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I am concerned in finding an explicit solution to the following functional equation, where $P$ is the unknown function:

$$P(q) = \lambda_1(q)+ \lambda_2(q)P(q+2 \chi_s),$$

where: $$\lambda_1(q)=\frac{\alpha_2(q) + \alpha_2(q+2\chi_s)}{\cosh q - \alpha_1(q)}$$ and: $$\lambda_2(q) = \frac{\cosh(q+2\chi_s)+\alpha_1(q+2\chi_s)}{\cosh q - \alpha_1(q)}$$ he auxiliary functions $\alpha_1$ and $\alpha_2$, are explicitly given by:

$$\alpha_1(q)= \alpha_{10}\sinh q + \frac{\alpha_{11}}{\sinh q},\ \ \ \alpha_2(q)= \frac{\alpha_{20}}{\sinh q}.$$

Besides, the real numbers $\alpha_{10}$, $\alpha_{11}$, $\alpha_{20}$ and $\chi_s$ are initial parameters of the problem, and are given real constants.

Iterating the original functional equation gives us a particular solution for P(q), expressed as an infinite series of the form:

$$P(q)= \lambda_1(q)+ \sum_{n=1}^{\infty}\lambda_1(q+2n\chi_s)\prod_{j=0}^{n-1}\lambda_2(q+2j\chi_s)$$

which can be verified to be a solution after substituting in the original equation. However, I would like to obtain an explicit solution written with a finite number of terms, not an infinite series. I wonder if this is actually possible, given its complexity. Any hint will be certainly acknowledged.

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Juan Gustavo Wouchuk Schmidt
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  • Your equation is a linear recurrence of the first order, so that you can split the solution in homogeneous and non-homogeneous parts. But unfortunately, the coefficients are horrible. –  Feb 18 '20 at 11:20
  • Dear Yves, thanks for your comment. .I'vve already tried it. Due to reasons related to the physics of the problem , a discussionwhich would be tooextense to include here, the solution to the homogeneous equation is always zero, and we are only left with the particular solution, given as the infinite series above. I would recast my question onto this otherone: would it be possible to obtain a particular solution written only with a finite number of terms instead of an infinite series as p? – Juan Gustavo Wouchuk Schmidt Feb 18 '20 at 18:24
  • Dear Yves, your suggestion is OK. I had already triedit long ago. Because of reasons that pertain to the physics of the problem driving us to the above functional equation, which would be out of scope here, the solution to the homogeneous equation is always zero, thus we are left again with the series displayed above. I would recast my question this way: would it be possible to sum the mentioned series, or to get another particular solution, in finite terms? Sorry for not claryfying mouch more the point, and thanks for your time, anyway. – Juan Gustavo Wouchuk Schmidt Feb 18 '20 at 18:29
  • IMO, the probability is below zero. –  Feb 18 '20 at 18:30
  • Dear Yves, I am of the opinion that ironies like probability is negative, do not help so much. Besides, we never know in advance of an ingenuous idea that may be useful to to get the desired solution. Anyway, thanks for your participation and for time invested in reading the posted question. Best. – Juan Gustavo Wouchuk Schmidt Feb 21 '20 at 10:26
  • That was just a kind hint that you risk to break your teeth on this problem. –  Feb 21 '20 at 10:27

1 Answers1

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A direct approch to solving the functional equation is by defining an iteration sequence, like : ($n\geq 1$, is an integer number defining the step level inside the iteration sequence) \begin{equation} P^{[n]}(q)= \lambda_1(q)+ \lambda_2(q)P^{[n-1]}(q+ 2\chi_s) \end{equation}

which is quite good for numerical evaluation, once we decide the seed function $P^{[0]}(q)$ for which, we usually choose:

$$P^{[0]}(q)=\frac{\lambda_1(q)}{1-\lambda_2(q)},$$

Unfortunately, the sequence has to be truncated at some high value n of the iteration index and the solution so obtained is still not very useful, because for certain initial conditions of the problem (which would take me way too far in the description of the physical phenomenon), giving rise to the above functional equation, very high values of n might be needed into obtain an accurate enough representation of P(q). This the reason why I struggle trying to get a finite form solution to the equation, expressible with a finite number of operations on the functions $\lambda_{1,2}(q)$ [1].

Hope this is doable.

Thanks again for your attention.

[1] "Analytical asymptotic velocities in linear Richtmyer-Meshkov-like flows" by F. Cobos Campos and J. G. Wouchuk, Physical Review E 90, 053007 (2014).

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Juan Gustavo Wouchuk Schmidt
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