Jacobi Differential Equation Research Articles

Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation

A system of third-order recurrence relations for the coefficients of polynomial eigenfunctions (PEFs) of a differential equation is solved. A recurrence relation for three consecutive PEFs and a formula for differentiating PEFs are obtained. We consider differential equations one of which generalizes the Hermite and Laguerre differential equations and the other is a generalization of the Jacobi differential equation. For these equations, we construct functions bringing them to self-adjoint form and find conditions under which these functions become weight functions. Examples are given where the PEFs for nonweight functions do not have real zeros.

Kramer-type sampling theorems associated with higher-order differential equations

For many decades, Kramer’s sampling theorem has been attracting enormous interest in view of its important applications in various branches. In this paper we present a new approach to a Kramer-type theory based on spectral differential equations of higher order on an interval of the real line. Its novelty relies partly on the fact that the corresponding eigenfunctions are orthogonal with respect to a scalar product involving a classical measure together with a point mass at a finite endpoint of the domain. In particular, a new sampling theorem is established, which is associated with a self-adjoint Bessel-type boundary value problem of fourth-order on the interval [0, 1]. Moreover, we consider the Laguerre and Jacobi differential equations and their higher-order generalizations and establish the Green-type formulas of the differential operators as an essential key towards a corresponding sampling theory.

On controlled Hamilton and Hamilton–Jacobi differential equations of higher-order

In this paper, we investigate the nonlinear dynamics associated with controlled Lagrangians involving higher-order derivatives. More precisely, we establish the controlled higher-order Hamilton ordinary differential equations (ODEs) and Hamilton–Jacobi partial differential equation (PDE) for the considered class of Lagrangians governed by higher-order derivatives of the state variables. Moreover, we formulate and prove an invariance result with respect to the state variable. In addition, in order to validate the theoretical results and to highlight their effectiveness, some illustrative applications are presented.

Singular boundary conditions for Sturm–Liouville operators via perturbation theory

AbstractWe show that all self-adjoint extensions of semibounded Sturm–Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in \{1,2\}$ . This characterization generalizes the well-known analog for semibounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as $$ \begin{align*} \boldsymbol{A}_{ {}_{\scriptstyle \Theta}}=\boldsymbol{A}_0+\mathbf{B}\Theta\mathbf{B}^*, \end{align*} $$ where $\boldsymbol {A}_0$ is a distinguished self-adjoint extension and $\Theta $ is a self-adjoint linear relation in $\mathbb {C}^d$ . The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form-bounded with respect to $\boldsymbol {A}_0$ , i.e., it belongs to $\mathcal {H}_{-1}(\boldsymbol {A}_0)$ , with possible “infinite coupling.” A boundary triple and compatible boundary pair for the symmetric operator are constructed to ensure that the perturbation is well defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations $\Theta $ .The merging of boundary triples with perturbation theory provides a more holistic view of the operator’s matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

Tukey Depths and Hamilton--Jacobi Differential Equations

The widespread application of modern machine learning has increased the need for robust statistical algorithms. This work studies one such fundamental statistical measure known as the Tukey depth. We study the problem in the continuum (population) limit. In particular, we derive the associated necessary conditions, which take the form of a first-order partial differential equation. We discuss the classical interpretation of this necessary condition as the viscosity solution of a Hamilton-Jacobi equation, but with a non-classical Hamiltonian with discontinuous dependence on the gradient at zero. We prove that this equation possesses a unique viscosity solution and that this solution always bounds the Tukey depth from below. In certain cases, we prove that the Tukey depth is equal to the viscosity solution, and we give some illustrations of standard numerical methods from the optimal control community which deal directly with the partial differential equation. We conclude by outlining several promising research directions both in terms of new numerical algorithms and theoretical challenges.

Asymptotic behavior of orthogonal polynomials. Singular critical case

Our goal is to find an asymptotic behavior as n→∞ of the orthogonal polynomials Pn(z) defined by Jacobi recurrence coefficients an (off-diagonal terms) and bn (diagonal terms). We consider the case an→∞, bn→∞ in such a way that ∑an−1<∞ (that is, the Carleman condition is violated) and γn:=2−1bn(anan−1)−1∕2→γ as n→∞. In the case |γ|≠1 asymptotic formulas for Pn(z) are known; they depend crucially on the sign of |γ|−1. We study the critical case |γ|=1. The formulas obtained are qualitatively different in the cases |γn|→1−0 and |γn|→1+0. Another goal of the paper is to advocate an approach to a study of asymptotic behavior of Pn(z) based on a close analogy of the Jacobi difference equations and differential equations of Schrödinger type.

