Solid cylinder transient 1-D.

R01 Solid cylinder 0 < r < b, with G = 0 (Dirichlet) at r = b.

$$\left\vert\vphantom{ \,r^{\prime },\tau }\right. \prime \tau \left.\vphantom{ \,r^{\prime },\tau }\right. {\frac{1}{\pi b^{2}}} \sum_{m=1}^{\infty } \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}}\right. \beta_{m}^{2} \alpha \tau \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}}\right]$$ =

$${\frac{J_{0}(\beta _{m}r/b)\,J_{0}(\beta _{m}r^{\prime }/b)}{\left[ J_{1}(\beta _{m})\,\right] ^{2}}}$$

with eigenvalues given by J₀($\beta_{m}^{}$) = 0.

R02 Solid cylinder 0 < r < b, with $\partial$G/$\partial$r = 0 (Neumann) at r = b.

$$\left\vert\vphantom{ \,r^{\prime },\tau }\right. \prime \tau \left.\vphantom{ \,r^{\prime },\tau }\right. {\frac{1}{\pi b^{2}}} \left[\vphantom{ 1+\sum_{m=1}^{\infty }\exp \left[ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}% \right] }\right. \sum_{m=1}^{\infty } \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}% }\right. \beta_{m}^{2} \alpha \tau \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}% }\right] \left.\vphantom{ 1+\sum_{m=1}^{\infty }\exp \left[ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}% \right] }\right.$$ =

$$\left.\vphantom{ \times \frac{J_{0}(\beta _{m}r/b)\,J_{0}(\beta _{m}r^{\prime }/b)}{% \left[ J_{0}(\beta _{m})\,\right] ^{2}}}\right. {\frac{J_{0}(\beta _{m}r/b)\,J_{0}(\beta _{m}r^{\prime }/b)}{% \left[ J_{0}(\beta _{m})\,\right] ^{2}}} \left.\vphantom{ \times \frac{J_{0}(\beta _{m}r/b)\,J_{0}(\beta _{m}r^{\prime }/b)}{% \left[ J_{0}(\beta _{m})\,\right] ^{2}}}\right]$$

with eigenvalues given by J₁($\beta_{m}^{}$) = 0.

R03 Solid cylinder 0 < r < b, with k$\partial$G/$\partial$r + hG = 0 (convection) at r = b.

$$\left\vert\vphantom{ \,r^{\prime },\tau }\right. \prime \tau \left.\vphantom{ \,r^{\prime },\tau }\right. {\frac{1}{\pi b^{2}}} \sum_{m=1}^{\infty } \left[\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}}\right. \beta_{m}^{2} \alpha \tau \left.\vphantom{ -\beta _{m}^{2}\alpha (t-\tau )/b^{2}}\right]$$ =

$${\frac{\beta _{m}^{2}J_{0}(\beta _{m}r/b)\,J_{0}(\beta _{m}r^{\prime }/b)}{\left[ J_{0}(\beta _{m})\,\right] ^{2}(B^{2}+\beta _{m}^{2})}}$$

with eigenvalues given by - $\beta_{m}^{}$J₁($\beta_{m}^{}$) + BJ₀($\beta_{m}^{}$) = 0 , and B = hb/k.