# Preface

This section discusses a simplified version of the Adomian decomposition method first concept of which was proposed by Randolf Rach in 1989 that was crystallized later in a paper published with his colleagues G. Adomian and R.E. Meyers. That is way this technique is frequently referred to as the Rach--Adomian--Meyers modified decomposition method (MDM for short). Initially, this method was applied to power series expansions, which was based on the nonlinear transformation of series by the Adomian--Rach Theorem. Similar to the Runge--Kutta methods, the MDM can be implemented in numerical integration of differential equations by one-step methods. In case of polynomials or power series, it shows the advantage in speed and accuracy of calculations when at each step the Adomian decomposition method allows one to perform explicit evaluations.

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## Glossary

# Falkner--Skan layer

# MATHEMATICA TUTORIAL for the First Course. Part VII: Tridiagonal linear systems

# 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

# Tridiagonal linear systems

The solution of linear systems of equations is one of the most important areas of computational mathematics. We are not going to present this topic in detail---it deserves a special course. Instead, we consider one particular and very important case when the leading matrix is tridiagonal:

**x**is an unknown

*n*-vector,

**b**is the given

*n*-vector, and

**A**is a tridiagonal \( n \times n \) matrix

*l*

_{i},

*d*

_{i}, and

*u*

_{i}to denote the lower-diagonal, diagonal, and upper-diagonal elements:

*l*

_{1}=0 and

*u*

_{n}=0. Under this notation, the augmented matrix corresponding to the system is

**l**,

**d**,

**u**, and

**b**, instead of the entire

*n*-by-

*n*matrix, which is mostly zeroes anyway.

This system of algebraic equations could be solved using standard Gaussian elimination procedure, which
actually reduces the problem to an upper triangular form. This stage is usually referred to as the forward
elimination (FE). Once it is completed, the second stage, which is called backward substitution (BS), involves finding actual
solution. Therefore, this algorithm is usually called **FEBS**, or in computational jargon
"progonka." In engineering,
the FEBS method is associated with the British scientist Llewellyn H. Thomas from Bell laboratories who solved
a simple Poisson problem (see following example) using this method in 1946. Historically, a prominent Soviet mathematician
Israel Moiseevich Gelfand (1913--2009) discovered FEBS algorithm in 1933 being a sophomore college student. He
personally refused to associate his name with FEBS because, in his opinion, it was a very simple application of
Gaussian elimination. Instead, he suggested to call FEBS algorithm as
"progonka (прогонка),"
and this slang is widely accepted.

Using the elimination procedure, the augmented matrix is reduced to an equivalent upper triangular form:

*x*

_{n-1}, and so on. Again, carrying out the backward substitution stage requires the assumption that each \( \delta_k \ne 0, \quad 1 \le k \le n . \)

```
FEBS (progonka) algorithm
```

% elimination stage

for i = 2 to n

d(i) = d(i) - u(i-1)*l(i)/d(i-1)

b(i) = b(i) - b(i-1)*l(i)/d(i-1)

endfor

% backward substitution

x(n) = b(n)/d(n)

for i=n-1 downto 1

x(i) = (b(i) - u(i)*x(i+1))/d(i)

endfor

Theorem: If the tridiagonal matrix **A** is diagonally dominant
(\( d_i > |l_i | + |u_i | > 0, \quad 1 \le i \le n; \) ), then FEBS algorithm
will succeed in producing the correct solution to the original linear system, within the limitations of rounding
error. ■

Example: Consider the Dirichlet problem on the interval [0,1]:

*f*is a known function, and we seek

*u*that satisfies the differential equation and the boundary conditions at

*x=0*and

*x=1*.

We divide the interval [0,1] into *n* equal subintervals \( x_{k-1} , x_k ] , \) according to

*u(x)*. We can use Taylor approximations:

*u*with

*U*, which yields

*n-1*values \( U_k = U (x_k ) , \quad 1 \le k \le n-1, \) that satisfy the recurrence:

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