# 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.