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\title{The computer algebra package {\sc Crack}} 
\author{Thomas Wolf \\                        
	School of Mathematical Sciences \\
	Queen Mary and Westfield College \\
	University of London \\
	London E1 4NS \\
	T.Wolf@maths.qmw.ac.uk
\\ \\
Andreas Brand \\Fakult\"{a}t f\"{u}r Mathematik und 
Informatik \\Friedrich Schiller Universit\"{a}t Jena \\ 07740 Jena
\\ Germany \\ maa@hpux.rz.uni-jena.de
}

\begin{document}
\maketitle
\tableofcontents                                  
\section{The purpose of {\sc Crack}}
The package {\sc Crack} attempts the solution of an overdetermined 
system of ordinary or partial differential
equations (ODEs/PDEs) with at most polynomial nonlinearities. 
Under `normal circumstances' the number of DEs which describe physical
processes matches the number of unknown functions which are involved.
Moreover none of those equations can be solved or integrated and
integrability conditions yield only identities. Although applying
the package
{\sc Crack} to such problems
directly will not be of much help usually, it is possible to
investigate difficult DE-systems indirectly by 
studying analytic properties which would be useful for their
solution. In this way overdetermined PDE-systems result.

Applications of {\sc Crack} include a program {\sc Conlaw}
for the computation of conservation laws of DEs, a program 
{\sc LiePDE} for the computation of infinitesimal symmetries of DEs
and a program {\sc ApplySym} for the computation of symmetry and 
similarity variables from infinitesimal symmetries.

\section{Technical details}  
\subsection{System requirements} 
The required system is {\sc Reduce}, version
3.6., strictly speaking the PSL version of {\sc Reduce} as distributed by
the Konrad Zuse Institut / Berlin. Work on compatibility issues
aims to provide a CSL {\sc Reduce} compatible version of {\sc Crack}
in near future (by the end of 1998).

Memory requirements depend crucially on the
application. 
The {\tt crack.rlg} file is produced from running 
{\tt crack.tst} in a 4MB session running {\sc Reduce}, version 3.6 under
{\sc Linux}. On the other hand 
it is not difficult to formulate problems that 
consume any amount of memory.

\subsection{Installation}
In a running {\sc Reduce} session either do \\
\verb+    in "crack.red"$ + \\
or, in order to speed up computation, either compile it with 
\verb+    on comp$ + \\
before the above command, or, generate a fast-loading compiled 
file once with \\
\verb+    faslout "crack"$ + \\
\verb+    in "crack.red"$ + \\
\verb+    faslend$ + \\
and load that file to run {\sc Crack} with \\
\verb+    load crack$ + 

\subsection{Updates / web demos}
%{\sc Crack} can be run from a web demo 
A web demo under the address
\verb+http://cathode.maths.qmw.ac.uk/demo.html+
that was created by Francis Wright and Arrigo Triulzi
allows to run problems of restricted size. 
The latest version is available from \\
\verb+ftp://ftp.maths.qmw.ac.uk/pub/tw/crack/+.
Publications related to {\sc Crack} can be found under  \\
\verb+http://www.maths.qmw.ac.uk/~tw/public2.html#2+.

\subsection{The files}
The following files are provided with {\sc Crack}
\begin{itemize}
\item {\tt crack.red} contains read-in statements of a number
of files {\tt cr*.red}.
\item {\tt crack.tst} contains test-examples.
\item {\tt crack.rlg} contains the output of {\tt crack.tst}.
\item {\tt crack.tex} is this manual.
\end{itemize}

