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Questions tagged [mathematical-modeling]

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modelling.

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Turing Pattern MATLAB code

I am very new in learning turing patterns. Can someone guide me with a sample MATLAB code to generate following type of figures: I have seen in papers that they mention those figures as 'snapshots of ...
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Modeling IP Problem

Hallco runs a day shift and a night shift. No matter how many units are produced, the only production cost during a shift is a setup cost. It costs \$8,000 to run the day shift and \$4,500 to run the ...
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Why are models often described as systems of ODE's rather than just the functions themselves?

Take for instance the following model describing an epedemic accounting for the number of susceptible people $S$ and the number of infected people $I$ $$\frac{dS(t)}{dt} = -aS(t) \cdot I(t)$$ $...
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Expectation Maximization (EM) for mixture of discrete distributions.

I am trying to get each components distribution from a mixture of discrete distribution but the model get stuck in m-step after just 1 or 2 iteration due to singular matrix error. I tried many ...
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Solution of the Lotka Volterra system of differential equations in $\mathbb{R^2}$ is cyclic

I'm currently reading an abstract about the Lotka Volterra equations and I've some questions. Please be aware I'm an amateur. First I will give you the system of differential equations: $$ x^{'} = ...
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How to calculate the amount of rain a person walk in [closed]

If a person walk in the rain ,think of this person as a cuboid,it has a length of a,width of b,height of c . The speed of the rain is u ,the precipitation of the rain is w.and the distance from the ...
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Rearranging an equation involving partial derivatives

I have the equation $\frac{\partial A}{\partial t}+cA^m\frac{\partial A}{\partial s}=0$. I need to rearrange so that the coefficient of the $\frac{\partial A}{\partial s}$ term is equal to 1. Since ...
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Trouble interpreting information to develop mathematical functions

Context: University coding assignment So far from this information, I've managed to develop two models: For the surface zone, I have $T(l)=\frac{-11}{45}l+24$ and for the deep zone, I have $T(l)=2$ ...
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How to model probability $x$ populations have recovered by day $k$?

Suppose the following scenario. $N$ patients are treated with an antibiotic, bringing the population of a certain bacteria in their guts down to negligible levels. I have the following 5 data points: ...
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Solving $\frac{dP}{dt}=aP-bP^2$ (population dynamics)

This equation represents the dynamics of a population. I am being asked to explain whether, according to this model, there is a possibility for unbounded growth or guaranteed decay. Usually, in order ...
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Finding leading order approximation to $\frac{d^2y}{dx^2}-\epsilon y=x$

The conditions provided were $y(0)=1$ and $y'(0)=1$. Since this equation is regular, I can neglect the term involving $\epsilon$, giving $\frac{d^2y}{dx^2}=x$. I tried to solve this in order to ...
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Hugoniot Locus given by parametric curves

I need to prove that the Hugoniot Locus of a point $\hat{u}$ of the equation $$u_t + f (u) _x = 0,\qquad f\in C^2$$ is the set of $n$ curves $$\begin{cases}\tilde{u}_p(\xi, \hat{u})=\hat{u}+\xi r_p(\...
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Select one of three computers to buy that will provide most economical gain

Person in company has to present proposition for purchase of one of three computers with same prices. Purchase of each computer working without failure for a year provides economical gain. Although ...
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Modeling a Time-Sensitive Product Decision

It appears to me that the question of how to cycle through time-sensitive products is often decided uncritically, taking almost for granted a First-In, First-Out (FIFO) algorithm. For example, the ...
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Formulating mathematical model that minimizes the largest deviation between the data and the line

Can someone help me and explain all the steps of solving this problem which is in the field of Model Fitting and Least-Squares Fit. Thank you. PROBLEM: For each of the following data sets, formulate ...
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Notation for set of sequences from another set that have a unique variable.

