Articles of mathematical modeling

How to plot a phase potrait of a system of ODEs (in Mathematica)

I am asked to use mathematica to plot the fixed-points of the following system $$ \frac{dN}{dt} = -\gamma N \left( 1 – \left( \beta M + N\right) \right) $$ $$\frac{dM}{dt} = M \left( 1 – \left( \alpha N + M\right) \right) $$ for the case where $\alpha = 2,\ \beta =2,\ \gamma=1$. I assume it […]

Population Dynamics model

I am currently searching for population dynamics models. Concerning animal population growth, I have found the following so far : Growth models for fish Predator-prey : Lotka-Voltera and Nicholson-Bayeux models Several species in competition for a resource Do you know any other mathematical models ? My aim is to simulate the dynamics of a human […]

Finding the heat flow across the curved surface of a cylinder

I have the following problem: The temperature at a point in a cylinder of radius $a$ and height $h$, and made of material with conductivity $k$, is inversely proportional to the distance from the centre of the cylinder. Find the heat flow across the curved surface of the cylinder. The solution says that $T = […]

Why don't elliptic PDE's have a time coordinate?

Usually second-order linear PDE’s are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the three most famous second-order PDE’s: Elliptic – Laplace’s equation $\nabla^2 u = 0$. Parabolic – the heat equation $u_t = \nabla^2 u$. Hyperbolic – the wave equation $u_{tt} […]

Combining probabilities from different sample spaces

I’m working on a project whereby I’m supposed to determine whether two objects (parts of moving machinery) will be in physical contact with one another given the uncertainty in their positions. I know the covariance of both parts so it wasn’t terribly difficult to formulate a probability density function. Integrating this over the size of […]

On the Cramér-Granville Conjecture and finding prime pairs whose difference is 666

Questions If $p= \text{NextPrime}[q]$ (the smallest prime greater than $p$), and $p-q = 666,$ what are $p$ and $q$? (There may be multiple choices. I am interested in finding one.) Cramér-Granville Conjecture: Defining $p_0=2$, and $p_n$ as the nth odd prime, and the nth prime gap as $g_n=p_{n+1}-p_n$, then $g_n< M \log(p_n)^2$ for some $M>1$. […]

Traffic flow modelling – How to identify fans/shocks?

A highway contains a uniform distribution of cars moving at maximum flux in the $x$-direction, which is unbounded in $x$. Measurements show that the car velocity $v$ obeys the relation: $v = 1 − ρ$, where ρ is the number of cars per unit length. An on-ramp is built into the highway in the region […]

Good book for mathematical modeling

Could you recommend/suggest a good book about mathematical modeling (Not advanced) with examples about classical mechanics, dynamics, aerodynamics, chemistry, electronics and etc?

When modelling a real world event by assuming it has probability p, what are we saying/assuming about how that event behaves?

There are countless books on statistics, and how to apply probability-theory to the real world. But I have never really understood what we are actually doing when we model a real world phenomenon with probability theory. If you have real world events, and say that you model the real world, and assign probabilities to those […]

How to start creating Mathematical models?

I know this is a quite general question , but I’ve always heard about creating mathematical models in economics, social sciences, engineering,… And I would want to know what are the starting points and roadmap to understand what a mathematical model is and how to create them. Thanks