Материал готовится,

пожалуйста, возвращайтесь позднее

пожалуйста, возвращайтесь позднее

All right.Today, I wanna finish off our discussion Fourier series. And sometimes we don’t really

finish the discussion of Fourier series because it will always be a touchstone for reference

for some of the other things that we do.

But I wanna finish up the discussion we started last time about using Fourier series to

solve the heat equation. And then I wanna talk about the transition from Fourier series to

the Fourier transform, and how one gets from the study of periodic phenomenon to the

study of non-periodic phenomenon, which is exactly what the Fourier transform is

concerned with, by means of limiting process, all right. So that’s where I wanna finish

up.

But first, let me go back to the discussion we started last time about the heat equation. So

this is the use of Fourier. This a classic example, the classic example one might say, of

Fourier series, and also shows in a particular case, a very general principle that we will be

seeing constantly throughout the course. That’s really one of the reasons why I wanna

talk about it.

So this is your Fourier series to solve the heat equation, one particular case of a heat

equation. And I’ll remind you what the setup is. We have a ring, we have a heated ring

like that with an initial temperature distribution of – we’re calling F of X. So the initial

temperature – so I’m thinking of X here as a spacial variable.

I wanted to mention a special variable, although the ring is sitting in two dimensions. So I

can think of the ring, if I want, as an interval from zero to one with the end points

identified.

At any rate, the fundamental – the important fact here is that since there is periodicity in

space, since the ring goes round and round and round, the function F is periodic as a

function of the spacial variable, the position on the ring. And we can normalize things to

assume the period is one, all right. So that’s how Fourier series comes into the picture.

So we take F to be periodic of period one. Then we let the U of XT – now, the

temperature is varying, both in position and in time. The temperature change is a function

in time, and the temperature is different at different points along the ring.

So I let U of XT be the temperature at a position X at time T. Then U is also a function –

it’s a periodic function in this spacial variable. U is a periodic function of X. So U XT is

periodic in X. That is U of X plus one T at any time T is U of X at the same instant of

time. When T is fixed, this periodic is a function of X.

And the physical situation is described by the heat equation, which is also referred to as

the defusion equation, and governs many similar processes that involve the diffusion of

something through something else [inaudible]. So heat through a region charged through

a wire is governed by this sort of equation. I mentioned these last time, holes through a

semiconductor. It really has quite a variety of applications.

And some of the techniques that we’re talking about here, although they’re specialized to

this case, can be applied in various forms to many different situations. So you have the

heat equation [inaudible] equation, so it relates to the derivatives in time to the

derivatives in X in space.

So it says UT is equal to one-half UXX. Here I’m just choosing the constant – there’s a

constant on the right-hand side of the heat equation. I’m just choosing things so the

constant is one-half just to simplify the calculations.

finish the discussion of Fourier series because it will always be a touchstone for reference

for some of the other things that we do.

But I wanna finish up the discussion we started last time about using Fourier series to

solve the heat equation. And then I wanna talk about the transition from Fourier series to

the Fourier transform, and how one gets from the study of periodic phenomenon to the

study of non-periodic phenomenon, which is exactly what the Fourier transform is

concerned with, by means of limiting process, all right. So that’s where I wanna finish

up.

But first, let me go back to the discussion we started last time about the heat equation. So

this is the use of Fourier. This a classic example, the classic example one might say, of

Fourier series, and also shows in a particular case, a very general principle that we will be

seeing constantly throughout the course. That’s really one of the reasons why I wanna

talk about it.

So this is your Fourier series to solve the heat equation, one particular case of a heat

equation. And I’ll remind you what the setup is. We have a ring, we have a heated ring

like that with an initial temperature distribution of – we’re calling F of X. So the initial

temperature – so I’m thinking of X here as a spacial variable.

I wanted to mention a special variable, although the ring is sitting in two dimensions. So I

can think of the ring, if I want, as an interval from zero to one with the end points

identified.

At any rate, the fundamental – the important fact here is that since there is periodicity in

space, since the ring goes round and round and round, the function F is periodic as a

function of the spacial variable, the position on the ring. And we can normalize things to

assume the period is one, all right. So that’s how Fourier series comes into the picture.

So we take F to be periodic of period one. Then we let the U of XT – now, the

temperature is varying, both in position and in time. The temperature change is a function

in time, and the temperature is different at different points along the ring.

So I let U of XT be the temperature at a position X at time T. Then U is also a function –

it’s a periodic function in this spacial variable. U is a periodic function of X. So U XT is

periodic in X. That is U of X plus one T at any time T is U of X at the same instant of

time. When T is fixed, this periodic is a function of X.

And the physical situation is described by the heat equation, which is also referred to as

the defusion equation, and governs many similar processes that involve the diffusion of

something through something else [inaudible]. So heat through a region charged through

a wire is governed by this sort of equation. I mentioned these last time, holes through a

semiconductor. It really has quite a variety of applications.

And some of the techniques that we’re talking about here, although they’re specialized to

this case, can be applied in various forms to many different situations. So you have the

heat equation [inaudible] equation, so it relates to the derivatives in time to the

derivatives in X in space.

So it says UT is equal to one-half UXX. Here I’m just choosing the constant – there’s a

constant on the right-hand side of the heat equation. I’m just choosing things so the

constant is one-half just to simplify the calculations.

Загрузка...

Выбрать следующее задание

Ты добавил

Выбрать следующее задание

Ты добавил