Boundary conditions

The corresponding Fourier series of initial function

Coefficient value

zero Dirichlet boundary conditions

$f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{A}_n}\sin \left(\frac{n\pi \left(x-a\right)}{b-a}\right)$

$g\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{\tilde{A}}_n}\sin \left(\frac{n\pi \left(x-a\right)}{b-a}\right)$

$A_n=\frac{2}{b-a}{\displaystyle {\int }_0^{b-a}f\left(x\right)\sin \left(\frac{n\pi \left(x-a\right)}{b-a}\right)\text{d}x}$

${\tilde{A}}_n=\frac{2}{b-a}{\displaystyle {\int }_0^{b-a}g\left(x\right)\sin \left(\frac{n\pi \left(x-a\right)}{b-a}\right)\text{d}x}$

zero Neumann boundary conditions

$f\left(x\right)=B_0+{\displaystyle \sum _{n=1}^{\infty }{B}_n}\cos \left(\frac{n\pi \left(x-a\right)}{b-a}\right)$

$g\left(x\right)={\tilde{B}}_0+{\displaystyle \sum _{n=1}^{\infty }{\tilde{B}}_n}\cos \left(\frac{n\pi \left(x-a\right)}{b-a}\right)$

$B_n=\frac{2}{b-a}{\displaystyle {\int }_0^{b-a}f\left(x\right)\cos \left(\frac{n\pi \left(x-a\right)}{b-a}\right)\text{d}x}$

${\tilde{B}}_n=\frac{2}{b-a}{\displaystyle {\int }_0^{b-a}g\left(x\right)\cos \left(\frac{n\pi \left(x-a\right)}{b-a}\right)\text{d}x}$

zero mixed boundary conditions

$f\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{C}_n}\sin \left(\frac{\left(n+1/2\right)\pi \left(x-a\right)}{b-a}\right)$

$g\left(x\right)={\displaystyle \sum _{n=0}^{\infty }{\tilde{C}}_n}\sin \left(\frac{\left(n+1/2\right)\pi \left(x-a\right)}{2\left(b-a\right)}\right)$

$C_n=\frac{2}{b-a}{\displaystyle {\int }_0^{b-a}f\left(x\right)\sin \left(\frac{\left(n+1/2\right)\pi \left(x-a\right)}{b-a}\right)\text{d}x}$

${\tilde{C}}_n=\frac{2}{b-a}{\displaystyle {\int }_0^{b-a}g\left(x\right)\sin \left(\frac{\left(n+1/2\right)\pi \left(x-a\right)}{b-a}\right)\text{d}x}$