In this topic, we will show you how can we use Fourier transform to solve initial value problem for wave equation in . Following is the problem

From the equation by taking Fourier transform to the both sides, we obtain

This is an ODE, the solution is given by

From the initial date we get

which implies that

.

From the initial date we see that

.

Thus, we obtain

.

Note that

.

Moreover,

.

Then

.

Since

and

then

.

Thus,

or equivalently,

.

This is the so-call D’ Alembert formula.

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Bài này nếu giải bằng hàm Green thì thế nào NQA nhỉ ?

Comment by viettran — March 24, 2009 @ 20:27

Nếu thế cần phải tìm hàm Green đã, việc này chắc ko dễ 😦

Comment by Ngô Quốc Anh — December 14, 2009 @ 17:25

hi , my name nguyen , i come from USA , nice to meet you ,

Comment by vo khanh nguyen — December 31, 2009 @ 16:54

Thanks for coming to my blog 🙂

Comment by Ngô Quốc Anh — January 2, 2010 @ 15:20

In the odd-dimensional space ()

http://arxiv.org/abs/0904.3252

and in the even case, implies by using Hadamard’s method of descent.

Comment by Tuan Minh — January 14, 2010 @ 23:40

Aha, that’s a good and new work, thanks Minh 🙂

Comment by Ngô Quốc Anh — January 14, 2010 @ 23:44

In fact, this idea is mentioned in the well-known book of Stein in the case (by using FT/Bessel function), Stein also gave an exercise for the general case. The Torchinsky’s inverse formula for is nice!

Comment by Tuan Minh — January 15, 2010 @ 1:53