what are some typical systems of equations generating from practical problems?

Question

I want to know some typical forms of system of equations generating from practical problems in engineering/economics/physics,etc.

Some examples or research articles would be good.

Specifically, I am looking for some examples of nonlinear system of equations generated from practical problems.

Thanks.

Answer 1

Have you done any linear or integer programming? Systems of equations come up all the time. In programming models, we seek to optimize $cx$ with respect to a constraint $Ax \leq b$, where $A$ is a matrix, $c$ and $b$ are vectors, and $x$ is the vector we are changing to maximize the system. The system of equations is $Ax \leq b$.

Linear programming is used all the time in economics for market optimization, insurance purchase optimizations, contractual optimizations, game theory, etc.

Linear programming and integer programming also come up in graph theory with network flows (which are practical in terms of fluid or electrical flows), shortest path algorithms, knapsack problems, and such.

Answer 2

Textbook example (not practical!):

A firm produces output $Q$ using capital $K$ and labor $L$. The technology is given by a production function $Q=F(K,L)$ the problem of the firm is to minimize the cost of producing $Q_0$ units of output. That is the firm solves: $\min r\,K+w\,L$ subject to $Q_0\le F(K,L)$. The first-order conditions of the Lagrangian $L(K,L)= rK+wL -\lambda F(K,L)$ are a non-linear system with 3 equations and 3 variables: $$ r -\lambda F_K(K,L)=0,$$ $$w-\lambda F_L(K,L)=0 \text{ and}$$ $$Q_0=F(K,L).$$ Here $w$ and $r$ are the price of labor and capital, they are given so they are not variables. The variables you solve for are $K,L$ and $\lambda$. We also make some assumptions of $F$: that is $F_K$ is decreasing for large $K$ and the same for $F_L$ and $L$.