I have been struggling to find the general solution of the following BVP of reaction-diffusion equation: $$\frac{\partial N}{\partial t}=\frac{\partial^2 N}{\partial x^2}+N(1-N)-\sigma N$$ $$N(0,x)=N_0(x)$$ $$N(t,0)=\alpha \hspace{2mm}, N(t,L)=\beta; t>0,0<x<L$$
I am trying to have some mathematical analysis to find stability of this problem from this BVP.And analyse the bifurcation digram.I linearized the PDE and found the fixed points $(N_{1}^*,N_{2}^*)=(0,0)\hspace{2mm} and\hspace{2mm} (N_{1}^*,N_{2}^*)=(1-\sigma)$.
For $(N_{1}^*,N_{2}^*)=(0,0)$ the eigenvalues are $\lambda_{1,2}=\frac{-c\pm\sqrt{c^2-4(1-\sigma)}}{2}$ which is asymptotically stable.For $(N_{1}^*,N_{2}^*)=(1-\sigma,0)$ the eigenvalues are $\lambda_{1,2}=\frac{-c\pm \sqrt{c^2+4(1-\sigma)}}{2}$
But what I really need is that how can I get a general solution from this BVP which satisfies the boundary condition? I tried both the separation of variable method and fourier sine function but I failed completely. I seems hard for me.And can I easily have general solution from eigenvalues?I really need help!