Boundary-value problem for linear hyperbolic system by Fourier series

Question

I am trying to solve the linear equations $$\partial_t \rho +\partial_x \varphi =0, \qquad \partial_t \varphi+\partial_x \rho = \alpha \rho +\beta \varphi,$$ where $\alpha$, $\beta$ are constants. The functions $\rho$, $\varphi$ are defined on $[0,T]\times [0,L]$. The initial conditions are $$\rho(0,x)=0, \qquad \varphi(0,x)=0$$ and the boundary conditions are $$\rho(t,0)=f(t), \qquad \varphi(t,L)=g(t).$$ I have been trying to obtain a Fourier series solution, but I always run into some sort of problem. Is it possible to obtain such a solution?

Answer 1

I don't think Fourier series are likely to work, but you might try a Laplace transform in $t$.

Answer 2

Applying $\partial_t$ to $\varphi_t + \rho_x = \alpha\rho + \beta\varphi$ and using $\rho_t = -\varphi_x$ leads to $$ \varphi_{tt} - \varphi_{xx} = \beta\varphi_t- \alpha\varphi_x \, . $$ The corresponding initial conditions are $\varphi(0,x) = 0$ and $\varphi_t(0,x) = 0$, and the boundary conditions are $\varphi_x(t,0) = -f'(t)$ and $\varphi(t,L) = g(t)$. Thus, there is a Neumann boundary condition at $x=0$ and a Dirichlet boundary condition at $x=L$. Using separation of variables $\varphi(x,t) = X(x) T(t)$, we have $$ \frac{T''}{T} - \beta\frac{T'}{T} = \frac{X''}{X} - \alpha \frac{X'}{X} = -\lambda \, , $$ i.e. $$ {T''} - \beta {T'} + \lambda T = 0 \qquad\text{and}\qquad {X''} - \alpha {X'} + \lambda X = 0 $$ for which the Fourier series approach could be applied. However, this might not be an easy task in the case where the boundary conditions are time-dependent (see e.g. this post, and chap. 4 of (1)). To tackle this case by using such an approach, one could expand the solution over the normal modes of the system.

Alternatively, one may use the method of characteristics, but this might not be straightforward due to the right-hand side (if $\alpha\neq 0$ or $\beta\neq 0$). You may have a look at (1), chap. 12.

Lastly, one may solve the problem by using Laplace transforms (see (1), chap. 13).

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(1) R. Habermann, Applied Partial Differential Equations: with Fourier Series and Boundary Value Problems, 5th ed. Pearson Education Inc., 2013.