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I've built a finite element solver to solve the transient diffusion-reaction equation $$\frac{\partial c}{\partial t} = D\frac{\partial^{2} c}{\partial x^{2}} - \lambda c + f$$ where $\lambda$ and $f$ are the reaction and source terms respectively. I've built the solver to be able to take any initial condition and Dirichlet/Neumann boundary conditions. However, I'm struggling to find analytical solutions that can help me benchmark my solver. Could someone point me in the right direction with finding some analytical solutions?

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Assume $D=1$. Assume a source term of the form $F(t,x)=-\mathrm e^{-\lambda t}x(1-x)$. Consider Dirichlet BCs on the interval $x\in[0,1]$. Now try $c(t,x)=\mathrm e^{-\lambda t}\phi(x)$ to get $$-\lambda \mathrm e^{-\lambda t}\phi(x)=\mathrm e^{-\lambda t}\phi''(x)-\lambda \mathrm e^{-\lambda t}\phi(x)-\mathrm e^{-\lambda t}x(1-x) \\ \implies \phi''(x)=x(1-x)$$ Which is solved by $$\phi(x)=\frac{-1}{12}(x^4-2x^3+x)$$ You can choose any $\lambda$ you want, $1$ is convenient. So, formally, the problem $$\begin{cases}\partial_t c=\partial_x^2c-c+F & (t,x)\in[0,\infty)\times [0,1] \\ c(0,x)=\frac{-1}{12}(x^4-2x^3+x)\\c(t,0)=c(t,1)=0\end{cases}$$ With $F(t,x)=-\mathrm e^{- t}x(1-x)$, is solved by $$c(t,x)=\mathrm e^{-t}c(0,x)$$

Hopefully this is good enough for testing purposes?

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  • $\begingroup$ Is it weird that the analytical solution matches my finite element solution exactly at different timescales? The finite element solution diffuses much quicker and reaches the same steady state at $$t=1$$ that the analytical solution would at $$t= 10$$. $\endgroup$
    – S0yboi
    Dec 9, 2022 at 22:19
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    $\begingroup$ There is no steady state. Observe $\partial_t c\neq 0$. $\endgroup$
    – K.defaoite
    Dec 9, 2022 at 22:20

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