# 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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# Hamming Method

In 1959, Richard Wesley Hamming from Bell Telephone Laboratories, Murray Hill, New Jersey, proposed a stable predictor-corrector method for ordinary differential equations, which now bears his name. He improved classical Milne--Simpson method by replacing unstable corrector rule by a stable one.

Journal of the ACM, Volume 6, Issue 1, Jan. 1959, pages 37-47.

*x*

_{0}= a) by using the predictor

*h*. Here

*y*

_{n}is an approximation to the true value \( \phi (x_n ) \) of the actual solution \( y = \phi (x) . \)

**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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