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Cell[TextData[{
"A new ",
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FontSlant->"Italic"],
"-based program for solving overdetermined systems of PDE"
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Cell["Stelios Dimas", "Author"],
Cell["\<\
Department of Mathematics, University of Patras, Rio, Greece
spawn@math.upatras.gr\
\>", "TextAboutAuthor"],
Cell["Dimitri Tsoubelis", "Author"],
Cell["\<\
Department of Mathematics, University of Patras, Rio, Greece
tsoubeli@math.upatras.gr\
\>", "TextAboutAuthor"],
Cell[TextData[{
"\tThe outline and the main features of a heuristic free algorithm \
implemented in ",
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" ",
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" for solving overdetermined systems of PDEs is decribed. This algorithm, \
which we call \"Seek&Solve\", is the core element of the command ",
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" of the Mathematica based symmetry analysis package ",
StyleBox["SYM",
FontSlant->"Italic"],
", developed by the authors recently ",
ButtonBox["[2]",
ButtonData:>"DiTsCyprus",
ButtonStyle->"Hyperlink"],
". Characteristic examples are presented, showing the effectiveness of the \
algorithm and, in particlar, its ability to deal with subsets of the initial \
system, corresponding to special parameter values."
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Cell[CellGroupData[{
Cell["Introduction", "SectionFirst"],
Cell[TextData[{
"Symmetry analysis is a well established tool in the field of differential \
equations of applied mathematics. Part of its success is due to the \
algorithmic way of determining the symmetries of an actual differential \
system, a task which is very tedious and error-prone when undertaken by hand. \
As a result, many symmetry-finding programs and specific packages based on \
various combuter algebra systems have already been developed. \n\tOne of the \
main components of every symmetry-finding package is the solver of \
overdetermined system of differential equations. An example of this kind of \
systems is given by the \"determining equations\" which arise when one \
imposes the linearized symmetry condition on the differential equation(s) \
under study. Even though the determining equations are linear, the total \
number of independent and dependent variables is usually so large that \
solving the corresponding equations is a very difficult task. As a result, \
the overdetermined system solver used in dealing with this task usually does \
not manage to obtain the most general solution.\n\tThere are two main \
categories of algorithms that can be employed in the construction of such an \
overdetermined system solver: the heurestic and the non-heurestic or \
heurestic free. By heurestic we mean an algorithm which tries to integrate \
the determining equations starting with certain assumptions regarding the \
structure of the saught solutions. For example, it is usually assumed that a \
particular subset of the solution functions are polynomials in the \
independent variables involved. By heurestic free, we mean an algorithm that \
attempts to integrate the determining system without making use of any ans\
\[ADoubleDot]tze. Usually, such an algorithm will first bring the system into \
a simpler form, called ",
StyleBox["canonical ",
FontSlant->"Italic"],
"or ",
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" form. Then, it will either seek the general solution of the system, \
including all those cases that correspond to special values of the parameters \
involved, [",
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solution space and the structure constants of the underlining algebra [",
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"]. The advantage of the heurestic free algorithms is the fact that they \
will always give the full set of solutions of the system under examination. \
However, the calculation of the canonical form can be a very tedious and \
time-consuming process depending exponentially on the complexity of the \
system. Moreover, because the methods used in obtaining the canonical form \
are based on the theory of differential algebra, e.g. differential Gr\
\[ODoubleDot]bner bases, they mostly work when the system consists of PDEs \
that are in polynomial form with respect to the uknown functions and their \
derivatives. The heurestic algorithms, on the other hand, are much faster \
but, in general, they do not give the complete solution space of the \
determining equations under investigation. \n\tIn this paper, we present the \
outline of the implementation we employed for the integration of \
overdetermined systems of equations. We call this ",
StyleBox["Mathematica-",
FontSlant->"Italic"],
"based algorithm \"Seek&Solve\", in order to stress the method of \
integration it employes and the Artificial Intelligence (AI) aspects of its \
structure. The capabilities of this implemention in SYM, under the command \
",
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known results. (The symmetry-finding package SYM can be obtained by \
contacting the authors)"
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Cell[TextData[{
"As its name denotes, the ``Seek&Solve\" algorithm is a heurestic free \
algorithm which employs a suite of search strategies to ``Seek\" the most \
appropriate equation to solve and then integrate \[LongDash]\"Solve\"\
\[LongDash] it using the built-in function ",
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" of ",
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FontSlant->"Italic"],
". Let us stress that, eventhough we use the latter function as the basis \
of our integrator, we have also further tweaked and augmented it so as to \
render it more effective in the problem at hand. The use of ",
StyleBox["Mathematica's",
FontSlant->"Italic"],
" internal solver offers the advandage of harnessing the capabilities of \
this ever expanding function. Furthermore, by adding to it extra integration \
rules we \"overload\" its purpose and overcome its current limitations. As a \
consequence, we do not have to make any a priori assumptions on the kind and \
form of the differential equations involved. \n\tIn addition to starting the \
integration process from the most appropriate equation, which it locates \
thanks to its pattern recognition capabilities, our algorithm simplifies the \
differential equations at hand and turns integro-differential equations to \
differential ones. Furthmore, it splits differential equations involving \
uknown functions with different arguments into simpler ones compatible with \
the originals. For example, the equation ",
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parameters, that might lead to alternative solutions, are found, the \
algorithm has the ability to auto-branch itself and examine each case \
separately in a sequence. Using parallel programming, the above \
auto-branching process can be readily implemented in a multiprocessor or grid \
enviroment. With each case assigned to a different unit, the full analysis of \
the original system of equations can be completed much faster than usual."
