\frac{d Q^{\downarrow}_{AQ}}{dt_D} = \frac{ B \cdot (p_i - p(t_D)) - Q^{\downarrow}_{AQ} \cdot p'_D(t_D)}{p_D(t_D) - t_D \cdot p'_D(t_D)}

p_D= \frac{370.529 \, \sqrt{t_D} +137.528 \, t_D + 5.69549 \, t_D^{1.5}}
{328.834 +265.488 \, \sqrt{t_D} + 45.2157 \, t_D +  t_D^{1.5} }

q^{\downarrow}_{AQ}(t)=\frac{d Q^{\downarrow}_{AQ}}{dt} 

p'_D= \frac
{716.441 + 46.7984 \, \sqrt{t_D} + 270.038 \, t_D + 71.0098 \, t_D^{1.5} }
{ 1296.86 \, \sqrt{t_D} + 1204.73 \, t_D + 618.618 \, t_D^{1.5} + 538.072 \, t_D^2 + 142.41 \, t_D^{2.5} }

t_D= \frac{\pi \, \chi \, t}{A_e}

Equation is not a solution of diffusion equation but an approximation of VEH model which does not involve convolution integral.

Q^{\downarrow}_{AQ}(t_{D,n}) = Q^{\downarrow}_{AQ}(t_{D, \, n-1}) + (t_{D,n}-t_{D,\, n-1}) \cdot 

 \frac{ B \cdot (p_i - p(t_{D,n})) - Q^{\downarrow}_{AQ}(t_{D, \, n-1}) \cdot p'_D(t_{D,n})}{p_D(t_{D,n}) - t_{D,\, n-1} \cdot p'_D(t_{D, n})}