Rational solutions of the mixed differential equation and its application to factorization of differential operators

1996 ◽  
Author(s):  
Mark van Hoeij
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Rational Solutions

A Study on Rational Solutions to a KP-like Equation

2015 ◽  
Vol 70 (4) ◽  
pp. 263-268 ◽  
Author(s):  
Yufeng Zhang ◽  
Wen-Xiu Ma
Keyword(s):  
Differential Equation ◽  
Prime Number ◽  
Nonlinear Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Differential Operators ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
Bilinear Equation

AbstractA KP-like nonlinear differential equation is introduced through a generalised bilinear equation which possesses the same bilinear form as the standard KP bilinear equation. By symbolic computation, nine classes of rational solutions to the resulting KP-like equation are generated from a search for polynomial solutions to the corresponding generalised bilinear equation. Three generalised bilinear differential operators adopted are associated with the prime number p=3.

scholarly journals Second-Order Differential Operators in the Limit Circle Case

2021 ◽  
Author(s):  
Dmitri R. Yafaev ◽  
◽  
◽  
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Measures ◽  
Limit Circle ◽  
Jacobi Operators ◽  
Limit Circle Case ◽  
Circle Case ◽  
Stieltjes Transforms

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

Spectral Theory for the Differential Equation Lu= λ Mu

1958 ◽  
Vol 10 ◽  
pp. 431-446 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Spectral Theory ◽  
Real Line ◽  
Differential Operators ◽  
Identity Operator ◽  
The Real ◽  
Ordinary Differential Operators ◽  
Extensive Bibliography

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L 2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

scholarly journals On first-order differential operators with Bohr-Neugebauer type property

1989 ◽  
Vol 12 (3) ◽  
pp. 473-476 ◽  
Author(s):  
Aribindi Satyanarayan Rao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Linear Operator ◽  
Real Line ◽  
Differential Operators ◽  
Bounded Solution ◽  
Almost Periodic ◽  
Bounded Linear ◽  
First Order ◽  
Type Property

We consider a differential equationddtu(t)-Bu(t)=f(t), where the functions u and f map the real line into a Banach space X and B: X→X is a bounded linear operator. Assuming that any Stepanov-bounded solution u is Stepanov almost-periodic when f is Bochner almost-periodic, we establish that any Stepanov-bounded solution u is Bochner almost-periodic when f is Stepanov almost-periodic. Some examples are given in which the operatorddt-B is shown to satisfy our assumption.

scholarly journals Dynamic Keynesian Model of Economic Growth with Memory and Lag

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 178 ◽  
Author(s):  
Vasily Tarasov ◽  
Valentina Tarasova
Keyword(s):  
Differential Equation ◽  
Economic Growth ◽  
Fractional Differential Equation ◽  
Differential Operators ◽  
Continuous Distribution ◽  
National Income ◽  
Fading Memory ◽  
Kummer Functions ◽  
Distributed Lag ◽  
Fractional Differential

A mathematical model of economic growth with fading memory and continuous distribution of delay time is suggested. This model can be considered as a generalization of the standard Keynesian macroeconomic model. To take into account the memory and gamma-distributed lag we use the Abel-type integral and integro-differential operators with the confluent hypergeometric Kummer function in the kernel. These operators allow us to propose an economic accelerator, in which the memory and lag are taken into account. The fractional differential equation, which describes the dynamics of national income in this generalized model, is suggested. The solution of this fractional differential equation is obtained in the form of series of the confluent hypergeometric Kummer functions. The asymptotic behavior of national income, which is described by this solution, is considered.

scholarly journals Multiparameter spectral theory of singular differential operators

1988 ◽  
Vol 31 (1) ◽  
pp. 49-66 ◽  
Author(s):  
B. P. Rynne
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Spectral Theory ◽  
Unbounded Domain ◽  
Differential Operators ◽  
Separation Of Variables ◽  
Singular Case ◽  
Ordinary Differential Operators ◽  
Adjoint Operators ◽  
Unbounded Intervals

In this paper we investigate certain aspects of the multiparameter spectral theory of systems of singular ordinary differential operators. Such systems arise in various contexts. For instance, separation of variables for a partial differential equation on an unbounded domain leads to a multiparameter system of ordinary differential equations, some of which are defined on unbounded intervals. The spectral theory of systems of regular differential operators has been studied in many recent papers, e.g. [1, 3, 6, 9, 19, 21], but the singular case has not received so much attention. Some references for the singular case are [7, 8, 10, 13, 14, 18, 20], in addition general multiparameter spectral theory for self adjoint operators is discussed in [3, 9, 19].

