Information Technology Reference
In-Depth Information
C [0
1] say 2 , and we want to find a solution
C [0
given a function
f
in
,
u
in
,
1]
such that
Tu
=
f,
(1)
B 0 u
=
u 0 ,...,B n− 1 u
=
u n− 1 .
2
2
e x D
Here
T
is a linear differential operator like
T
=
x
D
2
+1 and
B 0 ,...,B n− 1
u → u (0)
are boundary operators like
2
u
(1), whose number
n
should coincide
with the order of
T
. Furthermore, we require the boundary conditions to be such
that the solution
u
exists and is unique for every choice of
f
; in other words, we
consider only regular BVPs.
In my own understanding, BVPs are the prototype for a new kind of problem
in symbolic computation: Whereas “computer algebra” focuses on algorithmi-
cally solving for numbers (typical example: Grobner bases for triangularizing
polynomial systems) and “computer analysis” does the same for functions (typ-
ical example: differential algebra for solving differential equations), the proper
realm of “computer operator-theory” or symbolic functional analyis would be
solving for operators. See [17] for more details on this three-floor conception of
the algebraic part of symbolic computation.
Why are BVPs an instance of solving for operators ? The reason is that the
forcing function
f
in (1) is understood as a symbolic parameter: One wants to
have the solution
u
as a term that contains
f
as a free variable. In other words,
one needs an operator
G
that maps any given
f
to
u
. For making this explicit,
let us rewrite the traditional formulation (1) as
TG
=1
,
(2)
B 0 G
=0
,...,B n− 1 G
=0
.
Here 1 and 0 denote the identity and zero operator, respectively. Note also that I
have passed to homogeneous boundary conditions (it is always possible to reduce
a fully inhomogeneous problem to such a semi-inhomogeneous one).
The crucial idea of my solution method for (2) is to model the above op-
erators as noncommutative polynomials and to extract their algebraically rel-
evant properties into a collection of identities. For details, please refer to my
PhD thesis [14]; see also [16, 15]. The outcome ot this strategical plan is the
noncommutative polynomial ring ￿
#
{D, A, B, L, R}∪{f|f ∈ F
}
together
with a collection of 36 polynomial equalities. The indeterminates
D
,
A
,
B
,
L
,
R
and
stand for differentiation, integral, cointegral, left boundary value, right
boundary value and the multiplication operator induced by
f
may range
over any so-called analytic algebra (a natural extension of a differential algebra),
e.g. the exponential polynomials. The 36 polynomial identities express proper-
ties like the product rule of differentiation, the fundamental theorem of calculus,
and integration by parts.
f
;here
f
2 The smoothness conditions can be dispensed with by passing to distributions on
[0 , 1]. Of course one can also choose an arbitrary finite interval [ a, b ] instead of the
unit interval. See [15] for details.
Search WWH ::




Custom Search