Problem finding explicit formulation of series coefficient

Hi,

I am currently working on gas diffusion through membranes and I am having issues finding a coefficient for the ODE's solution, based on the method presented here.

My problem concerns the coefficient lambda, appearing after developping Fick's law, given by eq.16 :

tan(λ_{n}) = (k*S_{R} − δ_{R}λ_{n}(λ_{n} + S_{R}β_{n})) / (S_{R}δ_{R}λ_{n} + β_{n}(k*S_{R} − δ_{R}λ_{n}^{2})

where k = 1,

S_{R} remains a symbolic variable,

δ_{R }is a known constant,

β_{n }is a function of λ_{n},δ_{R},S_{R }and k

To perform a curve fit with my experimental data, I need to find an explicit formulation of λ_{n. }In practice, I can consider only the first term of the summation (λ_{1}) for sufficient accuracy.

How can I express explicitly λ_{1}? The tan() function is giving me a headache (Headbang)

Thank you,

edit : I would like to do something similar to the transformation in eqs 11-12 in this paper

Re: Problem finding explicit formulation of series coefficient

If $\beta_1$ is an unspecified function of $\lambda_1$ I don't see how you have any hope of even approximately solving for $\lambda_1$

Do you have bounds on $\lambda_1$ ?

On $\left(-\dfrac {\pi}{4}, \dfrac {\pi}{4}\right), \tan(x) \approx x$

Re: Problem finding explicit formulation of series coefficient

Hi,

β is a known function, sorry for the misunderstanding.

β = (δ_{R} * λn^{2} - k S_{R}) / (S_{R} * λn * δ_{R})

Could you give me hints on the method applied in the second paper (Aachib et al.) to transform the alpha parameter from eq. 11 to eq. 12? It would work for my purposes, but I don't see what the intermediate steps are ...

Thank you,