### 6.1 Summary table

Heat PDE $$\frac{\partial u}{\partial t}=k\frac{\partial ^{2}u}{\partial x^{2}}$$ in $$1D$$ (in a rod)

 Left side Right side $$u(x,0)$$ $$\lambda =0$$ $$\lambda >0$$ $$u\left ( 0\right ) =0$$ $$u\left ( L\right ) =0$$ triangle No $$\begin{array} [c]{l}\lambda _{n}=\left ( \frac{n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \\ X_{n}=B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}\end{array}$$ $$u\left ( 0\right ) =0$$ $$u\left ( L\right ) =0$$ $$100$$ No $$\begin{array} [c]{l}\lambda _{n}=\left ( \frac{n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \\ X_{n}=B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}\end{array}$$ $$u\left ( 0\right ) =T_{0}$$ $$u\left ( L\right ) =0$$ $$x$$ No $$\begin{array} [c]{l}\lambda _{n}=\left ( \frac{n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \\ X_{n}=B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =T_{0}-\frac{T_{0}}{L}x+\sum _{n=1}^{\infty }B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}\end{array}$$ $$\frac{\partial u\left ( 0\right ) }{\partial x}=0$$ $$\frac{\partial u\left ( L\right ) }{\partial x}=0$$ $$x$$ $$\begin{array} [c]{l}\lambda _{0}=0\\ X_{0}=A_{0}\end{array}$$ $$\begin{array} [c]{l}\lambda _{n}=\left ( \frac{n\pi }{L}\right ) ^{2},n=1,2,3,\cdots \\ X_{n}=A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}\end{array}$$ $$\frac{\partial u\left ( 0\right ) }{\partial x}=0$$ $$u\left ( L\right ) =T_{0}$$ $$0$$ No $$\begin{array} [c]{l}\lambda _{n}=\left ( \frac{n\pi }{2L}\right ) ^{2},n=1,3,5,\cdots \\ X_{n}=A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =T_{0}+\sum _{n=1,3,5,\cdots }^{\infty }A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}\end{array}$$ $$\frac{\partial u\left ( 0\right ) }{\partial x}=0$$ $$u\left ( L\right ) =0$$ $$f\left ( x\right )$$ No $$\begin{array} [c]{l}\lambda _{n}=\left ( \frac{n\pi }{2L}\right ) ^{2},n=1,3,5,\cdots \\ X_{n}=A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =\sum _{n=1,3,5\cdots }^{\infty }A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}\end{array}$$ $$u\left ( 0\right ) =0$$ $$\frac{\partial u\left ( L\right ) }{\partial x}=0$$ $$f\left ( x\right )$$ No $$\begin{array} [c]{l}\lambda _{n}=\left ( \frac{n\pi }{2L}\right ) ^{2},n=1,3,5,\cdots \\ X_{n}=B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =\sum _{n=1,3,5\cdots }^{\infty }B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}\end{array}$$ $$u\left ( 0\right ) =0$$ $$u\left ( L\right ) +\frac{\partial u\left ( L\right ) }{\partial x}=0$$ $$f\left ( x\right )$$ No $$\begin{array} [c]{l}\tan \left ( \sqrt{\lambda _{n}}L\right ) +\sqrt{\lambda _{n}}=0\\ X_{n}=B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =\sum _{n=1}^{\infty }B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}\end{array}$$ $$u\left ( 0\right ) +\frac{\partial u\left ( 0\right ) }{\partial x}=0$$ $$u\left ( 0\right ) =0$$ 0 $$\begin{array} [c]{l}\lambda _{0}=0\\ X_{0}=A_{0}\end{array}$$ $$\tan \left ( \sqrt{\lambda _{n}}L\right ) -\sqrt{\lambda _{n}}=0$$ $$u\left ( -1\right ) =0$$ $$u\left ( 1\right ) =0$$ $$f\left ( x\right )$$ No $$\begin{array} [c]{l}\sqrt{\lambda _{n}}=\frac{n\pi }{2}\qquad n=1,2,3,\cdots \\ X_{n}=A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) ,\sin \left ( \sqrt{\lambda _{n}}x\right ) \\ u\left ( x,t\right ) =\sum _{n=1,3,\cdots }^{\infty }A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) e^{-\lambda _{n}t}+\sum _{n=2,4,\cdots }^{\infty }B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) e^{-\lambda _{n}t}\end{array}$$

Heat PDE $$\frac{\partial u}{\partial t}=\alpha \frac{\partial ^{2}u}{\partial x^{2}}-\beta u$$ in $$1D$$ (in a rod) with $$\alpha ,\beta >0$$ for $$0<x<\pi$$

 Left side Right side initial condition $$\lambda =0$$ $$\lambda >0$$ analytical solution $$u\left ( x,t\right )$$ $$\frac{\partial u\left ( 0,t\right ) }{\partial x}=0$$ $$\frac{\partial u\left ( \pi ,t\right ) }{\partial x}=0$$ $$u\left ( x,0\right ) =x$$ $$\begin{array} [c]{l}\lambda _{0}=0\\ X_{0}=A_{0}\end{array}$$ $$\begin{array} [c]{l}\lambda _{n}=n^{2},n=1,2,3,\cdots \\ X\left ( x\right ) =A_{0}+\sum _{n=1}^{\infty }A_{n}\cos \left ( nx\right ) \end{array}$$ $$\frac{\pi }{2}+c_{0}\left ( e^{-\beta t}-1\right ) +\frac{2}{\pi }\sum _{n=1}^{\infty }\frac{\left ( \left ( -1\right ) ^{n}-1\right ) }{n^{2}}\cos \left ( nx\right ) e^{-\left ( n^{2}\alpha +\beta \right ) t}$$

(TO DO) Heat PDE for periodic conditions $$u\left ( -L\right ) =u\left ( L\right )$$ and $$\frac{\partial u\left ( -L\right ) }{\partial x}=\frac{\partial u\left ( L\right ) }{\partial x}$$

$\lambda _{n}=\left ( \frac{n\pi }{L}\right ) ^{2},n=1,2,3,\cdots$$u\left ( x,t\right ) =\overset{\lambda =0}{\overbrace{a_{0}}}+\overset{\lambda >0}{\overbrace{\sum _{n=1}^{\infty }A_{n}\cos \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}+\sum _{n=1}^{\infty }B_{n}\sin \left ( \sqrt{\lambda _{n}}x\right ) e^{-k\lambda _{n}t}}}$