Hermite Equation Research Articles

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation $$\frac{\partial u}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\frac{\partial^2}{\partial x^{2}_{i}}u,$$ but much less on those of the Hermite heat equation $$\frac{\partial U}{\partial t} = {\mathop\sum\limits^{n}_{i=1}}\left(\frac{\partial^2}{\partial x^{2}_{i}} - x^{2}_{i}\right) U$$ due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005).

The quantum harmonic oscillator on the sphere and the hyperbolic plane

A nonlinear model of the quantum harmonic oscillator on two-dimensional space of constant curvature is exactly solved. This model depends on a parameter λ that is related with the curvature of the space. First, the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a λ-dependent invariant measure d μ λ and expressing the Hamiltonian as a function of the Noether momenta. In the second part, the quantum superintegrability of the Hamiltonian and the multiple separability of the Schrödinger equation is studied. Two λ-dependent Sturm–Liouville problems, related with two different λ-deformations of the Hermite equation, are obtained. This leads to the study of two λ-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions Ψ m, n and the energies E m, n of the bound states are exactly obtained in both the sphere S 2 and the hyperbolic plane H 2.

The Hamilton-Jacobi method for arbitrary rheo-linear dynamical systems

The Hamilton-Jacobi method is briefly summarized and then applied to arbitrary rheo-linear systems with a single degree of freedom. Various means of finding a complete solution of the Hamilton-Jacobi equation and applying the Jacobi theorem to solve the canonical differential equations are discussed. Basic to the procedure is the separation of variables in the Hamilton-Jacobi equation which leads to a Riccati equation which must be solved for the particular rheonomic differential equation. Five different cases of complete solution are illustrated by a simple rheonomic example. As a direct application of the method, a number of canonical systems corresponding to many “named” equations are solved. They are the: Airy, Bessel, Chebyshev, Error function, Euler, Gegenbauer, Hermite, Hypergeometric, Kelvin, Kummer, Laguerre, Legendre, Jacobi, Mathieu, Spherical Bessel, Weber-Hermite and Whittaker equations. Finally, the conservation law is given for a forced and damped oscillator.

SOLVING SECOND ORDER DIFFERENTIAL EQUATIONS IN QUANTUM MECHANICS BY ORDER REDUCTION

Solving the famous Hermite, Legendre, Laguerre and Chebyshev equations requires different techniques of unique character for each equation. By reducing these differential equations of second order to a common solvable differential equation of first order, a simple common solution is provided to cover all the existing standard solutions of these named equations. It is easier than the method of generating functions and more powerful than the Frobenius method of power series.

Taylor polynomial solutions of linear differential equations

A matrix method, which is called Taylor-matrix method, for the approximate solution of linear differential equations with specified associated conditions in terms of Taylor polynomials about any point. The method is based on, first, taking truncated Taylor series of the functions in equation and then substituting their matrix forms into the given equation. Thereby the equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. To illustrate the method, it is applied to certain linear differential equations and the generalized Hermite, Laguerre, Legendre and Chebyshev equations given by Costa and Levine [Int. J. Math. Edu. Sci. Technol. 20 (1989) 1].

The Zeros of Special Functions from a Fixed Point Method

A scheme for the computation of the zeros of special functions and orthogonal polynomials is developed. We study the structure of the first order difference-differential equations (DDEs) satisfied by two fundamental sets of solutions of second order ODEs $y_n^{\prime\prime}(x)+A_{n}(x)y_n (x)=0$, n being the order of the solutions and An (x) a family of continuous functions. It is proved that, with a convenient normalization of the solutions, $T_{\pm1}(z)=z\pm \mbox{\rm sign}(d)\arctan(y_n (x(z))/y_{n\pm 1}(x(z)))$ are globally convergent iterations with fixed points $z(x_n^{(i)})$, $x_n^{(i)}$ being the zeros of yn(x); d is one of the coefficients in the DDEs and z(x) is a primitive of d. The structure of the DDEs is also used to set global bounds on the differences between adjacent zeros of functions of consecutive orders and to find iteration steps which guarantee that all the zeros inside a given interval can be found with certainty. As an illustration, we describe how to implement this scheme for the calculation of the zeros of arbitrary solutions of the Bessel, Coulomb, Legendre, Hermite, and Laguerre equations.

Resonant equations and special functions

Differential equations of the form Lf= g, where L is a differential operator, are termed Resonant if g satisfies Lg=0. In the case when L represents a linear harmonic oscillator, resonance occurs when the forcing term g has the same frequency as that of the unperturbed system. Resonance is associated with a transition from boundedness to unboundedness of the solution. We study the cases where L is the Legendre or Hermite operator. The first case arose in the context of supersymmetric Casimir operators for the di-spin algebra, and has solutions expressible in terms of singular functions, Legendre functions and polylogarithms. The non-singular polynomial parts of a certain class of solutions exhibit interesting properties. The non-resonant Hermite equation supports the theory of the quantum mechanical harmonic oscillator. A standard technique for its solution involves a Darboux/Infeld-Hull factorization of the Hamiltonian as a product of two first-order linear operators. The algebra of these operators can also be used to study the solutions of the resonant Hermite equation. A lowest order solution is found by elementary means, and then higher order solutions are generated by the repeated application of a ladder operator.

