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# Special Topics symbolic computation: Loewy Decomposition of Linear Differential Equations

Lecture No: 326.0SP, in blocked form in May 2015.

Loewy Decomposition of Linear Differential Equations
Fritz Schwarz, Fraunhofer Gesellschaft, Institut SCAI, Sankt Augustin
The central subject of this lecture is how to solve linear ordinary and
partial diﬀerential equations in closed form. The key concept applied for this
purpose is the factorization and the consequential decomposition of a given
diﬀerential equation as it has been initiated by Loewy.
In the ﬁrst part Loewy’s theory for linear ordinary diﬀerential equations
will be explained. The main result is the unique representation of any such
equation as product of completely reducible components of highest order, i.e.
its Loewy decomposition. For equations of order two or three this is utilized
for solving them in closed form whenever such a solution does exist. This
is illustrated by solving a large number of examples, supported by computer
algebra software.
In order to generalize Loewy’s result for partial diﬀerential equations it
must be expressed in the language of diﬀerential algebra. Therefore basic
concepts of this ﬁeld are introduced ﬁrst. They are applied for extending
Loewy’s theory to linear partial diﬀerential equations of order two or three
for an undetermined function depending on two variables. Based on these
results algorithms are developed for solving large classes of these equations for
which no other solution procedures seems to be available. They are applied
to numerous examples. The limitations that may be due to undecidability
of certain steps in this proceeding are also indicated.