Numeric integration with unknowns

The problem I have is a system of three non-linear equations with three unknowns. Each equation has a integration term. But the integration term has all the three unknowns in there. Some suggested to discretize. So I need to discretize all three unknowns. And solving the system become looping over the three variables and find when the equations hold at certain combination of the values. I am not sure if this is the right way to go or there are better options The three equations are $$L=N_a \int_S l(s)dF(s)$$ where $l(s)=(\gamma(1-\theta)Ak\frac{P_a}{q})^{\frac{1}{1-\gamma}}[\theta((\frac{\theta}{1-\theta})\frac{q}{r})^{\frac{\rho}{1-\rho}}+(1-\theta)s^{\frac{\theta}{1-\theta}}]^{\frac{\gamma-\rho}{\rho(1-\gamma)}}s^{\frac{\rho}{1-\rho}}$ $$K_a+K_n=K$$ , where $K_a$ is determined by a similar integral as above $K_a=N_a \int_S k(s)dF(s)$, and $k(s)$ has a similar function as $l(s)$ There a similar third equation.The three unknowns are $q$, $N_a$ and $K_n$. But the problem is the integrand $l(s)$ has $q$ in there.

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2 Answers

Without seeing your equations, one can offer nothing more than hypothetical advice.

One possible approach would be to differentiate the equation on both sides, obtaining, instead, a differential equation, which can then be solved using many methods. This, however, may not be possible, and you might be stuck with an integro-differential equation.

Alternatively, you could restrict the domain of your independent variables (either naturally, through restrictions invoked during the derivation of the problem, or heuristically based on some extrinsic expertise). Then, you could treat each integral as just some function, $F(x,y,z)$. Writing a numerical integration routine that allows you to compute $F(x,y,z)$ with sufficient accuracy is the same as computing the value $F(x,y,z)$ -- you can then use a standard root-finding method (e.g. Newton's method) to compute the solution. (This would be slow).

Without seeing the equations, however, it is impossible to advise you on what the best approach would be.

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I put the equations in there. I hope the question is clear now. – Yan Song Oct 31 '12 at 17:03

Perhaps you could differentiate relevant equations to convert them to a system of ODE instead, and then use standard methods?

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