I am dealing with an ODE of the form $$y^{(4)}(x) = -{\sigma}^4\bigl(y(x) + f(x)\bigr)$$ over $x\in[0, L]$ where the non-homogeneous term is a centered superposition of damped waves that can be written $$f(x) = \sum_j f_j(x) - M \quad\text{with}\quad f_j(x) = \Im(a_j\exp(iz_jx)),$$ and $M$ is the mean of the $f_j$s over the domain, $M:=\frac{1}{L}\int_0^L\sum_j f_j(x) dx$. The $a_j$s are complex to carry amplitude and phase at $x=0$, and $z_j := k_j + i\alpha_j$ with $k_j, \alpha_j >0$.
I can obtain a solution, which looks like (I am writing it from memory, some signs might be off) $$ y(x) = 2\Re\biggl[c_1\cosh\biggl(\frac{1+i}{\sqrt{2}}\sigma x\biggr)+c_2\sinh\biggl(\frac{1+i}{\sqrt{2}}\sigma x\biggr)\biggr] + \Im\sum_j A_j\exp(iz_jx) + M$$ with $A_j = -\frac{a_j}{1+{(z_j/\sigma)}^4}$. The complex $c_1, c_2$ can be determined from the boundary conditions, I don't think their expressions matter much here.
Now, I want to numerically evaluate this solution for various combinations of parameters. The index $j$ can take integer values in $(1..N)$, where $N$ is arbitrary but manageable. With some combinations, I quickly run into floating point arithmetic issue, and I thought that some judicious scaling could help^in reducing the number of free parameter. But I am rather new to this, and I do not know what scaling would be the most judicious.
For the independent variable $x$, $L$ would be a logical characteristic quantity that would restrain the domain to $[0, 1]$. But choosing the characteristic $1/\sigma$ would mean turn the factor of the zeroth order of the unknown function to 1, which might be a good thing as well. I might be leaning towards thinking the latter option makes more sense.
For the dependent variable $u$, I am completely lost. I guess each $f_j$ could have its own scaling term, but I probably do not want that? A natural nondimensionalised quantity for waves is the steepness $k A$, with $k$ wavenumber and $A$ (positive, real) amplitude. But here, I have the interplay of the lengthscale $1/\sigma$, the complex wavenumber $z_j$, the amplitudes and $1/\sigma$, $M$, the ratii $\alpha_j/k_j$, the coefficients $c_1$ and $c_2$ which involve the product $\sigma L$ as argument of hyperbolic functions and the $A_j$s, etc.
To be more specific, my question is: how to identify the correct lengthscale, and the correct number of lengthscales, used for nondimensionalising my dependent variable and the forcing terms, in order to make easier numerical manipulations on the solution.