Mathematical Methods of Physics: For Undergraduate

\begin{equation*} \mathscr{F}_{s}\{\frac{\partial u}{\partial t}\} =k\mathscr{F_{s}}\{\frac{\partial^{2}u}{\partial x^{2}}\} \end{equation*}

\begin{equation*} \int\limits_{0}^{\infty}\frac{\partial u}{\partial t}\sin sx\,dx = k \int\limits_{0}^{\infty}\frac{\partial^{2}u}{\partial x^{2}}\sin sx\,dx \end{equation*}

\begin{equation*} \frac{\partial}{\partial t}\int\limits_{0}^{\infty}u\sin sx\,dx = k\{\left.\frac{\partial u}{\partial x}\sin sx\right\vert_{0}^{\infty}-s\int\limits_{0}^{\infty}\frac{\partial u}{\partial x}\cos sx\,dx\} \end{equation*}

\begin{equation*} \frac{\partial}{\partial t}\bar{u_{s}}=k\{0-s\int\limits_{0}^{\infty}\frac{\partial u}{\partial x}\cos sx\,dx\} \end{equation*}

\begin{equation*} = ks\{\left.u\cos sx\right\vert_{0}^{\infty}+s \int\limits_{0}^{\infty}u\sin sx\,dx\} \end{equation*}

\begin{equation*} =ks\{u(0,t)\cos (0.s) - s\bar{u}_{s}\} \end{equation*}

\begin{equation} \frac{\partial}{\partial t}\bar{u}_{s} = ks^{2}\bar{u}_{s}\tag{6.2.8} \end{equation}

\begin{equation} \bar{u}_{s} = A e^{-ks^{2}t}\tag{6.2.9} \end{equation}

\begin{equation*} A=\bar{u}_{s}(s,0) = \int\limits_{0}^{\infty}u(x,0)\sin sx\,dx \end{equation*}

\begin{equation*} A = \int\limits_{0}^{1}1.\sin sx\,dx + \int\limits_{1}^{\infty}0.\sin sx\,dx \end{equation*}

\begin{equation*} A = -\left.\frac{\cos sx}{s}\right\vert_{0}^{1} = \frac{1-\cos s}{s} \end{equation*}

\begin{equation} \bar{u}_{s}=\frac{1-\cos s}{s}e^{-ks^{2}t}\tag{6.2.10} \end{equation}

\begin{equation} u(x,t) = \frac{2}{\pi}\int\limits_{0}^{\infty}\frac{1-\cos s}{s}e^{-ks^{2}t}\sin sx\,ds\tag{6.2.11} \end{equation}

\begin{equation} \frac{\partial^{2}y(x,t)}{\partial x^{2}}=\frac{1}{v^{2}}\frac{\partial^{2}y(x,t)}{\partial t^{2}}\tag{6.2.12} \end{equation}

\begin{equation} \mathscr{F}\{y(x,t)\} = Y(s,t) = \int\limits_{-\infty}^{\infty}y(x,t) e^{-isx}\,dx\tag{6.2.13} \end{equation}

\begin{equation*} \mathscr{F}\{\frac{\partial^{2}y}{\partial x^{2}}\} = \frac{1}{v^{2}}\mathscr{F}\{\frac{\partial^{2}y}{\partial t^{2}}\} \end{equation*}

\begin{equation*} (is)^{2}\mathscr{F}\{y(x,t)\} = \frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}Y(s,t) \end{equation*}

\begin{equation*} -s^{2}Y(s,t) = \frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}Y(s,t) \end{equation*}

\begin{equation} \frac{\partial^{2}Y}{\partial t^{2}}+v^{2}s^{2}Y =0\tag{6.2.14} \end{equation}

\begin{equation} Y(s,t) = C_{1}e^{+ivst}+C_{2}e^{-ivst}\tag{6.2.15} \end{equation}

\begin{equation} Y(s,0) = \int\limits_{-\infty}^{\infty}y(x,0) e^{-isx}\,dx = \int\limits_{-\infty}^{\infty}f(x) e^{-isx}\,dx=F(s)\tag{6.2.16} \end{equation}

\begin{equation} Y(s,0) = C_{1}+C_{2}\tag{6.2.17} \end{equation}

\begin{equation} F(s) = C_{1}+C_{2}\tag{6.2.18} \end{equation}

\begin{equation*} Y'(s,t) = \int\limits_{-\infty}^{\infty}\left(\frac{\partial y}{\partial t}\right)e^{-isx}\,dx = ivsC_{1}e^{ivst} -ivsC_{2}e^{-ivst} \end{equation*}

\begin{equation*} Y'(s,0) = \int\limits_{-\infty}^{\infty}\left(\frac{\partial y}{\partial t}\right)_{t=0}e^{-isx}\,dx = 0= ivs C_{1} -ivs C_{2} \end{equation*}

\begin{equation} \therefore \quad C_{1} = C_{2} \tag{6.2.19} \end{equation}

\begin{equation} Y(s,t) = \frac{F(s)}{2}\left[e^{+ivst}+e^{-ivst}\right]\tag{6.2.20} \end{equation}

\begin{equation*} \mathscr{F}^{-1}\{Y(s,t)\}=y(x,t) = \frac{1}{2\pi}\int\limits_{-\infty}^{\infty}Y(s,t)e^{isx\,ds} \end{equation*}

\begin{equation*} = \frac{1}{4\pi}\int\limits_{-\infty}^{\infty}F(s)\left[e^{ivst}+e^{-ivst}\right]\,ds \end{equation*}

\begin{equation*} =\frac{1}{4\pi}\int\limits_{-\infty}^{\infty}\{f(x')e^{-isx}\,dx'\}\{\left[e^{+ivst}+ e^{-ivst}\right]e^{isx}\}\,ds \end{equation*}

\begin{equation*} =\frac{1}{4\pi}\int\limits_{-\infty}^{\infty}\{f(x')\,dx'\}\{\int\limits_{-\infty}^{\infty}\left[e^{is(x-x' +vt)}+ e^{is(x-x'-vt)}\right]\,ds\} \end{equation*}

\begin{equation*} =\frac{1}{2}\int\limits_{-\infty}^{\infty}\{f(x')\,dx'\}\{\delta(x+vt-x') + \delta(x-vt-x')\} \end{equation*}

\begin{equation} \mathscr{F}^{-1}\{Y(s,t)\}=y(x,t) = \frac{1}{2}\left[f(x+vt) +f(x-vt)\right] \tag{6.2.21} \end{equation}

\begin{equation*} \delta (x) =\frac{1}{2\pi} \int\limits_{-\infty}^{\infty}e^{isx}\,ds \end{equation*}

\begin{equation*} \int\limits_{-\infty}^{\infty}f(x) \delta (a-x)\,dx =f(a)]. \end{equation*}

\begin{equation*} \mathscr{F}_{s}\{f(x)\} = F(s) = \int\limits_{0}^{\infty}f(x)\sin sx\,dx = \int\limits_{0}^{\infty}\frac{e^{-ax}}{x}\sin sx\,dx \end{equation*}

\begin{equation*} \frac{\,d}{\,ds}\left[F(s)\right] = \int\limits_{0}^{\infty}\frac{e^{-ax}}{x}(x\cos sx)\,dx = \int\limits_{0}^{\infty}e^{-ax}\cos sx\,dx = \frac{a}{a^{2}+s^{2}} \end{equation*}

\begin{equation*} F_{s}(s)= \int\limits_{0}^{\infty}\frac{a}{a^{2}+s^{2}}\,ds = \tan^{-1}\frac{s}{a}+c. \end{equation*}