Consider a heat equation $$ u_{t} = u_{xx}+f(t,x),\; (x,t) \in (0,L)\times(0,T] \\ u(0,x) = 0, \; x \in [0,L] \\ u(t,0) = 0, \; t \in [0,T] \\ u(t,L) = 0, \; t \in [0,T]. $$ If $f(t,x)$ is a regular function then solution of this system is given by [Tikhonov, Equations of Mathematical Physics]: $$ u(t,x) = \int\limits_{0}^{t} \int\limits_{0}^{L} G(x,\xi,t-\tau)f(\xi,\tau)d\xi d\tau, $$ where $$ G(x,\xi,t) = \frac{2}{l}\sum\limits_{n=1}^{\infty} \sin \left( \frac{\pi n}{L}x \right) \sin \left( \frac{\pi n}{L} \xi \right) \exp \left( -\left( \frac{\pi n }{L} \right)^2 t \right) $$ Tikhonov writes that this formula for solution may be extended to larger class of functions $f$. My question is how large is this class? Which class of distributions we can put in formula for solution? Is it possible for such distributions as $f(t,x)=h(t)\delta(x-x_0)$, where $h$ is a regular function?
Extension of formula for solution of heat equation on distributions
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pde
distribution-theory