How to transform a non-homogeneous Neumann boundary condition into a homogeneous for the wave equation?

Question

I have a initial/ boundary value problem for standard wave equation $$ \frac{\partial^2u}{\partial t^2}=c\frac{\partial^2u}{\partial x^2}, $$ where one of the boundary conditions is non-homogeneous: precisely $$ \frac{\partial u(0,t)}{\partial x}=e^t. $$ Question: is it possible to transform the above problem into a one with homogeneous boundary conditions?

In the Q&A "Solve wave equation and inhomogeneous Neumann Condition with eigenfunction expansion (Fourier Series Solution)" I saw a similar reduction, but I am unable to apply the method proposed there to my problem.

Answer 1

You should find a function $v(x, t) = A(t) x^2 + B(t) x + C(t)$ that satisfies boundary conditions (in your case $\partial_x v (0,t) = e^t$). And you can find solution of your problem as $u = v + w$. For function $w$ you get a problem for wave equation with homogenous boundary conditions.

Answer 2

Remember that the most general form of the solution of the 1D wave equation is $$u(x,t)=f(x-ct)+g(x+ct)$$ So, $$\partial_x u(x,t)=f'(x-ct)+g'(x+ct)$$ $$\partial_x u(0,t)=f'(-ct)+g'(ct)=e^t$$ Since we still have two unknown functions $f,g$ your condition is not yet enough to determine a unique solution.