We consider a time-dependent reaction-diffusion equation with a singularity arising from incompatible initial and boundary conditions: \[\begin{aligned} {2} u_t - u_{xx} + b(x,t) u & = f & \quad & \text{in} \ \ (0,\ell)\times(0,T], \\ \\ \textrm{subject to boundary conditions}\\ u(0,t) = \varphi_0(t), \ \ \ u(\ell,t) & = \varphi_\ell(t), && t\in (0,T], \\ \textrm{and the initial condition}\\ u(x,0) & = 0, && x\in (0,\ell)\,, \end{aligned}\] with \(\varphi_0(0) \neq 0\).

The discrepancy between initial and boundary conditions causes the formation of a singularity in the vicinity of the corner \((0,0)\). This singularity \(s\) can be characterised as the solution of \[\begin{aligned} {2} s_t - s_{xx} + b(0,0) s & = 0 & \quad & \text{in} \ \ (0,\infty)\times(0,T], \\ \\ \textrm{subject to the boundary condition}\\ s(0,t) & = \varphi_0(0), && t\in (0,T], \\ \textrm{and the initial condition}\\ s(x,0) & = 0, && x\in (0,\infty)\,. \end{aligned}\] This in turn can be given analytically using the error function. Then the interesting question is:

How can the remainder \(y=u-s\) be resolved numerically?

We derive bounds on the derivatives of the remainder \(y\) — under significantly less restrictive assumptions then previously assumed by other authors — and show how a numerical approximation can be obtained using an appropriately designed mesh.

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