Advanced Engineering Mathematics, Lecture 5.3: The transport and heat equations

Advanced Engineering Mathematics, Lecture 5.3: The transport and heat equations.

Let u(x,t) represents the displacement of a vibrating string or wire. We begin by showing how solutions to the PDE u_t-cu_x=0 are precisely u(x,t)=f(x+ct), for any function f(x), which are traveling waves to the left at speed C. Similarly, solutions to u_t+cu_x=0 have the form u(x,t)=f(x-ct), which are traveling waves to the left at speed C. Putting this together, we derive the wave equation, which is the PDE u_{tt}=c^2 u_{xx}. Next, we impose Dirichlet boundary conditions u(0,t)=u(L,t)=0, which represents a vibrating string of length L, where the endpoints are held fixed. We conclude by solving a specific instance of a boundary and initial value problem for the wave equation.

[nitzanbreitman]

Hey, Thanks for the videos. Are there going to be 6.x lectures?

[ozzyfromspace]

I’ve got a question: when is it better to use a Fourier Series or Fourier Transform?

[thomasjefferson]

Im sorry but if you havent watched the video on bvps then what are you doing here lol.