Solutions of a certain class of fractional differintegral equations

Recently, several authors demonstrated the usefulness of fractional calculus in obtaining particular solutions of a number of such familiar second-order differential equations as those associated with Gauss, Legendre, Jacobi, Chebyshev, Coulomb, Whittaker, Euler, Hermite, and Weber equations. The main object of this paper is to show how some of the latest contributions on the subject by Tu et al. [1], involving the associated Legendre, Euler, and Hermite equations, can be presented in a unified manner by suitably appealing to a general theorem on particular solutions of a certain class of fractional differintegral equations.

Spectral Hermite Approximations for the Actively Mode-Locked Laser

An approximation technique for the governing equations for the mode-locked laser is examined. The technique centers on a transformation of the governing equations in which the resulting equations closely resemble the Hermite equation. The approximation of the system is constructed through a linear combination of Hermite polynomials resulting in a Hermite-spectral method. The rate of decay of the resulting modes is examined for a simplified problem and difficulties in showing the stability of the method are also discussed. Numerical comparisons with a finite difference scheme are also presented.

The moments of the area under reflected Brownian bridgeconditional on its local time at zero

This paper develops a recursion formula for the conditional moments of the area under the absolute value of Brownian bridge given the local time at 0. The method of power series leads to a Hermite equation for the generating function of the coefficients which is solved in terms of the parabolic cylinder functions. By integrating out the local time variable, this leads to an integral expression for the joint moments of the areas under the positive and negative parts of the Brownian bridge.

The Hermite Equation and the Generalised Laplace Transform

AbstractThe second order Laplace equation (i.e. the linear o.d.e. with linear coefficients) may be reduced to one of a number of standard forms. One of these is the Hermite equation which has hitherto been thought not to be treatable by the Laplace transform. This is true because the defining integral for that transform diverges in this case. However this problem does not arise with a generalised Laplace transform. This paper gives the detailed working out of the theory from this point of view, and in the process derives results for the parabolic cylinder functions; these derivations are simpler than the standard ones or else produce these (known) results in a somewhat different form.

A method for the approximate solution of the second‐order linear differential equations in terms of Taylor polynomials

A matrix method is introduced for the approximate solution of the second‐order linear differential equation with specified associated conditions in terms of Taylor polynomials about any point. Examples are presented which illustrate the pertinent features of the method. Also it is applied to the generalized Hermite, Laguerre, Legendre and Chebyshev equations given by Costa and Levine.

Polynomial solutions of certain classes of ordinary differential equations

Several classes of ordinary differential equations which have polynomial solutions are studied. In particular, generalizations of the Hermite, Laguerre, Legendre, and Chebyshev equations are given for which such solutions exist. These solutions turn out to be generalizations of well‐known polynomials and enjoy similar properties.

Solutions of an infinite system of nonlinear differential equations related to the hermite equations of transfer of the Boltzmann equation

Abstract The equations of transfer of the Boltzmann equation for Maxwellian molecules can be solved recursively. For homogeneous one-dimensional systems these equations form a system o f ordinary nonlinear differential equations. For this system the recursion is carried out explicitely using a graphtheoretical method.

A Basic Analogue of Hermite's Equation

Using basic number and the analogues of differentiation and integration, a q-analogue of Hermite's equation is introduced. Series solutions are given, and it is shown that polynomial forms of these solutions are orthogonal with respect to basic integration. By reversing the series representation of these solutions, a basic analogue of the Hermite polynomial is obtained for which a generating function and a three-term recurrence relation are deduced. Finally, an orthogonality relation is given.

Computation of eigenvalues for second-order differential equations using imbedding techniques

The invariant imbedding method is used to provide algorithms for numerically computing eigenvalues for the Schrödinger equation and other eigenvalue problems. For the sake of illustration, the Hermite and Bessel equations are studied regarding their eigenvalues and, in the Hermite case, the asymptotic behavior of the eigenfunctions. The most salient feature is that the algorithms provided have good computational behavior in dealing with notoriously unstable problems.

Determination of Eigenvalues Using a Generalized Laplace Transform

The classical eigenvalue problems of mathematical physics are solved by means of a Laplace Transform extended, not over the interval (0, ∞) but over the interval of interest for the differential equation. The method is applied to the Hermite, Laguerre and Bessel equations and to the equation for the hypergeometric polynomials which include the Legendre, Tschebyscheff, and Jacobi polynomials as special cases.

A New Form of Solution of Hermite's Equation

A New Form of Solution of Hermite's Equation

Scientific Serials

American Journal of Mathematics, vol. xxi. No. 2, April.—On systems of multiform functions belonging to a group of linear substitutions with uniform coefficients, by E. J. Wilczynski. In this memoir, the author attempts to prove the existence of certain general functions, studied herein, he believes, for the first time. The existence of a large and important class of these functions is demonstrated by an indirect method, which con sists essentially in generalising the hypergeometric functions in a proper manner. The work is connected in a way with the researches of Fuchs, Schwarz and Neumann (on Riemann's theory of Abelian functions, and of Klein (Math. Ann., Bd. 41). Oskar Bolza states that his principal object, in his paper on the partial differential equations for the hyperelliptic θ and σ functions, is to replace part of Wiltheiss's work (Crelle, Bd. 99, and Math. Ann., Bd., 29, 31 and 33) by simpler and more direct proofs.—E. B. Van Vleck contributes an article on certain differential equations of the second order allied to Hermite's equation. The treatment is thorough, and the work is accompanied with numerous diagrams.

On Certain Differential Equations of the Second Order Allied to Hermite's Equation

On Certain Differential Equations of the Second Order Allied to Hermite's Equation