I have difficulty wrapping my head around how to write such notation. Here's my data: $j = 2$ $k = 10$ $X$ - combination, sequence of digits, ex: $(1, 2, 5)$ $Sx$ - sum of digits inside sequence $...
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Assignment for mathematical modeling of fishery population in known environment

My SO has an assignment in math modeling and her assignment reads "" how much fish can we harvest, in order to harvest an optimal amount of fish on longer period IF we know the optimal amount of fish ...
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Intuition: Time elimination ODEs

a common approach when calculating things like trajectories in systems of ODEs you take $\frac{\text{d}x}{\text{d}y}=\frac{f}{g}$, where $\frac{\text{d}x}{\text{d}t}=f$, and $\frac{\text{d}y}{\text{d}...
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MonteCarlo Random Walk Simulation - steps should be scaled by tmp ? (MATLAB)

I have a question regarding this code snippet that we changed for project 3 Monte-Carlo & Random Walk. monte-carlo-code-segment The code was changed to process all the stepNs. it loops through a ...
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1answer
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NDSolve and Plot report Replace Issue!

NDSolve and Plot Thanks Anyone know what's the issue? NDSolve[{x'[t] == x[t]*(1 - x[t]) - 9*x[t]*y[t]/(10*x[t] + 1), y'[t] == 0.3*y[t]*(1 - 10*y[t]/(10*x[t] + 1)), x[0] == 0.15, y[0] == 0.25}, {...
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What are some methods for finding global stability for this 3D system?

I am studying some biology system and arrives at this simplified dynamics: \begin{align} x_1' &= a_1 + a_2x_2 - a_1x_2 - a_4x_1 - a_5\frac{1+a_6x_3}{1+a_7x_3}x_1\\ x_2' &= a_5\frac{1+a_6x_3}{...
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Global stability question for system with a unique locally-asymptotically-stable steady state

I have an ordinary differential system of dimension larger than 2 that contains a locally-asymptotically-stable unique fixed point. Additionally, the system is bounded. Now, suppose that I can show ...
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SI model analysis

We have an SI model with susceptible individuals $$\frac {dS}{dt} = bM\left(1 - \frac M K\right) - \mu S - \beta SI$$ and infected individuals $$ \frac {dI}{dt} = \beta SI - (\alpha+\mu)I.$$ In this ...
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How to implement adaptive step size Runge-Kutta Cash-Karp?

Trying to implement an adaptive step size Runge-Kutta Cash-Karp but failing with this error: ...
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Using delay block in Simulink to account for flow rate in a fluid loop

I am trying to model a fairly simple cooling system loop where coolant flows over a battery to remove heat, then flows into a large reservoir where the coolant is mixed, coolant then flows out of the ...
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Sketching residual plots

A set of data has been analysed by fitting the model $Y_i = x_i^{'}\beta + \epsilon_i$, where $\epsilon_i$ follows a normal distribution with mean 0 and variance $\sigma^2$. Sketch a residual plot for ...
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Flexible grid update algorithm

I have a 2D square grid where the edges are line segments and the vertices have moved from their initial positions. Based on external constraints, the vertices are submitted to further motion, under ...
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Time-series modeling

I'm wondering what methods can be used to predict a future value using past values. I looked into linear regression modeling, but this doesn't allow for a time value. As an example, say I have an ...
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2answers
60 views

Logistic growth model

Find the solution of the model when $y(0)=100$ $\frac{dy}{dt}=2y(y-1)(3-y)$ This is what I have using partial fractions we get $1=A(y-1)(3-y)+B(2y)(3-y)+C(2y)(y-1)$ We then get that : $A=\frac{-1}...
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Variance of a fitted model

Show that for any linear model, $\sum_{i=1}^{n}\frac{\text{Var}(\widehat{Y_{i}})}{n} = \frac{p\sigma^2}{n}$. Wasn't too sure where to start here. I know that Bias($\widehat{\sigma^2}$)=$-\frac{p\...
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How to tell when a state variable violates a bound during a numerical ODE solve?

Is there a good way to tell when a state variable violates a bound during a numerical ODE solve? For example, say we're simulating an object flying through the air with an ODE solve, it'd be nice to ...
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Kuramoto model coupled equations query

Short query about the 'Kuramoto Model', which is a mathematical model of synchronized coupled oscillators. If we consider the $N=2$ case then the governing equations are $$\frac{d \theta_1}{dt} = \...
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Solving the normal mode equations of a 2D linear elastic solid

I am trying to solve the linear stability of a 2D elastic solid whose dynamics are given by the Navier equations, $\frac{\partial}{\partial x} \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\...
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Obtaining linear tridiagonal system from PDE in hydraulic fracturing

I'm trying to re-solve the governing equations in hydraulic fracturing modeling $$ \frac{\partial q}{\partial x} + \frac{2hC}{\sqrt{t-\tau(x)}} + \frac{\partial A}{\partial t} = 0 , \qquad 0<x&...
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1answer
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Prove that shock wave is weak solution of Burgers' equation (Riemann problem)

In math modeling studies, I need to prove that $$u(x,t)=\begin{cases}u_l\qquad x<st\\ u_r\qquad x>st\end{cases}$$ where $$s=(u_l+u_r)/2$$ is a weak solution for the Riemann problem of ...
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How to compute the number of years needed for the population be $52,000$ using the concept of logistic growth?

