Linear 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.

On the value distribution of linear differential polynomials and the solution growth of linear differential equations

2011 International Conference on Multimedia Technology ◽

10.1109/icmt.2011.6002551 ◽

2011 ◽

Author(s):

Li Yang

Keyword(s):

Differential Equations ◽

Value Distribution ◽

Solution Growth ◽

Linear Differential Polynomials ◽

Linear Differential

Commentary on: [L 49] Simplified Proof of the Expansions Theorem for Singular Second Order Linear Differential Equations [L 51] (with E. A. Coddington), On the Nature of the Spectrum of Singular Second Order Linear Differential Equations [L 59] The Expansion Theorem for Singular Self-Adjoint Linear Differential Operators

Selected Papers of Norman Levinson ◽

10.1007/978-1-4612-5332-7_21 ◽

1998 ◽

pp. 203-205

Author(s):

B. M. Levitan ◽

D. H. Sattinger

Keyword(s):

Differential Equations ◽

Differential Operators ◽

Second Order ◽

Expansion Theorem ◽

Order Linear ◽

Linear Differential Operators ◽

Linear Differential ◽

L 51

Bernstein Widths of Some Classes of Functions Defined by a Self-Adjoint Operator

Journal of Applied Mathematics ◽

10.1155/2012/495054 ◽

2012 ◽

Vol 2012 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Guo Feng

Keyword(s):

Differential Equations ◽

Iterative Method ◽

Differential Operators ◽

Sobolev Class ◽

Adjoint Operator ◽

Periodic Functions ◽

Linear Differential Operators ◽

Linear Differential ◽

Special Case

We consider the classes of periodic functions with formal self-adjoint linear differential operatorsWp(ℒr), which include the classical Sobolev class as its special case. Using the iterative method of Buslaev, with the help of the spectrum of linear differential equations, we determine the exact values of Bernstein width of the classesWp(ℒr)in the spaceLqfor1<p≤q<∞.

Pairs of non-homogeneous linear differential polynomials

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500004728 ◽

2006 ◽

Vol 136 (4) ◽

pp. 785-794 ◽

Cited By ~ 1

Author(s):

J. K. Langley

Keyword(s):

Differential Equations ◽

Rational Function ◽

Rational Functions ◽

Linear Differential Polynomials ◽

Linear Differential ◽

Image Position ◽

Homogeneous Linear

Let f be transcendental and meromorphic in the plane and let the non-homogeneous linear differential polynomials F and G be defined by where k,n ∈ N and a, b and the aj, bj are rational functions. Under the assumption that F and G have few zeros, it is shown that either F and G reduce to homogeneous linear differential polynomials in f + c, where c is a rational function that may be computed explicitly, or f has a representation as a rational function in solutions of certain associated linear differential equations, which again may be determined explicitly from the aj, bj and a and b.

Legendre's differential operators and application to linear differential equations

International Journal of Mathematical Education in Science and Technology ◽

10.1080/0020739820130206 ◽

1982 ◽

Vol 13 (2) ◽

pp. 155-157 ◽

Cited By ~ 3

Author(s):

Isaac I. H. Chen ◽

T. W. Barrett

Keyword(s):

Differential Equations ◽

Differential Operators ◽

Linear Differential

Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

Abstract and Applied Analysis ◽

10.1155/2012/451825 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11

Author(s):

Zhigang Huang

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Linear Differential Equation ◽

Entire Functions ◽

Small Functions ◽

Linear Differential ◽

The Relationship ◽

Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.