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UQ-Applied Mathematical Analysis
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1. Ordinary differential equations
- Slope fields of ordinary differential equations
- Slope and direction field plotter
- Euler's Method
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2. Systems of ODE
- Classification of critical point of linear systems
- Phase portrait of homogeneous linear first-order system DE
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3. Non-linear systems
- Lotka-Volterra model
- Non-linear system
- Lotka-Volterra predator-prey model
- Lotka-Volterra system
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4. Applications of Linear Second order Equations
- Simple Harmonic Motion
- Undamped Forced Oscillations
- Free Vibrations with Damping
- Forced Vibrations With Damping
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5. Attractors
- Lorenz attractor
- Rössler attractor
- Thomas' attractor
- Rabinovich–Fabrikant attractor
- Lorenz-84 attractor
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6. Velocity fields
- Velocity fields: Potential
- Systes of ODE with Particles 3D
- Uniform flow past a circular cylinder
- Uniform flow past a circular cylinder with circulation
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7. Fourier series
- Fourier series expansion
- Fourier series - Example 1
- Fourier series - Example 2
- Fourier series - Example 3
- Fourier series
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8. Heat equation
- Heat equation: visualisation tool
- Heat equation
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9. Diffusion
- Diffusion equation
- Diffusion
- Diffusion random
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10. Wave equation
- Wave equation: d'Alembert's formula
- Wave Equation on a Line
- Wave Equation on a Half Line
- Wave equation
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UQ-Applied Mathematical Analysis
Juan Carlos Ponce Campuzano, Sep 26, 2016

Table of Contents
- Ordinary differential equations
- Slope fields of ordinary differential equations
- Slope and direction field plotter
- Euler's Method
- Systems of ODE
- Classification of critical point of linear systems
- Phase portrait of homogeneous linear first-order system DE
- Non-linear systems
- Lotka-Volterra model
- Non-linear system
- Lotka-Volterra predator-prey model
- Lotka-Volterra system
- Applications of Linear Second order Equations
- Simple Harmonic Motion
- Undamped Forced Oscillations
- Free Vibrations with Damping
- Forced Vibrations With Damping
- Attractors
- Lorenz attractor
- Rössler attractor
- Thomas' attractor
- Rabinovich–Fabrikant attractor
- Lorenz-84 attractor
- Velocity fields
- Velocity fields: Potential
- Systes of ODE with Particles 3D
- Uniform flow past a circular cylinder
- Uniform flow past a circular cylinder with circulation
- Fourier series
- Fourier series expansion
- Fourier series - Example 1
- Fourier series - Example 2
- Fourier series - Example 3
- Fourier series
- Heat equation
- Heat equation: visualisation tool
- Heat equation
- Diffusion
- Diffusion equation
- Diffusion
- Diffusion random
- Wave equation
- Wave equation: d'Alembert's formula
- Wave Equation on a Line
- Wave Equation on a Half Line
- Wave equation
Slope fields of ordinary differential equations


Classification of critical point of linear systems
Consider the homogeneous linear first-order system differential equations
x'=ax+by y'=cx+dy
The following worksheet is designed to analyse the nature of the critical point of the linear system.

Lotka-Volterra model
Consider the Lotka-Voterra equations of interacting predator and prey systems
x'=x(a-cx-dy) y'=-y(b-ex)-h
where all of x, y, a, b, c, d, k, n are positive, and- x represents the number of prey
- y represents the number of predators
- a is the growth rate of the prey.
- b is the death rate of the predators independent of the prey.
- d is the is the rate of consumption of the prey per predator.
- a / c is the carrying capacity of the prey independent of the predators.
- e is the growth rate of the predator per prey consumed,
- h is prey harvesting.


Simple Harmonic Motion


Lorenz attractor
Note: Online version is slow. Download file for better performance.


Velocity fields: Potential
This simulation shows the potential surface of the vector field, with particles following the velocity vectors.
Enter an arbitrary potential function in the box Potential. Try for example:
- -(sqrt(x^2 + y^2) + 1 / sqrt(x^2 + y^2)) cos(atan2(y,x))
- ln(sqrt((x + 1)^2 + y^2)) - ln(sqrt((x - 1)^2 + y^2))
- 1/sqrt(x^2+y^2)
- sqrt(x^2+y^2)
- arctan2(y,x)
- x^2-y^2
- x-1/2y^2
- -1/x^2
- abs(x)
- -x

Fourier series expansion
Description
The following simulation shows the partial sum (up to 20 terms) of the Fourier series for a given function defined on the interval [a,b].
You can also check your calculations by entering the coefficients a0, a1 and b1.
Instructions:
Change the function and calculate its Fourier series. Then type the correct values of the terms a0, a1 and b1, rounded to two decimal places.
Remark: Activate the box Fourier series and increase, or decrease, the number of terms in the partial sum.


Heat equation: visualisation tool
Consider a thin rod of length with an initial temperature throughout and whose ends are held at temperature zero for all time . The temperature in the rod is determined from the boundary-value problem:
0<x<L and t>0;
t>0;
0<x<L.
In the following simulation, the temperature is graphed as a function of x for various fixed times.
Things to try:
- Change the initial condition u(x,0)=f(x).
- Explore the solutions by clicking on the buttons or type a number to show the graph.


Diffusion equation
Consider the heat equation
A particular solution of the heat equation is given by
The following worksheet shows a representation of this solution.
Use right click and drag the mouse to rotate the 3D view.


Wave equation: d'Alembert's formula
Consider the initial value problem for the wave equation on the whole number line:
with , with , with .


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