\subsection{The call}
{\sc Crack} is called by
\begin{tabbing}
  {\tt crack}(\=\{{\it equ}$_1$, {\it equ}$_2$, \ldots , {\it equ}$_m$\},  \\
	      \>\{{\it ineq}$_1$, {\it ineq}$_2$, \ldots , {\it ineq}$_n$\}, \\
	      \>\{{\it fun}$_1$, {\it fun}$_2$, \ldots , {\it fun}$_p$\},  \\
	      \>\{{\it var}$_1$, {\it var}$_2$, \ldots , {\it var}$_q$\});
\end{tabbing}     
	$m,n,p,q$ are arbitrary.
\begin{itemize}
\item
The {\it equ}$_i$ are identically vanishing partial differential expressions, 
i.e.\
they represent equations  $0 = {\it equ}_i$, which are to be solved for the 
functions ${\it fun}_j$ as far as possible, thereby drawing only necessary
conclusions and not restricting the general solution.
\item
The {\it ineq}$_i$ are algebraic or differential expressions which must 
not vanish identically for
any solution to be determined, i.e. only such solutions are computed for which
none of the expressions {\it ineq}$_i$ vanishes identically in all independent 
variables.
\item
The dependence of the (scalar) functions ${\it fun}_j$ on independent 
variables must be defined beforehand with {\tt DEPEND} rather than
declaring these functions 
as operators. Their arguments may  themselves only be identifiers
representing variables, not expressions.
Also other unknown functions not in ${\it fun}_j$ must not be represented 
as operators but only using {\tt DEPEND}.
\item
The functions ${\it fun}_j$ and their derivatives may only occur polynomially.
\item
The ${\it var}_k$ are further independent variables, which are not
already arguments  
of any of the ${\it fun}_j$. If there are none then the fourth argument is 
the empty list \{\}, although it does no harm to include arguments of
functions ${\it fun}_j$.
\item 
The dependence of the ${\it equ}_i$ on the independent variables and on
constants and functions other than ${\it fun}_j$ is arbitrary.
\item
{\sc Crack} can be run in automatic batch mode (by default) or
interactively with the switch {\tt OFF BATCH\_MODE}.
\end{itemize}

\subsection{The result}  
The result is a list of solutions 
\[      \{{\it sol}_1, \ldots \}  \]
where each solution is a list of 4 lists:
\begin{tabbing}
    \{\=\{${\it con}_1, \; {\it con}_2, \ldots , \; {\it con}_q$\}, \\
      \>\{${\it fun}_a={\it ex}_a, \;\;
{\it fun}_b={\it ex}_b, \ldots , \;\; {\it fun}_p={\it ex}_p$\},\=  \\
      \>\{${\it fun}_c, \;\; {\it fun}_d, \ldots , \;\; {\it fun}_r$\}, \> \\
      \>\{{\it ineq}$_1$, {\it ineq}$_2$, \ldots , {\it ineq}$_s$\}. \>  \}
\end{tabbing}
For example, in the case of a linear system as input, there is
at most one solution ${\it sol}_1$.

If {\sc Crack} finds a contradiction as e.g. $0=1$ then there exists no
solution and it returns the empty list \{\}. If {\sc Crack}
can factorize algebraically a non-linear equation then factors are set
to zero individually and different sub-cases are studied through
{\sc Crack} calling itself recursively.
If during such a recursive call a contradiction results,
then this sub-case will not have a solution but other sub-cases still
may have solutions.
The empty list is also returned if no solution exists 
which satisfies the inequalities
{\it ineq}$_i \neq 0.$ 

The expressions ${\it con}_i$ (if there are any), are the
remaining necessary and sufficient conditions for the functions
${\it fun}_c,\ldots,{\it fun}_r$ in the third list. Those
functions can be original functions from the equations to be
solved (of the second argument of the call of {\sc Crack}) or new
functions or constants which arose from integrations. 
The dependence of new functions on variables is declared with {\tt DEPEND}
and to visualize this dependence the algebraic mode function
${\tt FARGS({\it fun}_i)}$ can be used.
If there are no ${\it con}_i$ then all equations are solved and the
functions in the third list are unconstrained. The elements of the fourth
list are the expressions who have been assumed to be unequal zero
in the derivation of this solution.

The second list contains
equations ${\it fun}_i={\it ex}_i$ where each ${\it fun}_i$ is an
original function and ${\it ex}_i$ is the computed expression
for ${\it fun}_i$.

\subsection{Interactive mode, flags, parameters and the list of procedures}
Under normal circumstances one will try to have problems solved
automatically by {\sc Crack}. An alternative is to input
{\tt OFF BATCH\_MODE;} before calling {\sc Crack} and solve problems
interactively. In interactive mode it is possible to
\begin{itemize}
\item inspect data, like equations and their properties, unknown
functions, variables, identities, a statistics,
\item save, change, add or drop equations,
\item inspect and change flags and parameters which govern individual
modules as well as their interplay,
\item specify how to proceed, like doing
 \begin{itemize}
 \item one automatic step,
 \item one specific step,
 \item a number of automatic steps,
 \item a specific step as often as possible.
 \end{itemize}
\end{itemize}
To get interactive help one enters `h' or `?'.

