I am a student doing physics hons and have had very little experience in programming. This semester we are supposed to do a computational project in thermodynamics. I have to solve these two coupled diff eqns:

$$ \begin{aligned} ω (p,T ) &= \frac{p ^2}{2m} + \frac{2}{N}\sum_q f(p − q)\,n (q) − \frac{1}{N^2}\sum_{s,t} f(s − t)\,n(s)\,n(t) \quad\text{and} \\ n(p) &= \frac{1}{\exp\left[\cfrac{ω (p,T ) − \mu}{kT}\right] − 1} \end{aligned} $$

$\omega$ is the energy per boson.

$p$ is the momentum of a boson.

$n(p)$ is the number of bosons in the state with momentum $p$.

$f$ is a function of the form $$ f(p)=\frac{1}{2} \left[\epsilon_0- \frac{p^2}{2m}\right] $$

$\epsilon_0$ is elementary excitation energy at 0 K.

$N$ is total no of bosons.

$T$ is temp and $k$ is a constant.

Can someone guide me to any simple methods to generate some crude solution to this problem? Based on a paper: "Evaluation of specific heat for superfluid helium between 0 - 2.1 K based on nonlinear theory" By Shosuke Sasaki (arXiv:0807.1361v1 [cond-mat.other] 9 Jul 2008

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    $\begingroup$ Cannot understand your mathematics, are they differential equations? You are much more likely to get a good response if you rewrite your equations using latex. $\endgroup$ – boyfarrell Oct 23 '15 at 9:04
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    $\begingroup$ These are nonlinear algebraic equations and not ODEs. I'm not sure that the function is convex but you can check the Hessian. If it is convex, you can try to minimize $\omega$ using gradient descent. $\endgroup$ – Biswajit Banerjee Oct 23 '15 at 19:46

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