Let $x\in\mathbb{R}$,$t>0$, $$\frac{\partial u}{\partial t}+5\frac{\partial u}{\partial x}=e^{-t}\sin x,$$ $$u(x,0)=(1-x^2)_{+}.$$ Using Duhamel's principle to solve it.
By Duhamel's principle, the key point is to find $H(x,t)$ such that $$\frac{\partial H}{\partial t}+5\frac{\partial H}{\partial x}=0,$$ and $\int_\mathbb{R}H(x-y,0)f(y,t)dy=f(x,y),$ where $f(x,t)=e^{-t}\sin x$. Then $u(x,t)$ can be written as $$u(x,t)=\int_0^t\int_\mathbb{R}H(x-y,t-s)f(y,s)dyds+(1-x^2)_{+}.$$ How to construct such $H(x,t)$? Is it the $\delta$ function. But this is not reasonable.