Subsection 6.1.9 Choice of infinite Sine or Cosine Transforms to Boundary Value Problems
The choice of sine or cosine transform is decided by the form of boundary conditions.
\begin{equation*}
\mathscr{F}_{c}\{\frac{\partial^{2}u}{\partial x^{2}}\} = \int\limits_{0}^{\infty}\frac{\partial^{2}u}{\partial x^{2}}\cos sx\,dx = \left[\frac{\partial u}{\partial x}\cos sx\right]_{0}^{\infty} +s\int\limits_{0}^{\infty}\frac{\partial u}{\partial x}\sin sx\,dx
\end{equation*}
\begin{equation*}
=\left(-\frac{\partial u}{\partial x}\right)_{x=0}+s\left[u\sin sx\right]_{0}^{\infty} -s^{2}\int\limits_{0}^{\infty}u\cos sx\,dx \quad [\text{if}\quad \lim\limits_{x \to \infty}\frac{\partial u}{\partial x}=0.]
\end{equation*}
\begin{equation}
\therefore \quad \mathscr{F}_{c}\{\frac{\partial^{2}u}{\partial x^{2}}\}
= \left(-\frac{\partial u}{\partial x}\right)_{x=0} -s^{2}\bar{u_{c}} \tag{6.1.12}
\end{equation}
If
\begin{equation*}
\lim\limits_{x \to \infty}u=0 \quad \text{and}\quad \int\limits_{0}^{\infty}u\cos sx\,dx = \bar{u_{c}}=\mathscr{F}_{c}\{u\}
\end{equation*}
also,
\begin{equation*}
\mathscr{F}_{s}\{\frac{\partial^{2}u}{\partial x^{2}}\} = \int\limits_{0}^{\infty}\frac{\partial^{2}u}{\partial x^{2}}\sin sx \,dx
\end{equation*}
\begin{equation*}
= \left[\frac{\partial^{2}u}{\partial x^{2}}\sin sx\right]_{0}^{\infty}-s\int\limits_{0}^{\infty}\frac{\partial u}{\partial x}\cos sx\,dx
\end{equation*}
\begin{equation*}
=-s\left[\left.u\cos sx\right\vert_{0}^{\infty}+s\int\limits_{0}^{\infty}u\sin sx\,dx\right]
\end{equation*}
\begin{equation}
\therefore \mathscr{F}_{s}\{\frac{\partial^{2}u}{\partial x^{2}}\}=s(u)_{x=0}-s^{2}\bar{u_{s}}\tag{6.1.13}
\end{equation}
if
\begin{equation*}
\lim\limits_{x \to \infty}u =0\quad \text{and}\quad \int\limits_{0}^{\infty}u\sin sx\,dx = \bar{u_{s}}.
\end{equation*}
From equations (6.1.12) and (6.1.13), it follows that if we want to remove the term \(\frac{\partial^{2}u}{\partial x^{2}}\) from a given differential equation then we require
- \(\left(\frac{\partial u}{\partial x}\right)_{x=0}\) in cosine transform and
- \((u)_{x=0}\) in sine transform.
