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 \(\frac{\partial u\left ( -L\right ) }{\partial x}=\frac{\partial u\left ( L\right ) }{\partial x}\)

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