Flags and parameters are stored as symbolic fluid variables
which means that they can be accessed by {\tt lisp( ... )},
like {\tt lisp( print\_:=5 );} before calling {\sc Crack}.
{\tt print\_}, for example, is a measure of the maximal
length of expressions still to be printed on the screen
(the number of factors in terms).
A complete list of flags and parameters is given at the beginning of
the file {\tt crinit.red}. 

One more parameter shall be mentioned, which is the list of modules/procedures
called {\tt proc\_list\_}. In interactive mode this list can be looked
at with {\tt `p'} or be changed with {\tt `cp'}. This list defines
in which order the different modules/procedures are tried whenever
{\sc Crack} has to decide of what to do next. There are exceptions
to this rule possible. Some procedures, say
$P_1$, require after their execution another
specific procedure, say $P_2$, to be executed, independent of whether
$P_2$ would be next according to {\tt proc\_list\_}. This is managed by $P_1$
writing after its completion the procedure $P_2$
into a hot-list. This list is dealt with in the {\tt `to\_do'} step which
comes always first in {\tt proc\_list\_}. 
A way to have the convenience of running {\sc Crack} automatically
and still being able to break the fixed rhythm prescribed by {\tt
proc\_list\_} is to have the entry {\tt stop\_batch} in {\tt proc\_list\_}
and have {\sc
Crack} started in automatic batch mode. Then execution is continuing
until none of the procedures which come before {\tt stop\_batch} are
applicable any more so that {\tt stop\_batch} is executed next which will
stop automatic execution and go into interactive mode. This allows
either to continue the computation interactively, or to change the 
{\tt proc\_list\_} with {\tt `cp'} and to continue in automatic mode.

The default value of {\tt proc\_list\_} does not include all possible
modules because not all are suitable for any kind of overdetermined
system to be solved. The complete list is shown in interactive mode
under {\tt `cp'}. A few basic modules are described in the following
section. The efficiency of {\sc Crack} in automatic mode is very much
depending on the content of {\tt proc\_list\_} and the sequence of its
elements. Optimizing {\tt proc\_list\_} for a given task needs
experience which can not be formalized in a few simple rules and will
therefore not be explained in more detail here. The following remarks
are only guidelines.
\begin{description}    
\item[{\tt to\_do :}] hot list of steps to be taken next, should
always come first,
\item[{\tt subst\_level\_? :}] substitutions of functions by
expressions differing by their maximal allowed size and other
properties,
\item[{\tt separation :}] what is described as direct separation in the
next section,
\item[{\tt gen\_separation :}] what is as indirect separation in the
next section, only to be used for linear problems,
\item[{\tt quick\_integration :}] integration of very specific short equations,
\item[{\tt full\_integration :}] integration of equations which have
the chance to lead to a substitution,
\item[{\tt integration :}] any integration,
\item[{\tt factorization :}] splitting the computation into the
investigation of different subcases resulting from the algebraic
factorization of an equation, only useful for non-linear problems,
\item[{\tt undetlinode :}]  parametric solution of single under determined
linear ODE (with non-constant coefficients), only applicable for
linear problems,
\item[{\tt length\_reduction\_1 :}] length reduction by algebraic
combination, only for linear problems,
\item[{\tt length\_reduction\_2 :}] length reduction by differential
reduction,
\item[{\tt decoupling :}] steps towards the computation of a
differential Gr\"{o}bner Basis,
\item[{\tt add\_differentiated\_pdes :}] only useful for non-linear
differential equations with leading derivative occuring non-linearly,
\item[{\tt add\_diff\_star\_pdes :}] for the treatment of non-linear
indirectly separable equations,
\item[{\tt multintfac :}] to find integrating factors of for a system
of equations (very slow),
\item[{\tt alg\_solve\_deriv :}] to be used for equations quadratic in
the leading derivative,
\item[{\tt alg\_solve\_system :}] to be used if a (sub-)system of
equations shall be solved for a set of functions or their derivatives
algebraically, 
\item[{\tt subst\_derivative :}] substitution of a derivative of a
function everywhere by a new function if such a derivative exists
\item[{\tt undo\_subst\_derivative :}] undo the above substitution.
\end{description}

\section{Contents of the {\sc Crack} package}
The package {\sc Crack} contains a number of modules. 
The basic ones are for computing a pseudo differential Gr\"{o}bner
Basis (using integrability conditions in a systematic way), integrating
exact PDEs, separating PDEs, solving DEs containing functions of only
a subset of all variables and solving standard ODEs (of Bernoulli or
Euler type, linear, homogeneous and separable ODEs). These facilities
will be described briefly together with examples. The test file
{\tt crack.tst} demonstrates these and others.

