scholarly journals Splines and fractional differential operators

2020 ◽  
pp. 2040005
Author(s):  
Peter Massopust
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Operator Equations ◽  
B Splines ◽  
Linear Differential Operators ◽  
Fractional Differential ◽  
Linear Differential ◽  
Fractional Differential Operators ◽  
Integral Order

Several classes of classical cardinal B-splines can be obtained as solutions of operator equations of the form [Formula: see text] where [Formula: see text] is a linear differential operator of integral order. In this paper, we consider classes of generalized B-splines consisting of cardinal polynomial B-splines of complex and hypercomplex orders and cardinal exponential B-splines of complex order and derive the fractional linear differential operators that are naturally associated with them. For this purpose, we also present the spaces of distributions onto which these fractional differential operators act.

Explicit invariant solutions for invariant linear differential operators

1984 ◽  
Vol 98 (1-2) ◽  
pp. 149-166 ◽  
Author(s):  
Zofia Szmydt ◽  
Bogdan Ziemian
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Invariant Solution ◽  
Real Analytic Function ◽  
Linear Differential Operators ◽  
Invariant Distributions ◽  
Real Analytic Manifold ◽  
Real Analytic ◽  
Linear Differential

SynopsisLet F be a real analytic function on a real analytic manifold X. Let P be a linear differential operator on X such that , where Q is an ordinary differential operator with analytic coefficients whose singular points are all regular. For each (isolated) critical value z of F, we construct locally an F-invariant solution u of the equation Pu - v, v being an arbitrary F-invariant distribution supported in F−1(z). The solution u is constructed explicitly in the form of a series of F-invariant distributions.

scholarly journals Linear Differential Operators Invertible in the Integro-differential Sense

2019 ◽  
pp. 20-33
Author(s):  
V. N. Chetverikov
Keyword(s):  
Differential Operator ◽  
Integral Operator ◽  
Differential Operators ◽  
Spatial Variable ◽  
Diagonal Form ◽  
Operator Equations ◽  
Evolutionary Systems ◽  
Linear Differential Operators ◽  
The Matrix ◽  
Linear Differential

The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.

Boolean linear differential operators on elementary cellular automata

2014 ◽  
Vol 25 (06) ◽  
pp. 1450009
Author(s):  
Ángel Martín del Rey
Keyword(s):  
Differential Operator ◽  
Cellular Automata ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Linear Differential Operators ◽  
Elementary Cellular Automata ◽  
Linear Differential

In this paper, the notion of boolean linear differential operator (BLDO) on elementary cellular automata (ECA) is introduced and some of their more important properties are studied. Special attention is paid to those differential operators whose coefficients are the ECA with rule numbers 90 and 150.

scholarly journals Fiberwise linear differential operators

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Fabrizio Pugliese ◽  
Giovanni Sparano ◽  
Luca Vitagliano
Keyword(s):  
Vector Bundle ◽  
Differential Operator ◽  
Line Bundle ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Lie Algebroid ◽  
Lie Groupoid ◽  
Linear Differential Operators ◽  
Linear Differential ◽  
Definition Of

Abstract We define a new notion of fiberwise linear differential operator on the total space of a vector bundle E. Our main result is that fiberwise linear differential operators on E are equivalent to (polynomial) derivations of an appropriate line bundle over E ∗ {E^{\ast}} . We believe this might represent a first step towards a definition of multiplicative (resp. infinitesimally multiplicative) differential operators on a Lie groupoid (resp. a Lie algebroid). We also discuss the linearization of a differential operator around a submanifold.

scholarly journals Ger-type and Hyers-Ulam stabilities for the first-order linear differential operators of entire functions

2004 ◽  
Vol 2004 (22) ◽  
pp. 1151-1158 ◽  
Author(s):  
Takeshi Miura ◽  
Go Hirasawa ◽  
Sin-Ei Takahasi
Keyword(s):  
Entire Function ◽  
Differential Operator ◽  
Stability Problem ◽  
Differential Operators ◽  
Entire Functions ◽  
Ulam Stability ◽  
Nonzero Constant ◽  
First Order ◽  
Linear Differential Operators ◽  
Linear Differential

Lethbe an entire function andTha differential operator defined byThf=f′+hf. We show thatThhas the Hyers-Ulam stability if and only ifhis a nonzero constant. We also consider Ger-type stability problem for|1−f′/hf|≤ϵ.

scholarly journals Involutive Property of Resolutions of Differential Operators

1966 ◽  
Vol 27 (2) ◽  
pp. 419-427
Author(s):  
Masatake Kuranishi
Keyword(s):  
Vector Bundle ◽  
Differential Operator ◽  
Vector Space ◽  
Cross Sections ◽  
Vector Bundles ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
First Order ◽  
Linear Differential

Let E and E′ be C∞ vector bundles over a C∞ manifold M. Denote by Γ(E) (resp. by Γ(E′) the vector space of C∞ cross-sections of E (resp. of E′) over M. Take a linear differential operator of the first order D: Γ(E) → Γ(E′) induced by a vector bundle mapping σ(D): jl(E) ′ E′, where Jk(E) denotes the vector bundle of k-jets of cross-sections of E.

scholarly journals Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Oksana Bihun ◽  
Clark Mourning
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Hermite Polynomials ◽  
Pseudospectral Method ◽  
Matrix Representations ◽  
The Family ◽  
Linear Differential

Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization is based on a modification of pseudospectral matrix representations of linear differential operators proposed in the paper, which allows these representations to depend on two, rather than one, sets of interpolation nodes. The identities hold for every polynomial family pνxν=0∞ orthogonal with respect to a measure supported on the real line that satisfies some standard assumptions, as long as the polynomials in the family satisfy differential equations Apν(x)=qν(x)pν(x), where A is a linear differential operator and each qν(x) is a polynomial of degree at most n0∈N; n0 does not depend on ν. The proposed identities generalize known identities for classical and Krall orthogonal polynomials, to the case of the nonclassical orthogonal polynomials that belong to the class described above. The generalized pseudospectral representations of the differential operator A for the case of the Sonin-Markov orthogonal polynomials, also known as generalized Hermite polynomials, are presented. The general result is illustrated by new algebraic relations satisfied by the zeros of the Sonin-Markov polynomials.

scholarly journals The Use of Fractional B-Splines Wavelets in Multiterms Fractional Ordinary Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
X. Huang ◽  
X. Lu
Keyword(s):  
Differential Equations ◽  
Asymptotic Solution ◽  
Explicit Expression ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Differential Operators ◽  
Positive Real ◽  
B Splines ◽  
Fractional Differential ◽  
Linear Differential

We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit expression of the solution is given in this paper. At last, we show the asymptotic solution of the differential equations of fractional order and corresponding truncated error in theory.

scholarly journals A New Class of Analytic Normalized Functions Structured by a Fractional Differential Operator

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Najla M. Alarifi ◽  
Rabha W. Ibrahim
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Integral Operators ◽  
Real Variable ◽  
Open Unit ◽  
New Class ◽  
Science Technology ◽  
Fractional Differential Operator ◽  
Fractional Differential ◽  
Fractional Differential Operators

Newly, the field of fractional differential operators has engaged with many other fields in science, technology, and engineering studies. The class of fractional differential and integral operators is considered for a real variable. In this work, we have investigated the most applicable fractional differential operator called the Prabhakar fractional differential operator into a complex domain. We express the operator in observation of a class of normalized analytic functions. We deal with its geometric performance in the open unit disk.

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