# Introducing DifferentialEquations.jl

#### August 1 2016 in Differential Equations, FEM, Julia, Programming, Stochastics | Tags: DifferentialEquations.jl, julia | Author: Christopher Rackauckas

Edit: This post is very old. See this post for more up-to-date information.

Differential equations are ubiquitous throughout mathematics and the sciences. In fact, I myself have studied various forms of differential equations stemming from fields including biology, chemistry, economics, and climatology. What was interesting is that, although many different people are using differential equations for many different things, pretty much everyone wants the same thing: to quickly solve differential equations in their various forms, and make some pretty plots to describe what happened.

The goal of DifferentialEquations.jl is to do exactly that: to make it easy solve differential equations with the latest and greatest algorithms, and put out a pretty plot. The core idea behind DifferentialEquations.jl is that, while it is easy to describe a differential equation, they have such diverse behavior that experts have spent over a century compiling ... READ MORE

# Julia iFEM3: Solving the Poisson Equation via MATLAB Interfacing

#### January 24 2016 in FEM, Julia, MATLAB, Programming | Tags: FEM, julia, Poisson Equation | Author: Christopher Rackauckas

This is the third part in the series for building a finite element method solver in Julia. Last time we used our mesh generation tools to assemble the stiffness matrix. The details for what will be outlined here can be found in this document. I do not want to dwell too much on the actual code details since they are quite nicely spelled out there, so instead I will focus on the porting of the code. The full code is at the bottom for reference.

## The Ups, Downs, and Remedies to Math in Julia

At this point I have been coding in Julia for over a week and have been loving it. I come into each new function knowing that if I just change the array dereferencing from () to [] and wrap vec() calls around vectors being used as ... READ MORE

# Julia iFEM 2: Optimizing Stiffness Matrix Assembly

#### January 23 2016 in FEM, Julia, MATLAB, Programming | Tags: julia, optimization, sparse, vectorization | Author: Christopher Rackauckas

This is the second post looking at building a finite element method solver in Julia. The first post was about mesh generation and language bindings. In this post we are going to focus on performance. We start with the command from the previous post:

node,elem = squaremesh([0 1 0 1],.01)

which generates an array elem where each row holds the reference indices to the 3 points which form a triangle (element). The actual locations of these points are in the array node, and so node(1) gives the points in the (x,y)-plane for the th point. What the call is saying is that these are generated for the unit square with mesh-size .01, meaning ... READ MORE

# Julia iFEM1: Porting Mesh Generation

#### January 19 2016 in FEM, Julia, MATLAB, Programming | Tags: iFEM, julia, MATLAB, matplotlib, triangulation | Author: Christopher Rackauckas

My first project on the quest for a Julia finite element method is a simple homework problem. Just some background, this is for UC Irvine's graduate Computational PDEs 226B course where in the first quarter we did all sorts of finite difference methods and now is our first foray into finite element methods. The purpose of the project is to grasp the data structure enough to use simple tools (i.e. mesh creation and plotting) to create a finite element solver for Poisson's equation in 2D and check the performance differences. For those who haven't programmed much this is a great learning exercise, but being a pretty standard exercise the MATLAB code took no time to writeup and so I started thinking: how does Julia compare to MATLAB for solving simple PDEs with the finite element method?

Testing this would take ... READ MORE