\subsection{Pseudo Differential Gr\"{o}bner Basis}
This module (called `decoupling' in {\tt proc\_list\_})
reduces derivatives in equations by using other equations and it applies
integrability conditions to formulate additional equations which are
subsequently reduced, and so on.

%How this could work is demonstrated in the following example.
%The integrability condition for the system
%\[ \begin{array}{cccl}
%f = f(x,y), \; \; & f,_{x} & = & 1   \\
%		  & f,_{y} & = & (f-x-1/y)x - 1/y^2
%\end{array}  \]
%provides an algebraic condition for the function $f$
%which turns out not only to be necessary but also sufficient to solve both
%equations:
%\begin{eqnarray*}
% 0 = f,_{xy} - f,_{yx} & = & - xf,_x - f + 2x + 1/y \\
%		       & = & - f + x + 1/y \; \; \; \; \; \;
% \mbox{(with $f,_x$ from above)}
%\end{eqnarray*}
%\[ \rightarrow f = x + 1/y. \]
A general algorithm to bring a system of PDEs into a standard form
where all integrability conditions are satisfied by applying
a finite number of additions, multiplications and differentiations
is based on the general theory of involutive systems \cite{Riq,Th,Ja}.

Essential to this theory is a total ordering of partial derivatives
which allows assignment to each PDE of a {\em Leading Derivative} 
(LD) according to a chosen ordering of functions
and derivatives. Examples for possible orderings are 
\begin{itemize}
\item lex.\ order of functions $>$ lex.\ order of variables,
\item lex.\ order of functions $>$ total differential order $>$ lex.\ 
      order of variables,
\item total order $>$ lex.\ order of functions $>$ lex.\ order of variables
\end{itemize}
or mixtures of them by giving weights to individual functions and variables.
Above, the `$>$' indicate ``before'' in priority of criteria. The principle
is then to
\begin{itemize}
\item take two equations at a time and differentiate them as often as 
necessary to get equal LDs,
\item regard these two equations as algebraic equations in
the common LD and calculate the remainder w.r.t.\ the LD, i.e.\ to
generate an equation without the LD by the Euclidean algorithm, and
\item add this equation to the system.
\end{itemize}
Usually pairs of equations are taken first, such that only one must be
differentiated. If in such a generation step one of both equations is not
differentiated then it is called a
simplification step and this equation will be replaced by the new equation.

The algorithm ends if each combination of two equations yields only equations
which simplify to an identity modulo the other equations.
A more detailed description is given e.g. in \cite{Alex,Reid1}.

Other programs implementing this algorithm are described e.g. in
\cite{FS,Alex,Fush,Reid1} and \cite{Mans}.

In the interactive mode of {\sc Crack} it is possible to change the
lexicographical ordering of variables, of functions, to choose between
`total differential order' ordering of variables or lexicographical
ordering of variables and to choose whether lexicographical ordering
of functions should have a higher priority than the ordering of the
variables in a derivative, or not.

An example of the computation of a differential Gr\"{o}bner Basis is
given in the test file {\tt crack.tst}.

\subsection{Integrating exact PDEs}
The technical term `exact' is adapted for PDEs from exterior calculus and
is a small abuse of language but it is useful to characterize the kind of PDEs
under consideration.

The purpose of the integration module in {\sc Crack} is to  decide
whether a given differential
expression $D$ which involves unknown functions $f^i(x^j),\;\; 1\leq i\leq m$ 
of independent variables $x^j, 1\leq j\leq n$
is a total derivative of another expression $I$
w.r.t. some variable $x^k, 1\leq k\leq n$ 
\[ D(x^i,\; f^j,\; f^j,_p,\; f^j,_{pq}, \ldots) 
     = \frac{d I(x^i,\; f^j,\; f^j,_p,\; f^j,_{pq}, \ldots)}{d x^k}. \]
The index $k$ is
reserved in the following for the integration variable $x^k.$
With an appropriate function of integration $c^r,$
which depends on all variables except $x^k$ it is no loss of generality
to replace $0 = D$ by $0 = I + c^r$ in a system of equations.

