Search: ode

28 results

Modern 1

  Seam Carving


Seam carving is an algorithm for content-aware resizing of an image. This notebook presents the algorithm and tries to provide some insight into its workings.

Modern 3

  Neural Network From Scratch


Making a neural network from scratch and training the network using a dataset from scikit-learn.

  A quick introduction to the Julia programming language

fortran, basic, Julia

Julia has to some degree already cemented itself in the scientific community, and will most likely continue to expand in the coming years. It aims at taking the middle ground between Python on one side, and Fortran and C++ on the other. In this notebook we offer a quick introduction for those who wish to venture from Python to Julia.

  Quantum computing - Basics

basic, quantum computers, cat states

Uses the qiskit framework to run basic quantum circuits, both locally and on real quantum computers.

Biophysics 1

  Bak-Sneppen model in Julia

basic, Julia

The Bak-Sneppen model of evolution is a simple model describing the evolution of an ecosystem. It offers a surprising amount of insight given its simplicity. This notebook does not deep dive into the model, but illustrates the basics using the Julia programming language.

Ordinary Differential Equations 1

  Euler's Method

euler, differential equation, euler explicit method, set of odes, basic

A thorough walkthrough of the theoretical aspects of Euler's method. Also covers how to solve higher order ODEs.

Electromagnetism 1

  Magnetic Mirror

embedded runge-kutta pair, integration, set of odes

The trajectory of a charged particle propagating in a non-uniform magnetic field is calculated by solving the Lorentz force law using an embedded Runge-Kutta pair. The results show that the particle is mirrored.

Basics 1

  Simple implementation of Euler's method

explicit euler method, ode, implementation, basic

Basic notebook covering how to implement Euler's method, without much focus on theory

Statistical Mechanics 1

  Equilibrium Monte Carlo simulation of the 2D Ising model

magnetism, spin, monte carlo, fortran, f2py, metropolis, equilibrium, autocorrelation, ising

Using the Metropolis algorithm to approximate the magnetization and specific heat for a 2D Ising lattice.

Electromagnetism 1

  Uniform Magnetic Field

magnetism, explicit euler method, ode, Lorentz' law

Solving Lorentz force law for a charged particle traveling in a uniform magnetic field using Euler's method.

Quantum Mechanics 1

  Modeling Atoms, Molecules, and Crystals in One Dimension

spin, hemmer, eigenenergy, eigenstate, schrödinger equation, waves

Solving the time independent Schrödinger equation in one dimension using matrix diagonalisation for five different potentials.

Mechanics 1

  Projectile motion

explicit euler method, 4th order runge-kutta, ode, Big Bertha, set of odes

Computing the trajectory of a projectile moving through the air, subject to wind and air drag.

Astrophysics 2

  The Fate of Our Universe

space, gravity, ode

Solving the Friedmann equations to model the expansion of our universe.

  Precession of Mercury

space, einstein, gravity, 4th order runge-kutta, angular momentum, fortran, ode, extrapolation, f2py

Computes the precession of Mercury by linear extrapolation.

Mechanics 2

  Double Pendulum and Chaos

animation, differential equation, Lagrangian, Euler-Lagrange equations, chaos, phase space, odeint

Discusses the chaotic motion of the double pendulum using a phase-space diagram

  Simple Pendulum

animation, explicit euler method, ode

Simulates the simple pendulum and damped simple pendulum

Astrophysics 1

Curve Fitting 1

Thermodynamics 1

  One-dimensional Stationary Heat Equation

temperature, chebychev nodes

Solving the one-dimensional stationary heat equation with a Gaussian heat source by approximating the solution as a sum of Lagrange polynomials.

Biophysics 1

  The Fiber Bundle Bridge Model

polymer, hemmer, hansen, lennard-jones potential, sparse matrix, nygård, elasticity

Simulates the disentanglement of a polymer from a surface using an increasing electric field

Statistical Mechanics 1

  Ising Model in 1D and 2D

specific heat, magnetism, partition function, spin

Computing the internal energy, specific heat and magnetisation in the 1D and 2D Ising model. An analytical solution to the XY model is also provided.

Waves and Acoustics 1


simpson's method, newton, euler, eigenvalue, poisson's equation, integration, interpolation

Analyzing sloshing using a numerical approach based on a linear model, which reduces the problem to a Steklov eigenvalue problem.

Curve Fitting 1

  Polynomial Interpolation

newton, lagrange, interpolation, chebychev nodes, runges phenomenon

Using polynomial interpolation to interpolate a set of points and to approximate a function or a curve.

Ordinary Differential Equations 4

  Adaptive Runge-Kutta Methods

explicit euler method, 4th order runge-kutta, embedded runge-kutta pair, trapezoidal method, ode

Solving a first-order ordinary differential equation using Runge-Kutta methods with adaptive step sizes.

  Runge-Kutta Methods

explicit euler method, 4th order runge-kutta, ode

Solving a first-order ordinary differential equation using the Runge-Kutta method.

  Implicit Euler Method

euler, ode

Solving a first-order ordinary differential equation using the implicit Euler method (backward Euler method).

  Verlet Integration


Solving a second-order ordinary differential equation (Newton's second law) using Verlet integration.

Chemistry 1

  Lennard-Jones Potential

explicit euler method, lennard-jones potential

A simple physical model that approximates the interaction between a pair of neutral atoms or molecules.