Is the homogeneous BCs have no effect for solving PDE with D’Alembert

boundary value problemlinear-pdepartial differential equationswave equation

Suppose i have wave equation with 2 BCs (Boundary Conditions), one of them is Non homogeneous and 2 homogeneous ICs (Initial Conditions).

If I have a non-homogeneous boundary condition do I have to make it homogeneous first then it will make me able to solve PDE with the D'Alembert formula?

Does it mean that the homogeneous boundary conditions have no effect on the execution of the D'Alembert formula?

Please give me the best explanation. I have read 2 books, journals and site. But i still confused with the BCs, because some examples doesn't give that BC. But, many of them always included the ICs. So i can't find the answer. Please help me.

Best Answer

Ok, I got it!

You are right about what you do when you have a linear PDE with some inhomogeneous boundary conditions. You split $u(x,t)=v(x,t)+w(x, t)$ with:

  • $u$ the function you are searching for,
  • $v$ a function which verifies the BCs,
  • $w$ a function which verifies the PDE.

Then when you plug in the decomposition in the PDE, $w$ goes away and leaves only the $v$ with the BCs. The hard part is to find a suitable $v$, then.

The goal is to find a simple form for $v$, one which is automatically solution to the PDE because its derivatives of a certain order are 0, for instance.

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