Of course there
always exists a function $I$ with a total derivative equal to $D$ but
the question is whether for \underline{arbitrary} $f^i$ the integral
$I$ is functionally dependent only on the $f^i$ and their derivatives,
and \underline{not on integrals of $f^i.$} \\
\underline{Preconditions:} \\
$D$ is a polynomial in the $f^i$ and their derivatives. The number of
functions and variables is free. 
For deciding the existence of $I$ only, the explicit occurrence of the
variables $x^i$ is arbitrary. In order to actually
calculate $I$ explicitly, $D$ must have the property that all terms in $D$ 
must either contain an unknown function of $x^k$ or
must be formally integrable w.r.t. $x^k.$
That means if $I$ exists then 
only a special explicit occurrence of $x^k$ can prevent the
calculation of $I$ 
and furthermore only in those terms which do not contain
any unknown function of $x^k.$ 
If such terms occur in $D$ and $I$ exists then $I$ can still be expressed
as a polynomial in the $f^i, f^i,_j, \ldots$ and terms containing 
indefinite integrals with integrands explicit in $x^k.$ \\
\underline{Algorithm:} \\
Successive partial integration of the term with the highest
$x^k$-derivative of any $f^i.$ By that the 
differential order w.r.t. $x^k$ is reduced
successively. This procedure is always applicable because steps involve only
differentiations and the polynomial
integration $(\int h^n\frac{\partial h}{\partial x}dx =
h^{n+1}/(n+1))$ where $h$ is a partial derivative of some function
$f^i.$ For a more detailed description see \cite{WoInt}.\\
\underline{Stop:} \\
Iteration stops if no term with any $x^k$-derivative of any $f^i$ is left.
If in the remaining un-integrated terms any $f^i(x^k)$ itself occurs,
then $I$ is not expressible with $f^i$ and its derivatives only. In
case no $f^i(x^k)$ occurs then any remaining terms can contain $x^k$ only
explicitly. Whether they can be integrated depends on their formal
integrability. For their integration the {\sc Reduce} integrator is
applied. \\
\underline{Speed up:} \\
The partial integration as described above preserves derivatives with
respect to other variables. For example, the three terms $f,_x, f
f,_{xxx}, f,_{xxy}$ can not combine somehow to the same terms in the
integral because if one ignores $x$-derivatives then it is clear that
$f, f^2$ and $f,_y$ are like three completely different expressions
from the point of view of $x$-integrations.
This allows the following drastic speed up
for large expressions. It is possible to partition the complete sum of
terms into partial sum such that each of the partial sum has to be
integrable on its own. That is managed by generating a label for each
term and collecting terms with equal label into partial sums. The
label is produced by dropping all $x$-derivatives from all functions
to be computed and dropping all factors which are not powers of derivatives of
functions to be computed.

The partitioning into partial sums has two effects. Firstly, if the
integration of one partial sum fails then the remaining sums do not have
to be tried for integration. Secondly, doing partial integration for
each term means doing many subtractions. It is much faster to subtract
terms from small sums than from large sums.