Notes on the Riccati Equation

2005 ◽  
Vol 70 (7) ◽  
pp. 941-950 ◽  
Author(s):  
Eugene S. Kryachko
Keyword(s):  
Differential Equation ◽  
Normal Form ◽  
Riccati Equation ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Riccati Transformation ◽  
Rational Solutions ◽  
Order Polynomial ◽  
Linear Differential ◽  
The Relationship

The relationship between the Riccati and Schrödinger equations is discussed. It is shown that the transformation converting the Riccati equation into its normal form is expressed in terms of the roots of its algebraic part treated as a second-order polynomial. Together with the well-known Riccati transformation, a new transformation which also links the Riccati equation to the second-order linear differential equation is introduced. The latter is actually the Riccati transformation applied to an "inverse" Riccati equation. Two specific forms of the Riccati equation admitting the explicit particular rational solutions are obtained.

On the existence of nodal domains for elliptic differential operators

1983 ◽  
Vol 94 (3-4) ◽  
pp. 287-299 ◽  
Author(s):  
E. Müller-Pfeiffer
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Quadratic Form ◽  
Unbounded Domain ◽  
Dirichlet Boundary ◽  
Differential Operators ◽  
Dirichlet Boundary Conditions ◽  
Friedrichs Extension ◽  
Nodal Domains ◽  
Elliptic Differential Operators

SynopsisWe prove that the selfadjoint elliptic differential equation (1) has rectangular nodal domains if the quadratic form of the equation takes on negative values. The existence of nodal domains is closely connected with the position of the smallest point of the spectrum of the corresponding selfadjoint operator (Friedrichs extension). If the smallest point of a second order selfadjoint differential operator with Dirichlet boundary conditions is an eigenvalue, then this eigenvalue is strictly increasing when the (possibly unbounded) domain, where the coefficients of the differential operator are denned, is shrinking (Theorem 4).

On the study of the spectral properties of differential operators with a smooth weight function

2020 ◽  
pp. 25-47
Author(s):  
Sergey I. Mitrokhin
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Differential Operator ◽  
Spectral Properties ◽  
Differential Operators ◽  
Asymptotic Estimates ◽  
Third Order ◽  
Indicator Diagram ◽  
Estimates Of Solutions ◽  
Smooth Coefficients

In this paper we study the spectral properties of a third-order differential operator with a summable potential with a smooth weight function. The boundary conditions are separated. The method of studying differential operators with summable potential is a development of the method of studying operators with piecewise smooth coefficients. Boundary value problems of this kind arise in the study of vibrations of rods, beams and bridges composed of materials of different densities. The differential equation defining the differential operator is reduced to the solution of the Volterra integral equation by means of the method of variation of constants. The solution of the integral equation is found by the method of successive Picard approximations. Using the study of an integral equation, we obtained asymptotic formulas and estimates for the solutions of a differential equation defining a differential operator. For large values of the spectral parameter, the asymptotics of solutions of the differential equation that defines the differential operator is derived. Asymptotic estimates of solutions of a differential equation are obtained in the same way as asymptotic estimates of solutions of a differential operator with smooth coefficients. The study of boundary conditions leads to the study of the roots of the function, presented in the form of a third-order determinant. To get the roots of this function, the indicator diagram wasstudied. The roots of this equation are in three sectors of an infinitely small size, given by the indicator diagram. The article studies the behavior of the roots of this equation in each of the sectors of the indicator diagram. The asymptotics of the eigenvalues of the differential operator under study is calculated. The formulas found for the asymptotics of eigenvalues allow us to study the spectral properties of the eigenfunctions of the differential operator under study.

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