I'm trying to solve the following Cauchy problem in ${\rm \Bbb R}$ without using the fundamental solution. $$ \begin{cases} u_t = u_{xx} &\text{ for }\;\,(x,t)\in\Bbb R\times\{ 0<t<\infty\}\\ u|_{t=0}=g, \end{cases} $$ where $g(x)$ is defined by $$ g(x)= \begin{cases} 0, & x < 0 \\ 1, & x > 0 \\ \frac{1}{2} & x=0 \end{cases} $$ I have a hint to look for a solution in the form $u(x,t)=\phi\big(\frac{x}{\sqrt{t}}\big)$, but I don't know how to apply this hint and get started! Any help, or skeleton of a solution would be appreciated!
Edit: I think we can use the hint to write $\phi$ as an ODE, but then it would be a function of both x and t, so I don't know if this would help.
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Hint: \begin{align} \int_{0}^{\infty}\partiald{\mrm{u}\pars{x,t}}{t}\expo{-st}\dd t & = \int_{0}^{\infty}\partiald[2]{\mrm{u}\pars{x,t}}{x}\expo{-st}\dd t \\[5mm] -\mrm{g}\pars{x} +s\hat{\mrm{u}}\pars{x,s}& = \partiald[2]{\hat{\mrm{u}}\pars{x,t}}{x}\,,\quad \left\{\begin{array}{rcl} \ds{\hat{\mrm{u}}\pars{x,s}} & \ds{\equiv} & \ds{\int_{0}^{\infty}\mrm{u}\pars{x,t}\expo{-st}\dd t} \\[2mm] \ds{\mrm{u}\pars{x,t}} & \ds{=} & \ds{\int_{c - \infty\ic}^{c + \infty\ic}}\hat{\mrm{u}}\pars{x,s}\expo{ts}{\dd s \over 2\pi\ic} \end{array}\right. \end{align}