$$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial^2 t}$$
with an initial condition and boundary conditions $$T(x,0)=T_0$$ $$T(L,t)=T_0$$ $$-k\left.\frac{\partial T}{\partial x}\right|_{x=0}=2A\cos^2\left(\frac{\omega t}{2}\right)=A\cos(\omega t+1)$$
The solution for this problem is
$$T(x,t)-T_0=\frac Ak\sqrt{\frac\alpha\omega}\exp\left (-\sqrt{\frac{\omega}{2\alpha}}x\right)\\ \times\cos\left(\omega t-\sqrt{\frac{\omega}{2\alpha}}x-\frac\pi4\right)-\frac Ak(x-L)$$
I'm trying to solve Fourier heat conduction equation with the boundary conditions shown above.But the solution I'm getting contains eigenvalues and Fourier series which the solution doesn't contain. More interestingly the solution contains a phase terms. I got this from a paper but the paper doesn't contain any derivation. I searched in the internet but couldn't find how they solved it. I would appreciate any help!