# Can anyone help me in minimizing the following integral equation and finding the roots ?

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AKANKSHA TRIPATHI on 15 May 2024
Edited: Torsten on 25 May 2024
Here i am posting an integral equation for which i want the minimization code and methods.
Ω =λ^ 4 (σ 2 + ~π2 − v 2 ) 2 − cσ + νqT∫ (d ^3k)/ (2π) ^3 {ln(1 − nq(T, µ)) + ln(1 − nq¯(T, µ))} equation(1)
we have to minimize the above equation with respect to sigma(σ) and in the scond term,there is integration on the momentum vector k, where integration term also contain the sigma variable in the following ways
nq(T, µ) = 1/ (1 + exp((sqrt(k^2+g^2(σ^2+π^2)) − µ)/T ))
nq¯(T, µ))= 1/ (1 + exp((sqrt(k^2+g^2(σ^2+π^2)) +µ)/T ))
where all the other notation like pi,lambda,vq,T are the some parametrs which we can fix like
lambda=87.6 Mev
T = 500(temperature)
vq=12
g=3.8
mu=100
c=13.6
pi=0
so, we have to find ∂Ω/∂σ and corresponding value of sigma at which above equation (1) is minimized .
AKANKSHA TRIPATHI on 25 May 2024
I have tried all these optimizers like fminsearch, fmincon but not getting any result . These are not working on the above equation. Without knowing the σ value we can't calculate the Ω because during the calculation of numerical integration σ is required. Please can you provide some initial steps how can i proceed it in any other ways.
Torsten on 25 May 2024
Edited: Torsten on 25 May 2024
Without knowing the σ value we can't calculate the Ω because during the calculation of numerical integration σ is required.
I repeat for the third time:
The optimizer provides numerical values for σ for which you have to evaluate the function Ω(σ). These σ values will be different test values that the solver uses to move towards the optimal σ_opt.
So if you can evaluate Ω for a given value of σ, you can use any optimizer to solve your minimization problem.
Did you plot the function Ω(σ) in the meantime ?
Do you have a function like
function Omega = fun(sigma)
Omega = λ^ 4 (σ 2 + ~π2 v 2 ) 2 cσ + νqT(d ^3k)/ (2π) ^3 {ln(1 nq(T, µ)) + ln(1 nq¯(T, µ))} ;
end
?
Please can you provide some initial steps how can i proceed it in any other ways.
There is no other way.
Maybe you will understand what I mean if you take a look at the examples provided for the optimizers.