with no Music - Duration: 1:24.

Hello , My name is Angel Cisneros

I am from Mobile, AL and I am 20 years old.

My school is Gentry in Talladega, AL.

What i doing today?

i am dancing without the music.

Who's filming on me?

That's my girlfriend,

Her First name is Victoria

Her last name is Umfleet

Begin

a Dance