I am solving this PDE $$2u_{xx}=u_t$$ $$u(x,0)=x$$ $$u(0,t)=u_x(4,t)=0$$.
I found this solution $$u(x,t)=\sum_0^{\infty}\left(D_n\sin\left(\frac{n\pi+\frac{\pi}{2}}{4}x\right)\exp\left(\left(\frac{4}{n\pi+\frac{\pi}{2}}\right)^2t\right)\right)$$
Then, I need to solve the fourier series bellow
$$u(x,0)=\sum_0^{\infty}\left(D_n\sin\left(\frac{n\pi+\frac{\pi}{2}}{4}x\right)\right)=x$$
For the period $L=4$. I don't know how to deal with $\frac{\pi}{2}$ inside the sine. How should I solve?