Advanced Engineering Mathematics, Lecture 6.4: Solving PDEs with Fourier Transforms

Advanced Engineering Mathematics, Lecture 6.4: Solving PDEs with Fourier Transforms

The Fourier transform takes a function f(x) of position and outputs a function of frequency, ω. It turns x-derivatives into multiplication by iω, and so it can be used to solve an ODE by turning it into an algebraic equation. The Fourier transform can turn a PDE in the multivariate function u(x,t) into an ODE in û(ω,t), where ω can be regarded as a constant. In this lecture, we show how the Fourier transform of a Gaussian function is another Gaussian. Then we solve a second order ODE with a generic forcing term using Fourier transforms. Finally, we solve a Cauchy problem for the heat equation on the real line with a Fourier transform, which we did in an earlier lecture using a different method.

[gsgp]

These video series are excellent.
So far you showed a variety of ways to solve ODES and PDEs.
Now can you show solution maps which summarize the most appropriate ways of solution to solve the differential equations for each of different sort of ODEs and PDEs you presented in this videos playlist?

[b.amitesh]

Sir, example 3 u(x, 0)=f(x), then after how it has converted into f(s)

[ozzyfromspace]

If you don’t wanna assume your solutions are Schwartz Functions, you can always assume they are of exponential order. It’s pretty much the same thing

[fasikawondimu]

Hi! Professor. Currently I am doing research on PDE problems with Robin conditions. But, I face doubt on how to impose Robin boundary conditions into MATLAB. Would you help help me?

[abrahanraymundom]

I like your videos, What virtual whiteboard do you use?