# Preface

This
tutorial is made solely for the purpose of education and it is 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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## Glossary

# Finite Difference Schemes

Methods involving difference quotient approximations for derivatives can be used for solving certain second-order boundary value problems. Consider the Dirichlet boundary value problem for the linear differential equation

*x*

_{j}and the resulting equations are substituted into the given differential equation. This yields

*h*

^{3}) to obtain

*h*

^{2}and then collecting terms involving

*x*

_{j-1},

*x*

_{j}, and

*x*

_{j+1}and arranging them in a system of linear equations:

There is a special scheme, called progonka or forward elimination and back substitution (FEBS) to solve algebraic equations with tridiagonal matrices. This scheme, which is widely used in numerical simulations, was first discovered by a prominent Soviet mathematician Israel Moiseevich Gel'fand (1913--2009) in his student work. He personally never claimed his authority for the discovery because he thought it was just easy (from his point of view) application of Gauss elimination. In engineering, the FEBS method is sometimes accosiated with Llewellyn H. Thomas from Bell laboratories who used it in 1946.

**Example**.

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