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Does someone know if it is possible to solve this type of coupled differential equations with NDSolve?:

enter image description here The dot indicates a $\Lambda$-derivation. I am just asking if it is possible to do this in principle, before i waste 6 hours trying. Which I will attempt to do if noone answers.

EDIT

Ok, so i tried to solve a somewhat similar but simpler equation system:

$\partial_{x1}y_1(x1,x2)=y_1(x1,x2)$, $\partial_{x1}y_2(x1)=y_1(x1,x1)$

with initial conditions: $y1(0,x2)=\delta_{0,x2} \ , y2(0)=0$

My code is

s = NDSolve[{D[y1[x1, x2], x1] == y1[x1, x2], D[y2[x1], x1] == y1[x1, x1], y2[0] == 0, y1[0, x2] == KroneckerDelta[x2, 0]}, y2, {x1, 0, 30}]

The Problem i encounter here is that NDSolve gives me an error that initial conditions need to be numbers.

EDIT

If I add {x2,0,30} I get the error "the arguments should be ordered consistently", this is the error I want to avoid..

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    $\begingroup$ This system of equations can be solved numerically. I do not understand the notation $[k_1<->k_3,k_2<->k_4]$. What does this mean? $\endgroup$ – Alex Trounev Sep 10 '18 at 11:48
  • $\begingroup$ @AlexTrounev I guess it means the previous term shall be repeated with the indicated perturbations of indices applied. $\endgroup$ – Henrik Schumacher Sep 10 '18 at 12:16
  • $\begingroup$ @AGP As Alex already said: It should be possible to solve the system numerically. A first step would be the translation of these expression into Mathematica. And it is also not clear to me what $k_1,\cdots$ are supposed to be let alone $\gamma_0$, $\gamma_1$, $\gamma_2$, $\beta$. $\endgroup$ – Henrik Schumacher Sep 10 '18 at 12:20
  • $\begingroup$ Ok so for some reason I need reputation to comment? So this answer is actually a comment: Henrik is right about the permutations. The $\gamma$'s are so called vertexfunctions, $\beta=1/k_bT$ and the k's are frequencies, this is a physics problem, specifically a Renormalization Group problem. And i just found the edit option. yey. $\endgroup$ – A G P Sep 10 '18 at 12:32
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    $\begingroup$ @AGP I have converted your non-answer into a real comment. I'm not sure why you have two different accounts with the same name. Try to get them merged because you can always comment under your own questions. To comment everywhere however, you need 50 reputation points. $\endgroup$ – halirutan Sep 10 '18 at 14:47

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