The function $u(x,t)$ is defined in $0<x<1$ and obeys the pde:
$$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} - u$$
Alongside BCs:
$$u_x = 0\ when\ x =0$$
and
$$u_x = -1\ when\ x =1$$
Alongside an IC
$$u(x,0) = u_0(x)$$
I need to find a series solution that obeys sturm liouville properties as well as derive an orthogonality condition. My general solution was calculated as:
$$u(x,t) = \sum_{n=1}^{\infty} B_n \exp(-n^2\pi^2 t)\sin(n\pi x) $$
Which I'm fairly certain is wrong. Where have I gone wrong?
Your equations are $$ u_t = u_{xx}-u \\ u_x(0,t)=0,\;\; u_{x}(1,t)=-1,\\ u(x,0)=u_0(x). $$ The condition $u_x(1,t)=-1$ is inhomogeneous, which poses a problem for separation of variables. The functions $e^{x}, e^{-x}$ are solutions of $u_{xx}-u=0$. So the function $$-\cosh(x)/\sinh(1)$$ is a solution of $$ u_{xx}-u=0, u_{x}(0)=0, u_{x}(1)=-1. $$ Therefore $v=u+\cosh(x)/\sinh(1)$ is a solution of $$ v_t = v_{xx}-v \\ v_{x}(0,t)=0,\;\; v_{x}(1,t)=0 \\ v(x,0)=u(x,0)+\cosh(x)/\sinh(1)=u_0(x)+\cosh(x)/\sinh(1). $$ This is solvable with separation of variables.