# NDSolve PDE of function u[x,t], boundary condition needs to feed an expression of t into u[x,t] as x

I am trying to obtain a range of numerical solutions of the following convection-diffusion-reaction equation:

$$\frac{\partial u(x,t)}{\partial t}+0.01\frac{\partial u(x,t)}{\partial x}=-10^{-12}\frac{\partial^2u(x,t)}{\partial x^2}+10 ^6\cdot \left(u(x,t)\right)^2+\left(10^6\cdot10^{-8}\cdot10^{-8}-10^{-3}(10^{-8}+10^{-8})\right)\cdot u(x,t)+10^6\cdot 10^{-8} \cdot 10^{-8}$$

With the initial and boundary conditions:

$$u(0,t)=0\\u(x,0)=0\\\frac{\partial u(x,t)}{\partial x}|_{x=0.01t}=0$$

I used NDSolve as the following:

eqn = D[u[x, t], t] + 0.01*D[u[x, t], x] == -10^-12*D[u[x, t], x, x] + 10^6*(u[x, t])^2 + (10^6*10^-8*10^-8 - 10^-3*(10^-8 + 10^-8)) u[x, t] + 10^6*10^-8*10^-8
uval = NDSolve[{eqn, u[0, t] == 0, u[x, 0] == 0, (D[u[x, t], x] /. x -> 0.01 t) == 0}, u, {x, 0, 10}, {t, 0, 600}]


As you can see, the boundary condition $$\frac{\partial u(x,t)}{\partial x}|_{x=0.01t}=0$$ feeds $$0.01t$$ as $$x$$ into the partial derivative of $$u(x, t)$$ with respect to $$x$$. But running this in Wolfram Mathematica gives the following error:

NDSolve::conarg: The arguments should be ordered consistently.


Apparently NDSolve doesn't accept feeding an independent variable as another independent variable into a multivariable function. How do I solve this?

• Flux conditions are applied on an elemental basis. For example, a 1-D problem would be applied to a LineElement that is specified with 2 points. What is the meaning of applying a zero flux condition at a singular point? Commented Aug 19, 2020 at 14:54
• You may want to check for sign errors as you have a negative diffusion coefficient. Also, are your trying to represent a moving boundary with your flux condition? Commented Aug 19, 2020 at 20:21
• @TimLaska Yes, this is a model of Lateral-Flow Immuno-Assay, which involves the movement of a liquid through a membrane with a constant velocity, and hence indicating a moving boundary. About the issues you have mentioned, I also find the negative diffusion coefficient weird, but this equation is directly adopted from a published research paper... So I don't really know if it's my problem or the authors'. (The title of that paper is "A mathematical model to predict the optimal test line location and sample volume for lateral flow immunoassays") Commented Aug 20, 2020 at 0:14
• Unfortunately, I cannot find a pdf of the paper and I am also traveling for the next few days. What I gathered online is that the LFIA is chromatography like problem consisting of a mobile and stationary phase (the stationary phase has at least a test and control region). You can break this into a coupled problem as I did on this adsorption problem answer 226609 . Commented Aug 20, 2020 at 1:02
• @TimLaska Thank you, I will take a look. Commented Aug 20, 2020 at 7:59