46,873 research outputs found
Factorization method for second order functional equations
We apply general difference calculus in order to obtain solutions to the
functional equations of the second order. We show that factorization method can
be successfully applied to the functional case. This method is equivariant
under the change of variables. Some examples of applications are presented.Comment: 22 pages, examples and new section added, several correction
Solution of heat equation by a novel implicit scheme using block hybrid preconditioning of the conjugate gradient method
The main goal of the study is the approximation of the solution to the Dirichlet boundary value problem (DBVP) of the heat equation on a rectangle by developing a new difference method on a grid system of hexagons. It is proved that the given special scheme is unconditionally stable and converges to the exact solution on the grids with fourth order accuracy in space variables and second order accuracy in time variable. Secondly, an incomplete block factorization is given for symmetric positive definite block tridiagonal (SPD-BT) matrices utilizing a conservative iterative method that approximates the inverse of the pivoting diagonal blocks by preserving the symmetric positive definite property. Subsequently, by using this factorization block hybrid preconditioning of the conjugate gradient (BHP-CG) method is applied to solve the obtained algebraic system of equations at each time level
Factorization method and general second order linear difference equation
This paper addresses an investigation on a factorization method for
difference equations. It is proved that some classes of second order linear
difference operators, acting in Hilbert spaces, can be factorized using a pair
of mutually adjoint first order difference operators. These classes encompass
equations of hypergeometic type describing classical orthogonal polynomials of
a discrete variable
Wiener-Hopf solution for impenetrable wedges at skew incidence
A new Wiener-Hopf approach for the solution of impenetrable wedges at skew incidence is presented. Mathematical aspects are described in a unified and consistent theory for angular region problems. Solutions are obtained using analytical and numerical-analytical approaches. Several numerical tests from the scientific literature validate the new technique, and new solutions for anisotropic surface impedance wedges are solved at skew incidence. The solutions are presented considering the geometrical and uniform theory of diffraction coefficients, total fields, and possible surface wave contribution
Raising and lowering operators, factorization and differential/difference operators of hypergeometric type
Starting from Rodrigues formula we present a general construction of raising
and lowering operators for orthogonal polynomials of continuous and discrete
variable on uniform lattice. In order to have these operators mutually adjoint
we introduce orthonormal functions with respect to the scalar product of unit
weight. Using the Infeld-Hull factorization method, we generate from the
raising and lowering operators the second order self-adjoint
differential/difference operator of hypergeometric type.Comment: LaTeX, 24 pages, iopart style (late submission
Semiconjugate Factorizations of Higher Order Linear Difference Equations in Rings
We study linear difference equations with variable coefficients in a ring
using a new nonlinear method. In a ring with identity, if the homogeneous part
of the linear equation has a solution in the unit group of the ring (i.e., a
unitary solution) then we show that the equation decomposes into two linear
equations of lower orders. This decomposition, known as a semiconjugate
factorization in the nonlinear theory, generalizes the classical operator
factorization in the linear context. Sequences of ratios of consecutive terms
of a unitary solution are used to obtain the semiconjugate factorization. Such
sequences, known as eigensequences are well-suited to variable coefficients;
for instance, they provide a natural context for the expression of the
classical Poincar\'{e}-Perron Theorem. We discuss some applications to linear
difference equations with periodic coefficients and also derive formulas for
the general solutions of linear functional recurrences satisfied by the
classical special functions such as the modified Bessel and Chebyshev.Comment: Application of nonlinear semiconjugate factorization theory to linear
difference equations with variable coefficients in rings; 29 pages,
containing the main theory and more than 8 examples worked out in detai
Adaptive high-order splitting schemes for large-scale differential Riccati equations
We consider high-order splitting schemes for large-scale differential Riccati
equations. Such equations arise in many different areas and are especially
important within the field of optimal control. In the large-scale case, it is
critical to employ structural properties of the matrix-valued solution, or the
computational cost and storage requirements become infeasible. Our main
contribution is therefore to formulate these high-order splitting schemes in a
efficient way by utilizing a low-rank factorization. Previous results indicated
that this was impossible for methods of order higher than 2, but our new
approach overcomes these difficulties. In addition, we demonstrate that the
proposed methods contain natural embedded error estimates. These may be used
e.g. for time step adaptivity, and our numerical experiments in this direction
show promising results.Comment: 23 pages, 7 figure
- âŠ