# Preface

This is a
tutorial 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* 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.

Finally, the commands in this tutorial are all written in bold black font,
while *Mathematica* output is in regular fonts. This means that you can
copy and paste all comamnds into *Mathematica*, change the parameters and
run them. You, as the user, are free to use the scripts
to your needs for learning how to use the *Mathematica* program, and have
the
right to distribute this tutorial and refer to this tutorial as long as
this tutorial is accredited appropriately.

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# 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]

## Fixed Point Iteration

## Bracketing Methods

## Secant Methods

## Euler's Methods

## Heun Method

## Runge-Kutta Methods

## Runge-Kutta Methods of order 2

## Runge-Kutta Methods of order 3

## Runge-Kutta Methods of order 4

## Polynomial Approximations

## Error Estimates

## Adomian Decomposition Method

## Modified Decomposition Method

## Multistep Methods

## Multistep Methods of order 3

## Multistep Methods of order 4

## Milne Method

## Hamming Method

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