\underline{Example :} \\
We apply the above algorithm to
\begin{equation}
D := 2f,_yg' + 2f,_{xy}g + gg'^3 + xg'^4 + 3xgg'^2g'' = 0
\label{D}
\end{equation}
with $f = f(x,y), \, g = g(x), \, '\equiv d/dx.$
Starting with terms containing $g$
and at first with the highest derivative $g,_{xx},$ the steps are
\[
\begin{array}{rcccl}
\int 3xgg,_x^2g,_{xx} dx 
& = & \int d(xgg,_x^3)
    & - & \int \left( \partial_x(xg) g,_x^3\right) dx \\ \\
& = & xgg,_x^3 & - & \int g,_x^3(g + xg,_x) dx,
\end{array} \]
\[ I := I + xgg,_x^3 \]
\[ D := D - g,_x^3(g + xg,_x) - 3xgg,_x^2g,_{xx} \]
The new terms $- g,_x^3(g + xg,_x)$ are of lower order than $g,_{xx}$ 
and so in the expression $D$ the maximal order of $x$-derivatives 
of $g$ is lowered. The conditions that $D$ is exact are the following.
\begin{itemize}
\item The leading derivative must occur linearly before each partial
integration step. 
\item After the partial integration of the terms with first order
$x$-derivatives of $f$ the remaining $D$ must not contain $f$ 
or other derivatives of $f$, because such terms cannot
be integrated w.r.t.\ $x$ without specifying $f$.
\end{itemize}
The result of $x$- and $y$-integration in the above example is
(remember $g=g(x)$)
\begin{equation}
0 = 2fg + xygg,_x^3 + c_1(x) + c_2(y) \; \; (=I). \nonumber
\end{equation}
{\sc Crack} can now eliminate $f$ and substitute
for it in all other equations. \\
\underline{Generalization:} \\
If after applying the above basic algorithm, terms are left which contain
functions of $x^k$ but each of these functions depends only on a subset of
all $x^i, 1\leq i\leq n,$ then a generalized version of the above algorithm
can still provide a formal expression for the integral $I$
(see \cite{WoInt}). The price consists of
additional differential conditions, but they are equations in less variables
than occur in the integrated equation. Integrating for example 
\begin{equation}
\tilde{D} = D + g^2(y^2 + x\sin y + x^2e^y)       \label{Dnew}
\end{equation}
by introducing as few 
new functions and additional conditions as possible gives as the integral 
$\tilde{I}$
\begin{eqnarray*}
\tilde{I} & = & 2fg + xygg,_{x}^{3} + c_1(x) + c_2(y) \\
  &   & + \frac{1}{3}y^3c_3'' - \cos y(xc_3'' - c_3)
+ e^y(x^2c_3'' - 2xc_3' + 2c_3)          
\end{eqnarray*}
with $c_3 = c_3(x), \, '\equiv d/dx$ and the single additional 
condition $g^2 = c_3'''.$
The integration of the new terms of (\ref{Dnew}) is
achieved by partial integration again, for example
\begin{eqnarray*}
\int g^2x^2 dx & = & x^2\int g^2 dx - \int (2x\!\int g^2 dx) dx \\
& = & x^2\int g^2 dx - 2x\int\!\!\int g^2 dx 
+ 2 \int\!\!\int\!\!\int g^2 dx \\
& = & x^2c_3'' - 2xc_3' + 2c_3.
\end{eqnarray*}
\underline{Characterization:} \\
This algorithm is a decision algorithm which does not involve any
heuristic. 
After integration the new equation is still a polynomial
in $f^i$ and in the new constant or function of integration.
Therefore the algorithms for bringing the system into standard form can still 
be applied to the PDE-system 
after the equation $D = 0$ is replaced by $I = 0.$

The complexity of algorithms for bringing a PDE-system into a standard
form depends nonlinearly on the order of these equations because of
the nonlinear increase of the number of different leading derivatives
and by that the number of equations generated intermediately by such
an algorithm. It therefore in general pays off to integrate equations 
during such a standard form algorithm.  

If an $f^i,$ which depends on all variables, can be eliminated after an 
integration, then depending on its length 
it is in general helpful to substitute $f^i$ in other equations and
to reduce the number of equations and functions by one. This is especially
profitable if the replaced expression is short and 
contains only functions of less variables than $f^i.$ \\
\underline{Test:} \\
The corresponding test input is
\begin{verbatim}
depend f,x,y;
depend g,x;
crack({2*df(f,y)*df(g,x)+2*df(f,x,y)*g+g*df(g,x)**3
       +x*df(g,x)**4+3*x*g*df(g,x)**2*df(g,x,2)
       +g**2*(y**2+x*sin y+x**2*e**y)},
      {},{f,g},{});
\end{verbatim}
The meaning of the {\sc Reduce} command {\tt depend} is to declare that $f$
depends in an unknown way on $x$ and $y$. For more details on the
algorithm see \cite{WoInt}.

\subsection{Direct separation of PDEs}
As a result of repeated integrations the functions in the 
remaining equations have less and less variables. It therefore may happen
that after a substitution an equation results where at least one variable
occurs only explicitly and not as an argument of an unknown function.
Consequently all coefficients of linearly independent expressions in this
variable can be set to zero individually. \\
{\em Example:}  \\
$f = f(x,y), \;\; g = g(x), \;\; x,y,z$ are independent variables. 
The equation is
\begin{equation} 
0 = f,_y + z(f^2+g,_x) + z^2(g,_x+yg^2) \label{sep}
\end{equation}
$x$-separation? $\rightarrow$ no  \\
$y$-separation? $\rightarrow$ no  \\
$z$-separation? $\rightarrow$ yes: $0 \,=\, f,_y \,=\, f^2+g,_x \,=\,
g,_x+yg^2$ \\
$y$-separation? $\rightarrow$ yes: $0 = g,_x = g^2\;\;$ 
(from the third equation from the $z$-separation)   

