10,804 research outputs found
An optimal algorithm for certain boundary value problem
AbstractThe O(h4) finite-difference scheme for the second derivative u″(x) leads to a coherent pentadiagonal matrix which is factorized into two tridiagonal matrices. This factorization is used to derive an optimal algorithm for solving a linear system of equations with the pentadiagonal matrix. As an application, a nonlinear system of ordinary differential equations is approximated by an O(h4) convergent finite-difference scheme. This scheme is solved by the implicit iterative method applying the algorithm at each iteration. A Mathematica module designed for the purpose of testing and using the method is attached
Constructive factorization of LPDO in two variables
We study conditions under which a partial differential operator of arbitrary
order in two variables or ordinary linear differential operator admits a
factorization with a first-order factor on the left. The factorization process
consists of solving, recursively, systems of linear equations, subject to
certain differential compatibility conditions. In the generic case of partial
differential operators one does not have to solve a differential equation. In
special degenerate cases, such as ordinary differential, the problem is finally
reduced to the solution of some Riccati equation(s). The conditions of
factorization are given explicitly for second- and, and an outline is given for
the higher-order case.Comment: 16 pages, to be published in Journal "Theor. Math. Phys." (2005
Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems
We consider solution operators of linear ordinary boundary problems with "too
many" boundary conditions, which are not always solvable. These generalized
Green's operators are a certain kind of generalized inverses of differential
operators. We answer the question when the product of two generalized Green's
operators is again a generalized Green's operator for the product of the
corresponding differential operators and which boundary problem it solves.
Moreover, we show that---provided a factorization of the underlying
differential operator---a generalized boundary problem can be factored into
lower order problems corresponding to a factorization of the respective Green's
operators. We illustrate our results by examples using the Maple package
IntDiffOp, where the presented algorithms are implemented.Comment: 19 page
Methods in Mathematica for Solving Ordinary Differential Equations
An overview of the solution methods for ordinary differential equations in
the Mathematica function DSolve is presented.Comment: 13 page
Invariant Form of BK-factorization and its Applications
Invariant form of BK-factorization is presented, it is used for factorization
of the LPDOs equivalent under gauge transformation and for construction of
approximate factorization simplifying numerical simulsations with corresponding
LPDEs of higher orderComment: 11 pages, 7 figure
Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra
We present algorithms to factorize weighted homogeneous elements in the first
polynomial Weyl algebra and -Weyl algebra, which are both viewed as a
-graded rings. We show, that factorization of homogeneous
polynomials can be almost completely reduced to commutative univariate
factorization over the same base field with some additional uncomplicated
combinatorial steps. This allows to deduce the complexity of our algorithms in
detail. Furthermore, we will show for homogeneous polynomials that
irreducibility in the polynomial first Weyl algebra also implies irreducibility
in the rational one, which is of interest for practical reasons. We report on
our implementation in the computer algebra system \textsc{Singular}. It
outperforms for homogeneous polynomials currently available implementations
dealing with factorization in the first Weyl algebra both in speed and elegancy
of the results.Comment: 26 pages, Singular implementation, 2 algorithms, 1 figure, 2 table
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