[Math] How to use D’Alembert formula for Neumann boundary conditions on a finite interval

partial differential equationswave equation

I have a PDE to solve that I am not sure how to do. I know how to solve this using D'Alembert's formula for Dirichlet boundary conditions but I do not know how to solve it for Neumann boundary conditions.

Evaluate $u(\frac{5}{a},0.3)$ if $u$ satisfies $$u_{tt}=a^2u_{xx}, \quad 0<x<1,\quad t>0$$ $$u(0,x)=x(1-x), \quad u_t(0,x)=0, \quad 0\le x\le 1$$ $$u_x(t,0)=u_x(t,1)=0, \quad t>0.$$

Best Answer

D'Alembert's formula requires an appropriate extension of the initial values to the entire line $\mathbb R$.

  • for Dirichlet conditions, you would use odd periodic extension: first $f(-x)=-f(x)$, then $f(x+2m)=f(x)$ for all integers $m$.
  • for Neumann conditions, use even periodic extension: first $f(x)=-f(x)$, then $f(x+2m)=f(x)$ for all integers $m$.

Dealing with the periodic extension is very painful if you need a complete formula for $u$. But here you are just asked to find the value at $x=0.3$ and $t=5/a$. This means $$\frac12 (f(0.3+5)+f(0.3-5))$$ Use the periodicity and even-ness: $$f(5.3)=f(3.3)=f(1.3)=f(-0.7) = f(0.7)$$ and so on.

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