If $z^2$ had been replaced in (\ref{sep}) by a third
function $h(z)$ then direct separation would not have been possible.
The situation changes if $h$ is a parametric function which is
assumed to be independently given and which should not be
calculated, i.e.\ $f$ and $g$ should be calculated for any
arbitrary given $h(z)$. Then the same separation could have been
done with an extra treatment of the special case $h,_{zz} = 0,$
i.e.\ $h$ linear in $z$. This different treatment of unknown functions
makes it necessary to input explicitly the functions to be
calculated as the third argument to {\sc Crack}. The input
in this case would be
\begin{verbatim}
depend f,x,y;
depend g,x;
depend h,z;
crack({df(f,y)+z*f**2+(z+h)*df(g,x)+h*y*g**2},{},{f,g},{z});
\end{verbatim}
The fourth parameter for {\sc Crack} is necessary to make clear that
in addition to the variables of $f$ and $g$, $z$ is also an independent
variable.
 
If the flag {\tt independence\_} is not {\tt nil} then {\sc Crack} will
stop if linear independence of the explicit expressions of the
separation variable (in the example $1,z,z^2$) is not clear and ask 
interactively whether separation should be done or not.

\subsection{Indirect separation of PDEs}
For the above direct separation a precondition is that at least one
variable occurs only explicitly or as an argument of parametric
functions.  The situation where each variable is an argument of at least
one function but no function contains all independent variables of an
equation needs a more elaborate treatment.

The steps are these 
\begin{itemize}
 \item A variable $x_a$ is chosen which occurs in as few functions as possible.
 This variable will be separated directly later which
 requires that all unknown functions $f_i$ containing $x_a$ are to be
 eliminated. Therefore, as long as $F:=\{f_i\}$ is not empty do the following:
 \begin{itemize}
  \item Choose the function $f_i(y_p)$ in $F$ with the smallest number of
  variables $y_p$ and with $z_{ij}$ as those variables on which $f_i$ does 
  not depend.
  \item Identify all different products $P_{ik}$ of powers of 
  $f_i$-derivatives and of $f_i$ in the equation. 
  Determine the $z_{ij}$-dependent factors $C_{ik}$ of the coefficients 
  of $P_{ik}$ and store them in a list.
  \item For each $C_{il}$ ($i$ fixed, $l=1,\ldots)$ choose a $z_{ij}$ and :
  \begin{itemize}
   \item divide by $C_{il}$ the equation and all following elements 
         $C_{im}$ with $m>l$ of this list, such that these elements are
         still the actual coefficients in the equation after the division,
   \item differentiate the equation and the $C_{im}, m>l$ w.r.t.\ $z_{ij}$
  \end{itemize}
 \end{itemize}
 \item The resulting equation no longer contains any unknown function of $x_a$
 and can be separated w.r.t.\ $x_a$ directly in case $x_a$ still occurs
 explicitly. In both cases the equation(s) is (are) free of $x_a$ afterwards 
 and inverting the sequence of integration and multiplication 
 of all those equations (in case of direct separability) will also result
 in an equation(s) free of $x_a.$ More exactly, the steps are
 \begin{itemize}
  \item multiplication of the equation(s) and the $C_{im}$ with 
        $m<l$ by the elements
  of the $C_{ik}$-lists in exactly the inverse order,
  \item integration of these exact PDEs and the $C_{im}$ w.r.t.\ $z_{ij}$.
 \end{itemize}
 \item The equations originating that way are used to evaluate those
 functions which do not depend on $x_a$ and which survived in the above
 differentiations. Substituting these functions in the original equation,
 may enable direct separability w.r.t. variables on which the $f_i$
 do not depend on.
 \item The whole procedure is repeated for another variable $x_b$ if the
 original DE could not be separated directly and still has the property that 
 it contains only functions of a subset of all variables in the equation.
\end{itemize}
The additional bookkeeping of coefficients $C_{ik}$ and their updating by
division, differentiation, integration and multiplication is done to use
them as integrating factors for the backward integration.
The following example makes this clearer. The equation is
\begin{equation}
0 = f(x) g(y) - \frac{1}{2}xf'(x) - g'(y) - (1+x^2)y. \label{isep}
\end{equation}
The steps are (equal levels of indentation in the example correspond to
those in the algorithm given above)
\begin{itemize}
 \item $x_1:=x, \, F=\{f\}$
 \begin{itemize}
  \item Identify $f_1:=f, \; \; \; \; \; y_1:=x, \; \; \; \; \; z_{11}:=y$ 
  \item and $P_1=\{f',f\}, \; \; \; \; \; C_1=\{1,g\}$
  \begin{itemize}
   \item Divide $C_{12}$ and 
	 (\ref{isep}) by $C_{11}=1$ and differentiate w.r.t. $z_{11}=y:$
	 \begin{eqnarray}
	 0 & = & fg' - g'' - (1+x^2)   \label{isep2}  \\
	 C_{12} & = & g'    \nonumber
	 \end{eqnarray}
 \item Divide (\ref{isep2}) by $C_{12}=g'$ and differentiate w.r.t. $z_{11}=y:$
\[ 0 = - (g''/g')' - (1+x^2)(1/g')' \]