On Jacobi polynomials $${\mathscr {P}}_{k}^{(\alpha ,\beta )}$$ and coefficients $$c_{j}^{\ell }(\alpha ,\beta )$$ $$\left( k\ge 0, \ell =5,6; 1\le j\le \ell ; \alpha ,\beta &gt; -1\right)$$

In their paper, Awonusika and Taheri proved a spectral identity that relates the even ( $$2\ell$$ )th, $$\ell \ge 1$$ , derivatives of Jacobi polynomials $$P_{k}^{(\alpha ,\beta )}(\cos \theta )$$ , $$k\ge 0; \alpha ,\beta >-1$$ (evaluated at $$\theta =0$$ ), to the $$\ell$$ th-degree polynomials $${\mathscr {R}}_{\ell }^{(\alpha ,\beta )}\left( \lambda _{k}\right) :=R_{\ell}( \lambda _{k}^{\alpha ,\beta })$$ with constant coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ ( $$1\le j\le \ell$$ ; $$\alpha ,\beta >-1$$ ), called Jacobi coefficients; the numbers $$\lambda _{k}^{\alpha ,\beta }=k(k+\alpha +\beta +1)$$ ( $$k\ge 0$$ ) are the eigenvalues of the associated Jacobi operator. These Jacobi coefficients appear in the Maclaurin spectral expansion of the Schwartz kernels of functions of the Laplacian on rank one compact symmetric spaces. The first Jacobi coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ ( $$1\le j\le \ell \le 4$$ ) were computed using the qth, $$1\le q\le \ell ,$$ derivative formula for Jacobi polynomials $$P_{k}^{(\alpha ,\beta )}(t)$$ (evaluated at $$t=1$$ ) and the basic properties of the Gamma function. In this paper, we use the Jacobi differential equation to generate a recursion formula that explicitly computes the polynomials $${\mathscr {R}}_{\ell }^{(\alpha ,\beta )}\left( \lambda _{k} \right)$$ and the coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ . To illustrate the recursion formula we compute the higher coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ for $$\ell =5,6;1\le j\le \ell$$ . Remarkably, the local Jacobi coefficients $${\mathsf {c}}^{q}_{j}(\alpha ,\beta )$$ $$(1\le j\le q\le \ell )$$ (which appear in the computation of $$c^{\ell }_{j}(\alpha ,\beta )$$ ) coincide with the Jacobi–Stirling numbers of the first kind $${{\mathsf {j}}}{{\mathsf {s}}}_{j}^{q}(\alpha ,\beta )$$ , and this is a new phenomenon in the Maclaurin spectral analysis of the Laplacian on rank one compact symmetric spaces. The Jacobi coefficients $$c^{\ell }_{j}(\alpha ,\beta )$$ are useful in the description of constants appearing in the power series expansion of any spectral functions involving Jacobi polynomials.

Fast algorithms for the multi-dimensional Jacobi polynomial transform

We use the well-known observation that the solutions of Jacobi's differential equation can be represented via the non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial transforms. More explicitly, it follows from this observation that the matrix corresponding to the discrete Jacobi transform is the Hadamard product of a numerically low-rank matrix and a multi-dimensional discrete Fourier transform (DFT) matrix. The application of the Hadamard product can be carried out via r d fast Fourier transforms (FFTs), where r = O ( log ⁡ n log ⁡ log ⁡ n ) and d is the dimension, resulting in a nearly optimal algorithm to compute the multidimensional Jacobi polynomial transform.

Differential Equation-Based Specification of Turbulence Integral Length Scales for Cavity Flows

A new modeling approach has been developed that explicitly accounts for expected turbulent eddy length scales in cavity zones. It uses a hybrid approach with Poisson and Hamilton–Jacobi differential equations. These are used to set turbulent length scales to sensible expected values. For complex rim-seal and shroud cavity designs, the method sets an expected length scale based on local cavity width which accurately accounts for the large-scale wakelike flow structures that have been observed in these zones. The method is used to generate length scale fields for three complex rim-seal geometries. Good convergence properties are found, and a smooth transition of length scale between zones is observed. The approach is integrated with the popular Menter shear stress transport (SST) Reynolds-averaged Navier–Stokes (RANS) turbulence model and reduces to the standard Menter model in the mainstream flow. For validation of the model, a transonic deep cavity simulation is performed. Overall, the Poisson–Hamilton–Jacobi model shows significant quantitative and qualitative improvement over the standard Menter and k–ε two-equation turbulence models. In some instances, it is comparable or more accurate than high-fidelity large eddy simulation (LES). In its current development, the approach has been extended through the use of an initial stage of length scale estimation using a Poisson equation. This essentially reduces the need for user objectivity. A key aspect of the approach is that the length scale is automatically set by the model. Notably, the current method is readily implementable in an unstructured, parallel processing computational framework.

