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