![]() ![]() Let’s take for example the classic ode45. MATLAB documents its ODE solvers very well, there’s a similar interface for using each of the different methods, and it tells you in a table in which cases you should use the different methods.īut the modifications to the methods goes even further. The MATLAB ODE Suite does extremely well at hitting these goals. Instead of focusing on efficiency, they key for this group is to have a clear and neatly defined (universal) interface which has a lot of flexibility. The idea is pretty simple: users of a problem solving environment (the examples from his papers are MATLAB and Maple) do not have the same requirements as more general users of scientific computing. Shampine also had a few other papers at this time developing the idea of a “methods for a problem solving environment” or a PSE. MATLAB’s differential equation solver suite was described in a research paper by its creator Lawerance Shampine, and this paper is one of the most highly cited SIAM Scientific Computing publications. You can find it here (click for PDF):ĭue to its popularity, let’s start with MATLAB’s built in differential equation solvers. If you just want a quick summary, I created a table which has all of this information. You will see at the end that DifferentialEquations.jl does offer pretty much everything from the other suite combined, but that’s no accident: our software organization came last and we used these suites as a guiding hand for how to design ours.) Quick Summary Table (Full disclosure, I am the lead developer of DifferentialEquations.jl. I hope that by giving you the details for how each suite was put together (and the “why”, as gathered from software publications) you can come to your own conclusion as to which suites are right for you. This is a good way to reflect upon what’s available and find out where there is room for improvement. What I would like to do is take the time to compare and contrast between the most popular offerings. For the field of scientific computing, the methods for solving differential equations are what’s important. As we state there, students are strongly encouraged to verify that the proposed solution indeed satisfies the requisite equation and supplementary conditions.Many times a scientist is choosing a programming language or a software for a specific purpose. Accordingly, we have not reproduced the steps of the verification process in every case, rather content with the illustration of some basic cases of verification in the text. ![]() The subject of differential equations is particularly well-suited to self-study, since one can always verify by hand calculation whether or not a given proposed solution is a bona fide solution of the differential equation and initial conditions. However, in our teaching we have found that it is helpful to have further documentation of the various solution techniques introduced in the text. In an ideal world this volume would not be necessary, since we have systematically worked to make the text unambiguous and directly useful, by providing in the text worked examples of every technique which is discussed at the theoretical level. ![]() The purpose of this companion volume to our text is to provide instructors (and eventu ally students) with some additional information to ease the learning process while further documenting the implementations of Mathematica and ODE. ![]()
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