Hermite Equation Research Articles

Recurrence Relations and General Solution of the Exceptional Hermite Equation

Recurrence Relations and General Solution of the Exceptional Hermite Equation

The initial heat distribution problem associated with the Ornstein–Uhlenbeck and Hermite equations

The initial heat distribution problem associated with the Ornstein–Uhlenbeck and Hermite equations

Constructing reliable approximations of the random fractional Hermite equation: solution, moments and density

We extend the study of the random Hermite second-order ordinary differential equation to the fractional setting. We first construct a random generalized power series that solves the equation in the mean square sense under mild hypotheses on the random inputs (coefficients and initial conditions). From this representation of the solution, which is a parametric stochastic process, reliable approximations of the mean and the variance are explicitly given. Then, we take advantage of the random variable transformation technique to go further and construct convergent approximations of the first probability density function of the solution. Finally, several numerically simulations are carried out to illustrate the broad applicability of our theoretical findings.

Generating Resonant and Repeated Root Solutions to Ordinary Differential Equations Using Perturbation Methods

In the study of ordinary differential equations (ODEs) of the form $\hat{L}[y(x)]=f(x)$, where $\hat{L}$ is a linear differential operator, two related phenomena can arise: resonance, where $f(x)\propto u(x)$ and $\hat{L}[u(x)]=0$, and repeated roots, where $f(x)=0$ and $\hat{L}=\hat{D}^n$ for $n\geq 2$. We illustrate a method to generate exact solutions to these problems by taking a known homogeneous solution $u(x)$, introducing a parameter $\epsilon$ such that $u(x)\rightarrow u(x;\epsilon)$, and Taylor expanding $u(x;\epsilon)$ about $\epsilon = 0$. The coefficients of this expansion $\frac{\partial^k u}{\partial\epsilon^k}\big{|}_{\epsilon=0}$ yield the desired resonant or repeated root solutions to the ODE. This approach, whenever it can be applied, is more insightful and less tedious than standard methods such as reduction of order or variation of parameters. We provide examples of many common ODEs, including constant coefficient, equidimensional, Airy, Bessel, Legendre, and Hermite equations. While the ideas can be introduced at the undergraduate level, we could not find any existing elementary or advanced text that illustrates these ideas with appropriate generality.

Some Spectral Properties of Multiplicative Hermite Equation

We reconstruct the Multiplicative Hermite Equation from multiplicative Sturm-Liouville equation. A new representation of eigenfunctions for the constructed problem are obtained by the power series solution technique. While making these solutions, multiplicative Hermite polynomials were used strongly. We get a generator for multiplicative Hermite polynomials and construct integration representations for these polynomials. Finally, some spectral properties of the multiplicative Hermite problem are examined in detail.

On Continued Fractions and Their Applications

Continued fractions constitute a very important subject in mathematics. Their importance lies in the fact that they have very interesting and beautiful applications in many fields in pure and applied sciences. This review article will reveal some of these applications and will reflect the beauty behind their uses in calculating roots of real numbers, getting solutions of algebraic Equations of the second degree, and their uses in solving special ordinary differential Equations such as Legendre, Hermite, and Laguerre Equations; moreover and most important, their use in physics in solving Schrodinger Equation for a certain potential. A comparison will also be given between the results obtained via continued fractions and those obtained through the use of well-known numerical methods. Advances in the subject will be discussed at the end of this review article.

Generalized Hermite polynomials for the Burgers hierarchy and point vortices

Rational solutions of equations for the Burgers hierarchy are considered. Using self-similar variables this hierarchy is reduced to the family of nonlinear ordinary differential equations. Then the family is transformed to the hierarchy of non-autonomous linear differential equations by means of the Cole-Hopf formula. This hierarchy is a generalization of the second-order equation for Hermite polynomials. It is shown that every member of the hierarchy for ordinary differential equation has the solution in the form of polynomials. Properties of solutions of generalized Hermite equations in the form the special polynomials are studied. A recursion relation that can be used for finding corresponding polynomials for every member is given. It is proved that the well-known property for Hermite polynomials connecting two polynomials can be used for all polynomials of the generalized Hermite hierarchy. It is shown that the Cole-Hopf transformation is a direct consequence of the differential connection between two special polynomials in the hierarchy of Hermite equations. A derivation of the generalized Tkachenko equations is given for polynomials of the generalized Hermite hierarchy whose roots correspond to point vortices in the background flow.

Generalized Null Lagrangians for Equations with Special Function Solutions

A method to derive general standard and null Lagrangians for second-order differential equations whose solutions are special function of mathematical physics is presented. The general null Lagrangians are used to find the corresponding general gauge functions. All derived Lagrangians are new and in special cases they reduce to those published in the literature. The obtained results are applied to the Bessel, Hermite and Legendre equations, which have many applications in physics, applied mathematics and engineering.

Some Identities Involving Two-Variable Partially Degenerate Hermite Polynomials Induced from Differential Equations and Structure of Their Roots

In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially degenerate Hermite polynomials. Finally, we study the symmetric properties of the structure of the roots of the two-variable partially degenerate Hermite equations.

