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.

Method of Forced Monotonicity for Conjugate type Boundary Value Problems for Ordinary Differential Equations

1988 ◽  
Vol 31 (1) ◽  
pp. 79-84
Author(s):  
P. W. Eloe ◽  
P. L. Saintignon
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Linear Differential Operator ◽  
Boundary Value ◽  
Linear Differential ◽  
Type Boundary ◽  
Sign Conditions ◽  
Conjugate Type

AbstractLet I = [a, b] ⊆ R and let L be an nth order linear differential operator defined on Cn(I). Let 2 ≦ k ≦ n and let a ≦ x1 < x2 < … < xn = b. A method of forced mono tonicity is used to construct monotone sequences that converge to solutions of the conjugate type boundary value problem (BVP) Ly = f(x, y),y(i-1) = rij where 1 ≦i ≦ mj, 1 ≦ j ≦ k, mj = n, and f : I X R → R is continuous. A comparison theorem is employed and the method requires that the Green's function of an associated BVP satisfies certain sign conditions.

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 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.

Mutually conjugate solutions of formally self-adjoint differential equations

1979 ◽  
Vol 83 (1-2) ◽  
pp. 93-101 ◽  
Author(s):  
J. M. Hill ◽  
R. V. Nillsen
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Linear Differential Operator ◽  
Adjoint Operator ◽  
Linear Differential ◽  
Factorization Result

SynopsisLet L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n + 1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (−1)nQ*Q or L = (−1)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,−, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which yi and yn+j are mutually conjugate if i ≠ j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pólya.

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 New Approach for Solving Partial Differential Equations Based on Collocation Method

2020 ◽  
Vol 18 ◽  
pp. 118-128
Author(s):  
Alaa Almosawi ◽  
Luma N. M. Tawfiq
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Operator ◽  
Differential Equations ◽  
Collocation Method ◽  
Linear Differential Operator ◽  
Partial Differential ◽  
Restrictive Assumption ◽  
New Approach ◽  
Linear Differential

In this paper, a new approach for solving partial differential equations was introduced. The collocation method based on LA-transform and proposed the solution as a power series that conforming Taylor series. The method attacks the problem in a direct way and in a straightforward fashion without using linearization, or any other restrictive assumption that may change the behavior of the equation under discussion. Five illustrated examples are introduced to clarifying the accuracy, ease implementation and efficiency of suggested method. The LA-transform was used to eliminate the linear differential operator in the differential equation.

Linear Differential Polynomials

2017 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Lou van den Dries ◽  
Joris van der Hoeven
Keyword(s):  
Differential Equations ◽  
Differential Operators ◽  
Linear Differential Operator ◽  
Linear Operators ◽  
Linear Differential Equations ◽  
Differential Polynomials ◽  
Linear Differential Operators ◽  
Differential Field ◽  
Linear Differential Polynomials ◽  
Linear Differential

This chapter introduces the reader to linear differential polynomials. It first considers homogeneous differential polynomials and the corresponding linear operators before proving various basic results on them. In particular, it describes the property of a linear differential operator over a differential field K of defining a surjective map K → K, along with the transformation of a system of linear differential equations in several unknowns to an equivalent system of several linear differential equations in a single unknown. The chapter also discusses second-order linear differential operators, diagonalization of matrices, differential modules, linear differential operators in the presence of a valuation, and compositional conjugation. It concludes with an analysis of the Riccati transform and Johnson's Theorem.

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 Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to $$2\times 2$$ Hypergeometric Operators

2021 ◽  
Author(s):  
C. Calderón ◽  
M. M. Castro
Keyword(s):  
Differential Operator ◽  
Orthogonal Polynomials ◽  
Differential Operators ◽  
Jacobi Polynomials ◽  
Rodrigues Formula ◽  
The Family ◽  
Structural Formulas ◽  
Hypergeometric Type ◽  
Derivatives Of ◽  
Three Term Recurrence Relation

AbstractWe give some structural formulas for the family of matrix-valued orthogonal polynomials of size $$2\times 2$$ 2 × 2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.

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