I'm trying to solve a boundary value problem (Laplace's equation within a rectangle), with a peculiar combination of boundary conditions: one side of the rectangle has a homogeneous Dirichlet boundary condition applied to half of it, and an inhomogeneous Neumann boundary condition applied to the other half.
I was able to "reduce" the problem to the following: determine the functions $f(x)$ and $g(x)$ defined for $0\le x \le 1$, such that $$\int\limits_0^1f(x)\sin(n\pi x)dx = \frac 1 n\tanh(n\pi)\int\limits_0^1g(x)\sin(n\pi x)dx$$ for all integers $n \ge 1$, with the constraints $$f(x)=0\text{, for }x>\frac 1 2$$ $$g(x)=1\text{, for }x<\frac 1 2$$
The problem is essentially a relation between the Fourier (sine) series coefficients of $f$ and $g$ (so we can calculate one if we know the other), but I'm not sure how to utilize the constraints to come up with a unique solution. The equation can also be cast into the form $$\int\limits_0^1\left[f(x) - \frac 1 n\tanh(n\pi)g(x)\right]\sin(n\pi x)dx = 0$$ although maybe this isn't particularly useful. I don't know how to proceed, and would appreciate any help or resources. Thanks.