Special Functions of Mathematical Physics: A Unified Lagrangian Formalism

Lagrangian formalism is established for differential equations with special functions of mathematical physics as solutions. Formalism is based on either standard or non-standard Lagrangians. This work shows that the procedure of deriving the standard Lagrangians leads to Lagrangians for which the Euler–Lagrange equation vanishes identically, and that only some of these Lagrangians become the null Lagrangians with the well-defined gauge functions. It is also demonstrated that the non-standard Lagrangians require that the Euler–Lagrange equations are amended by the auxiliary conditions, which is a new phenomenon in the calculus of variations. The existence of the auxiliary conditions has profound implications on the validity of the Helmholtz conditions. The obtained results are used to derive the Lagrangians for the Airy, Bessel, Legendre and Hermite equations. The presented examples clearly demonstrate that the developed Lagrangian formalism is applicable to all considered differential equations, including the Airy (and other similar) equations, and that the regular and modified Bessel equations are the only ones with the gauge functions. Possible implications of the existence of the gauge functions for these equations are discussed.

Differential Equations Associated with Two Variable Degenerate Hermite Polynomials

In this paper, we introduce the two variable degenerate Hermite polynomials and obtain some new symmetric identities for two variable degenerate Hermite polynomials. In order to give explicit identities for two variable degenerate Hermite polynomials, differential equations arising from the generating functions of degenerate Hermite polynomials are studied. Finally, we investigate the structure and symmetry of the zeros of the two variable degenerate Hermite equations.

Hidden SUSY features, and Dawson’s function and the wrong-sign Hermite functions

We propose that some problems in mathematical physics show non-typical supersymmetric (SUSY) features which may still need to be thoroughly explored. In particular, we first show that the Pöschl-Teller problem posses some inherent ladder peculiarities which are not considered in the usual SUSY approach. Then we show that Dawson’s function and its derivatives show a SUSY type relation with the wrong-sign Hermite equation, found in a general factorization of the simple harmonic oscillator, which includes Mielnik’s SUSY partners as particular cases.

From Conformal Group to Symmetries of Hypergeometric Type Equations

We show that properties of hypergeometric type equations become transparent if they are derived from appropriate 2nd order partial differential equations with constant coefficients. In particular, we deduce the symmetries of the hypergeometric and Gegenbauer equation from conformal symmetries of the 4- and 3-dimensional Laplace equation. We also derive the symmetries of the confluent and Hermite equation from the so-called Schr\"odinger symmetries of the heat equation in 2 and 1 dimension. Finally, we also describe how properties of the ${}_0F_1$ equation follow from the Helmholtz equation in 2 dimensions.

Hypergeometric Type Functions and Their Symmetries

We give a systematic and unified discussion of various classes of hypergeometric type equations: the hypergeometric equation, the confluent equation, the F_1 equation (equivalent to the Bessel equation), the Gegenbauer equation and the Hermite equation. In particular, we discuss recurrence relations of their solutions, their integral representations and discrete symmetries.

The second solution of the Hermite equation and the monomiality formalism

The Hermite polynomials can be defined through a second order differential equation with non constant coefficients, admitting two solutions one of which of non polynomial nature. The properties of this solution are studied here using an algebraic formalism, employing the techniques developed within the framework of the monomiality principle.

On spectral properties of finite population processor shared queues

We consider sojourn or response times in processor-shared queues that have a finite population of potential users. Computing the response time of a tagged customer involves solving a finite system of linear ODEs. Writing the system in matrix form, we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large. This corresponds to finite population models where the total population is \(N\gg 1\). Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Hermite equation. The dominant eigenvalue leads to the tail of a customer’s sojourn time distribution.

Exactly Solvable Schrödinger Operators

We systematically describe and classify one-dimensional Schrödinger equations that can be solved in terms of hypergeometric type functions. Beside the well-known families, we explicitly describe two new classes of exactly solvable Schrödinger equations that can be reduced to the Hermite equation.

Fractional operators and some special functions

This paper considers the Riemann–Liouville fractional operator as a tool to reduce linear ordinary equations with variable coefficients to simpler problems, avoiding the singularities of the original equation. The main result is that this technique allow us to obtain an extension of the classical integral representation of the special functions related with the original differential equations. In particular, we will use as examples the cases of the well-known Generalized, Gauss and Confluent Hypergeometric equations, Laguerre equation, Hermite equation, Legendre equation and Airy equation.

Quantum corrections to static solutions of the sine-Gordon and Nahm models via a generalized zeta function

We consider one-dimensional Yang-Mills-Nahm and sine-Gordon models in terms of a class of nonlinear Klein-Gordon-Fock equations. We perform a semiclassical quantization of the models using a generalized zeta function and construct a representation of the quantum theory in terms of the diagonal of the Green’s function for the heat equation with an elliptic potential via solutions of the Hermite equation. We formulate an alternative approach based on Baker-Akhiezer functions for the KP equation. We evaluate quantum corrections to the action of the Nahm and sine-Gordon models. We study the fields from the class of elliptic functions. We take extra variables of arbitrary dimensions into account for possible applications of quantized sine-Gordon solitons in solid state physics via the Frenkel-Kontorova model or other models. For the Nahm model, whose field is represented via an elliptic (lemniscate) integral by construction, the Yang-Mills field mass coincides with the correction evaluated as a hyperelliptic integral.

Approximation of analytic functions by Hermite functions

We solve the inhomogeneous Hermite equation and apply this result to estimate the error bound occurring when any analytic function is approximated by an appropriate Hermite function.