Saw tooth IC exact solution: $$ u(x, t) = \frac{x {\mathrm e}^{-\frac{x^{2}}{4 \mu \left(t+1\right)}} \mathrm{erf}\! \left(\frac{t-x+1}{2 \mathrm{\sqrt{\mu t}}\, \sqrt{t+1}}\right) \mathrm{\sqrt{\pi \mu t}} \sqrt{t+1}+2 \left(t+1\right) {\mathrm e}^{-\frac{x^{2}+t-2 x+1}{4 \mu t}} \mu+x {\mathrm e}^{-\frac{x^{2}}{4 \mu \left(t+1\right)}} \mathrm{erf}\! \left(\frac{t+x+1}{2 \mathrm{\sqrt{\mu t}}\, \sqrt{t+1}}\right) \mathrm{\sqrt{\pi \mu t}} \sqrt{t+1}-2 \left(t+1\right) {\mathrm e}^{-\frac{x^{2}+t+2 x+1}{4 \mu t}} \mu-4 \left(t+1\right)^{2} \sinh\! \left(\frac{x}{2 \mu t}\right) {\mathrm e}^{-\frac{x^{2}+t+1}{4 \mu t}} \mu}{\mathrm{\sqrt{t}}\, \sqrt{\pi}\, \mathrm{\sqrt{\mu}}\, \left(t+1\right)^{{3}/{2}} \left({\mathrm e}^{-\frac{x^{2}}{4 \mu \left(t+1\right)}} \mathrm{erf}\! \left(\frac{t-x+1}{2 \mathrm{\sqrt{\mu t}}\, \sqrt{t+1}}\right)+{\mathrm e}^{-\frac{x^{2}}{4 \mu \left(t+1\right)}} \mathrm{erf}\! \left(\frac{t+x+1}{2 \mathrm{\sqrt{\mu t}}\, \sqrt{t+1}}\right)+{\mathrm e}^{-\frac{1}{4 \mu}} \sqrt{t+1}\, \left(\mathrm{erf}\! \left(\frac{x-1}{2 \mathrm{\sqrt{\mu t}}}\right)-\mathrm{erf}\! \left(\frac{x+1}{2 \mathrm{\sqrt{\mu t}}}\right)+2\right)\right)} $$

Ramp down IC exact solution: $$ u(x, t) = \frac{{\mathrm e}^{-\frac{x^{2}+t-2 x}{4 \mu \left(t-1\right)}} \mathrm{\sqrt{\pi \mu t}} \sqrt{1-t}\, \left(x-1\right) \mathrm{erf}\! \left(\frac{t-x}{2 \mathrm{\sqrt{\mu t}} \sqrt{1-t}}\right)+\left(x-1\right) \mathrm{\sqrt{\mu}}\, \mathrm{erf}\! \left(\frac{x-1}{2 \mathrm{\sqrt{\mu t}} \sqrt{1-t}}\right) \mathrm{\sqrt{t}}\, \sqrt{1-t}\, \sqrt{\pi}\, {\mathrm e}^{-\frac{x^{2}+t-2 x}{4 \mu \left(t-1\right)}}-2 t \mu \left(t-1\right) {\mathrm e}^{-\frac{x^{2}+t-2 x+1}{4 \mu t}}+\mathrm{\sqrt{\mu t}} {\mathrm e}^{\frac{t-2 x}{4 \mu}} \left(\mathrm{\sqrt{t}}-1\right)^{2} \sqrt{\pi}\, \left(\mathrm{\sqrt{t}}+1\right)^{2} \mathrm{erf}\! \left(\frac{t-x}{2 \mathrm{\sqrt{\mu t}}}\right)+2 t \mu \left(t-1\right) {\mathrm e}^{-\frac{x^{2}}{4 \mu t}}+\mathrm{\sqrt{\mu t}} {\mathrm e}^{\frac{t-2 x}{4 \mu}} \left(\mathrm{\sqrt{t}}-1\right)^{2} \sqrt{\pi}\, \left(\mathrm{\sqrt{t}}+1\right)^{2}}{\sqrt{\pi \mu} \sqrt{1-t}\, \mathrm{\sqrt{t}}\, \left(t-1\right) \left(-{\mathrm e}^{-\frac{x^{2}+t-2 x}{4 \mu \left(t-1\right)}} \mathrm{erf}\! \left(\frac{t-x}{2 \mathrm{\sqrt{\mu t}} \sqrt{1-t}}\right)-{\mathrm e}^{-\frac{x^{2}+t-2 x}{4 \mu \left(t-1\right)}} \mathrm{erf}\! \left(\frac{x-1}{2 \mathrm{\sqrt{\mu t}} \sqrt{1-t}}\right)+\left({\mathrm e}^{\frac{t-2 x}{4 \mu}} \mathrm{erf}\! \left(\frac{t-x}{2 \mathrm{\sqrt{\mu t}}}\right)+{\mathrm e}^{-\frac{1}{4 \mu}} \mathrm{erf}\! \left(\frac{x-1}{2 \mathrm{\sqrt{\mu t}}}\right)+{\mathrm e}^{\frac{t-2 x}{4 \mu}}+{\mathrm e}^{-\frac{1}{4 \mu}}\right) \sqrt{1-t}\right)} $$