# Preface

This tutorial was made solely for the purpose of education and it was designed for students taking Applied Math 0330. It is primarily for students who have very little experience or have never used *Mathematica* and programming before and would like to learn more of the basics for this computer algebra system.
As a friendly reminder, don't forget to clear variables in use and/or the kernel. The *Mathematica* commands in this tutorial are all written in **bold black font**,
while *Mathematica* output is in normal font.

Finally, you can copy and paste all commands into your *Mathematica* notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License
(GPL). You, as the user, are free to use the scripts for your needs to learn the *Mathematica* program, and have
the right to distribute this tutorial and refer to this tutorial as long as
this tutorial is accredited appropriately. The tutorial accompanies the
textbook *Applied Differential Equations.
The Primary Course* by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

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## Glossary

# Milne--Simpson Method

Another popular predictor-corrector scheme is known as the Milne or Milne--Simpson method. See

Milne, W. E., Numerical Solutions of Differential Equations, Wiley, New York, 1953.

Its predictor is based on integration of the slope function *f(t, y(t))* over the interval \( \left[ x_{n-3} , x_{n+1} \right] \) and then applying the Simpson rule:

*f(t, y(t))*based on four mesh points \( (x_{n-3} , f_{n-3} ), \ (x_{n-2} , f_{n-2} ), \ (x_{n-1} , f_{n-1} ), \ (x_{n} , f_{n} ). \) It is integrated over the interval \( \left[ x_{n-3} , x_{n+1} \right] . \) This produces the Milne predictor:

*h*

^{5}). The Milne corrector is developed similarly.

*p*

_{n+1}. The Milne--Simpson multistep method is sometimes unstable. The Milne method converges when

**Example**. Let us start with the Riccati equation \( y' = x^2 + y^2 \) subject to the initial condition \( y(0) =-1 . \) Its solution is expressed through Bessel functions:

*x = 2.223378383*as the figure shows

Plot[d[x], {x, 0, 5.5}, PlotStyle -> Thick]

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