To use the concept of logistic growth, it is needed to identify the maximum or the limit of population. In this problem, I can't identify what will be the maximum of the population should I use. The ...
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26 views

Steady States and fractional Population

If I'm assuming that I have a population of size $N(t)$ that is growing, can my steady states be fractions? I'm quite confused because how can a population be a fraction? Note that the differential ...
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How to find a curve whose value changes for a certain percentage?

I've had to find a curve $y(x)$ whose value is $N\%$ smaller than $y(x)$ for $y(x+1)$ where $x $ is an integer.I've been told that this differential equation finds the family of such curves (this was ...
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Show what is the limit of the function in the following ODE system

In the following ODEs system: $$ \begin{array}{ll} \displaystyle \frac{d}{dt}u_1=-\lambda_1 u_1, \\ \displaystyle \frac{d}{dt}u_i=\lambda_{i-1}u_{i-1} -\lambda_i u_i, \quad i=2,\dots, N-1\\ \...
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Is there a formula to calculate the rents of properties in Monopoly boardgame?

I'm writing a game of Monopoly for a personal project in a C++ class this semester and instead of hard-coding in each rent value of a property I was wondering if there's a formula to get them. For ...
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Bernoulli random variables and two player games

With hockey playoffs upon us, I have been thinking about ways of modeling the results of athletic events. Suppose team A wins 50% of its games against average teams, and team B wins 50% of its games ...
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Is there a mathematical or logical theory of mathematical modelling?

This question will be admittedly a bit vague, since I am inquiring about the existence of a theory that I am not sure exists, and if it does exists I have only a vague notion of what it might look ...
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Combining Gaussian estimates over different variables

I have real-world quantities $(X, Y, Z)$ whose joint distribution I would like to model as a multivariate Gaussian $G = (\mu, \Sigma)$. By some method, I come up with an estimate $G_{x, y} = (\mu_{x, ...
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How to fit models to data?

I am working on a project in which I am comparing how well and exponential, logistic and Gompertz model fit to tumour growth data. I have the equations for all three models but I don't know how to go ...
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Relation of linear models

Let's assume that \begin{equation} Y_t -Y_{t-1} = \beta(X_t - X_{t-1}) + \alpha + \varepsilon_t \end{equation} where $Y_t -Y_{t-1}$, $X_t-X_{t-1}$, $\varepsilon_t$ fulfill all assumptions of linear ...
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54 views

Modelling cancer with ordinary differential equations

How can I model cancer spread and cancer treatment using first order differential equations? I will be investigating this topic in my high school math paper, but am unsure of where to begin.
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scaling points of 4 hexagons to plot on top of each other

I posted this question in Cross Validated, but I think my question is best fitted here. If you don't think so please feel free to comment so that I find the right place. Here is the link to question ...
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1answer
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Model for population growth and finding the equilibrium solutions

A model for population growth is given by: $$\frac{dN}{dt} = f(N) = r N \left( 1-\frac{N}{K} \right) \left( \frac{N}{U}-1 \right) $$ where $r,\ U,$ and $K$ are positive parameters and $U < K$. (a)...
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Mathemathical model equation PDE

I am studying models in DPE an the professor give us this problem: $\begin{cases}u_t+au_x=f(x,t)\\ u(x,0)=g(x)\end{cases}$ I've studied the transport equation and the Burger's equation. About heat ...
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My question is about finding best ADVANCED [beyond basic] math book for studying mathematical models in real world (like in nature)? [closed]

I know calculus, somewhat. I need to practice more. I was very good extra-ordinary in math in beginning of high school (2 years). Then, in 3rd year, I was in a school that I was only financially poor ...