}], "Text"],
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Cell[TextData[{
"Below, we give the algorithm, divided in 6 steps, which we have \
incorporate it into our package. These 6 steps form a closed loop that \
terminates only if no equation that can be solved remains. The implementation \
in ",
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"."
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StyleBox["Step 1: ",
FontSlant->"Italic"],
"Search for integro-differential equations and turn them into differential. \
Moreover, split the equations to simpler ones if possible, as described \
above."
}], "Text",
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Cell[TextData[{
StyleBox["Step 2: ",
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"Simplify the system, with cross-differentiation when possible, and search \
for any special values of the parameters, if any, that satisfy any of the \
equations. For example, the equation ",
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Cell[TextData[{
StyleBox["Step 3: ",
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multiple solutions, choose the first one and put on queue the rest."
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Cell[TextData[{
StyleBox["Step 4: ",
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(the number of unkown functions involved and the length). "
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Cell[TextData[{
StyleBox["Step 5: ",
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Again, in case there are multiple solutions, pick the first one and put on \
queue the rest."
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Cell[TextData[{
StyleBox["Step 6: ",
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Cell[CellGroupData[{
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Cell["\<\
As mentioned previously, the goal of a heurestic free algorithm is \
to transform the system of determining equations into one that it is easier \
to classify and solve. The advantage of \"Seek&Solve\" is that it realizes \
the above two step process automatically, without any external intervention. \
In other words, it keeps simplifying the system of equations while integrates \
it. In particular, \"Seek&Solve\" detects the special values of the \
parameters that must be treated seperately at different stages of the \
solution process, a fact that makes this algorithm much more time-efficient \
than existing ones.\
\>", "Text"]
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In the examples that follow we give some characteristic \
applications of the implementation of the \"Seek&Solve\" algorithm in the SYM \
symmetry-finding package.\
\>", "Text"],
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" The determining equations that stem from the non classical analysis of \
the Huxley equation ",
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u\_\(\(,\)\(xx\)\) + 2 \(\( u\^2\)(1 - u)\)\)]],
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2 \[Xi]\_\(\(,\)\(xu\)\) + 2 \[Xi]\ \[Xi]\_\(\(,\)\(u\)\) =
0, \[IndentingNewLine]2 \[Eta]\_\(\(,\)\(xu\)\) - \[Xi]\_\(\(,\)\(xx\)\
\) - \((2 \[Eta] + 6 u\^3 - 6 u\^2)\) \[Xi]\_\(\(,\)\(u\)\) +
2 \[Xi]\ \[Xi]\_\(\(,\)\(x\)\) + \[Xi]\_t =
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" (Chazy equation). Its worth noting that in the above reference onother \
case was found, namely ",
Cell[BoxData[
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". As remarked by the authors, the latter case is a mere artifact of the \
calculations and is covered by the general case for ",
Cell[BoxData[
\(TraditionalForm\`\[Beta] \[NotEqual] 3\)]],
". The use of SYM leads to the following results automatically whithout \
ever encountering the artificial ",
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" case.\n1. ",
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Cell[CellGroupData[{
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Cell[TextData[{
"We thank European Social Fund (ESF), Operational Program for Educational \
and Vocational Training II (EPEAEK II) and particularly the program ",
StyleBox["Irakleitos",
FontSlant->"Italic"],
", for funding the above work. We would also like to thank the University \
of Patras and particularly the program ",
StyleBox["Karatheodory",
FontSlant->"Italic"],
" 2001 for funding the above work during its initial stage of \
development."
}], "Text"],
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[1] Wolfram Reasearch, Inc., Mathematica, Version 5.x, Wolfram \
Reasearch, Inc., Illinois, 2003.\
\>", "Reference",
CellTags->"Wolfram"],
Cell[TextData[{
" [2] Dimas S., Tsoubelis D., SYM: A new \
symmetry--finding package for Mathematica,in Proceedings of 10th \
International Conference in Modern Group ANalysis (Octomber,24--30,2004, \
Larnaca,Cypus), Editors N. H. Ibragimov, C. Sophocleous, P. A. Damianou, ",
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"[3] Reid G. J., A triangularization algorithm which determines the Lie \
symmetry algebra of any system of PDEs, ",
StyleBox["J. Phys. A: Math. Gen",
FontSlant->"Italic"],
", 1990, V. 23, L853\[Dash]L859. "
}], "Reference",
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"[4] Reid G. J., Algorithms for reducing a system of PDEs to standard form, \
determining the dimension of its solution space and calculating its Taylor \
series solution, ",
StyleBox["Eur. J. Appl. Math.",
FontSlant->"Italic"],
", 1991, V. 2, 293\[Dash]318. "
}], "Reference",
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Cell[TextData[{
"[5] Reid G. J., Finding symmetries of differential equations without \
integrating determining equations, ",
StyleBox["Eur. J. Appl. Math.",
FontSlant->"Italic"],
", 1991, V. 2, 319\[Dash]340. "
}], "Reference",
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Cell[TextData[{
"[6] E. L. Mansfield, P. A. Clarkson, Applications of the Differential \
Algebra Package ",
StyleBox["diffgrob2",
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" to Classical Symmetries of Differential Equations, ",
StyleBox["J. Symb. Comput.",
FontSlant->"Italic"],
", 1997, V. 23, 517\[Dash]533."
}], "Reference",
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"[7] E. L. Mansfield, Algorithms for Symmetric Differential Systems, ",
StyleBox["Found. Comput. Math.",
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", 2001, V. 1, 335\[Dash]383."
}], "Reference",
CellTags->"Mansfield"],
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[8] P. E. Hydon, Symmetry Methods for Differential Equations \
\[Dash] A Beginner's Guide, Cambridge University Press, 2000.
\
\>", "Reference",
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