  \end{itemize}
 \end{itemize}
 \item Direct separation w.r.t.\ $x$ and integration:
 \[\begin{array}{rclclcl}
  x^2: 0 & = & (1/g')' & \Rightarrow & c_1g' =  1 & \Rightarrow &
	g = y/c_1 + c_2 \\
  x^0: 0 & = & (g''/g')' & \Rightarrow & c_3g' = g'' & \Rightarrow &
	c_3 = 0
 \end{array} \]
 \item Substitution of $g$ in the original DE
       \[0 = (y/c_1+c_2)f - \frac{1}{2}xf' - 1/c_1 - (1+x^2)y \]
       provides a form which allows {\sc Crack} standard methods to succeed
       by direct separation w.r.t.\ $y$
 \[\begin{array}{rclcl}
  y^1: 0 & = & f/c_1 - 1 - x^2               & \Rightarrow & f'  =  2c_1x \\
  y^0: 0 & = & c_2f - \frac{1}{2}xf' - 1/c_1 & \Rightarrow & 0   =  
       c_2c_1(1+x^2) - c_1x^2 - 1/c_1
 \end{array}\]
       and direct separation w.r.t.\ $x$:
 \begin{eqnarray*}
 x^0:  0 & = & c_2c_1 - c_1    \\
 x^2:  0 & = & c_2c_1 - 1/c_1   \\
    & \Rightarrow &  0 = c_1 - 1/c_1   \\
    & \Rightarrow & c_1 = \pm 1 \Rightarrow c_2 = 1.
 \end{eqnarray*}
\end{itemize}
We get the two solutions $f = 1 + x^2, g = 1 + y$ and 
$f = - 1 - x^2, g = 1 - y.$ The corresponding input to {\sc Crack} would be
\begin{verbatim}
depend f,x;
depend g,y;
crack({f*g-x*df(f,x)/2-df(g,y)-(1+x**2)*y},{},{f,g},{});
\end{verbatim}
 
\subsection{Solving standard ODEs}
For solving standard ODEs the package {\sc ODESolve} by Malcalm MacCallum and
Francis Wright  
\cite{Mal} is applied. This package is distributed with {\sc Reduce} 
and can be used independently of {\sc Crack}. The syntax of
{\sc ODESolve} is quite similar to that of {\sc Crack} \\
\verb+depend+ {\it function}, {\it variable}; \\
\verb+odesolve(+ODE, {\it function}, {\it variable});  \\
In the present form (1998) it solves standard first order ODEs
(Bernoulli and Euler type, with separable variables, $\ldots$) and linear
higher order ODEs with constant coefficients. 
An improved version is currently under preparation by Francis Wright.
The applicability of {\sc ODESolve} is 
increased by a {\sc Crack}-subroutine which recognizes such PDEs in which
there is only one unknown function of all variables and all occurring
derivatives of this function
are only derivatives w.r.t. one variable of only one partial derivative.
For example the PDE for $f(x,y)$
\[ 0 = f,_{xxy} + f,_{xxyy} \]
can be viewed as a first order ODE in $y$ for $f,_{xxy}.$

\section{Acknowledgement}
Francis Wright contributed a module that provides simplifications
of expressions involving symbolic derivatives and integrals. Also, {\sc Crack}
makes extensive use of the {\sc Reduce} program {\sc ODESolve} written
by Malcolm MacCallum and Francis Wright.

Arrigo Triulzi provided a module to support the use of different total
orderings of derivatives in doing pseudo differential Gr\"{o}bner
basis computations.

Work on this package has been supported by the Konrad Zuse
Institut/Berlin through a fellowship of T.W..  Winfried
Neun and Herbert Melenk are thanked for many discussions and 
constant support.

Anthony Hearn provided free copies of {\sc Reduce} to us as a
{\sc Reduce} developers group which also is thankfully acknowledged.

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\end{document}
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