Determined position dependent Mass of the Rosen-Morse potential and its Bound state

In this paper, Schrodinger equation has been solved analytically with position dependent mass by the Rosen-Morse potential. The position dependent mass defined as the function 1/(1 — tanh(ηx)). Then corresponding position dependent mass substituted into Schrodinger equation. After that the obtained equation compared with associated Jacobi differential equation. Therefore, the eigenvalue and eigenfunctionhave been calculated, and from there the bound state has been found in terms of quantum numbers and the coefficients of potential. Also, the wave function has been obtained in terms of the associated Jacobi equation by μ = —3 and v = 2.

A Note on Meromorphic Solutions of Linear Partial Differential Equations of Second Order

We will prove a uniqueness theorem for meromorphic solutions of linear partial differential equations of second order with polynomial coefficients associated with the Jacobi differential equations and their generalizations.

Differential Equation Specification of Integral Turbulence Length Scales

A Hamilton–Jacobi differential equation is used to naturally and smoothly (via Dirichlet boundary conditions) set turbulence length scales in separated flow regions based on traditional expected length scales. Such zones occur for example in rim-seals. The approach is investigated using two test cases, flow over a cylinder at a Reynolds number of 140,000 and flow over a rectangular cavity at a Reynolds number of 50,000. The Nee–Kovasznay turbulence model is investigated using this approach. Predicted drag coefficients for the cylinder test-case show significant (15%) improvement over standard steady RANS and are comparable with URANS results. The mean flow-field also shows a significant improvement over URANS. The error in re-attachment length is improved by 180% compared with the steady RANS k-ω model. The wake velocity profile at a location downstream shows improvement and the URANS profile is inaccurate in comparison. For the cavity case, the HJ–NK approach is generally comparable with the other RANS models for measured velocity profiles. Predicted drag coefficients are compared with large eddy simulation. The new approach shows a 20–30% improvement in predicted drag coefficients compared with standard one and two equation RANS models. The shape of the recirculation region within the cavity is also much improved.

Relativistic Schr&amp;#246;dinger Wave Equation for Hydrogen Atom Using Factorization Method

In this investigation a simple method developed by introducing spin to Schrodinger equation to study the relativistic hydrogen atom. By separating Schrodinger equation to radial and angular parts, we modify these parts to the associated Laguerre and Jacobi differential equations, respectively. Bound state Energy levels and wave functions of relativistic Schrodinger equation for Hydrogen atom have been obtained. Calculated results well matched to the results of Dirac’s relativistic theory. Finally the factorization method and supersymmetry approaches in quantum mechanics, give us some first order raising and lowering operators, which help us to obtain all quantum states and energy levels for different values of the quantum numbers n and m.

A NOTE ON FOCAL POINT COMPARISON IN RIEMANNIAN GEOMETRY

Using a comparison fo the jacobi differential equations instead of volume comparison by Itokawa in [3], the focal radius is estimated under the suitable Ricci and mean curvature conditions.

On meromorphic solutions of linear partial differential equations of second order

This paper is concerned with entire and meromorphic solutions of linear partial differential equations of second order with polynomial coefficients. We will characterize entire solutions for a class of partial differential equations associated with the Jacobi differential equations, and give a uniqueness theorem for their meromorphic solutions in the sense of the value distribution theory, which also applies to general linear partial differential equations of second order. The results are complemented by various examples for completeness.

Explicit solutions of Jacobi and Gauss differential equations by means of operators of fractional calculus

Judging by the remarkably large number of recent publications on fractional calculus and its applications in several widely diverse areas of mathematical, physical and engineering sciences, the current popularity and importance of the subject of fractional calculus cannot be overemphasized. Motivated by some of these potentially useful developments, many authors have recently demonstrated the usefulness of fractional calculus in the derivation of explicit particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. The main object of the present paper is to show how several interesting contributions on this subject, involving a certain class of ordinary differential equations associated with (for example) the celebrated Gauss and Jacobi differential equations, can be obtained (in a unified manner) by suitably applying some general theorems on explicit particular solutions of a family of linear ordinary fractional differintegral equations.

Raising and Lowering of Generalized Hulthén Potential from Supersymmetry Approaches

We obtain the exact bound states of the generalized of Hulthen potential with negative energy levels using an analytic approach. In order to obtain bound states, we use the associated Jacobi differential equation. Using the supersymmetry approach to quantum mechanics, we show that these bound states, via four pairs of first order differential operators, represent four types of ladder equations. Two types of these supersymmetric structures suggest derivation of algebric solutions for the bound states using two different approaches.

Factorization method and stability of ϕ4 and Sine–Gordon theory

In this paper, we consider ϕ 4 and Sin–Gordon theory and also we construct the general form of the stability equation. In order to discuss the stability, we use factorization method. Using the associated Jacobi differential equation , we obtain the exactly bound states of the ϕ 4 and Sine–Gordon theory. According to the supersymmetry approach, these bound states are represented by two pairs of first order differential operators.

SUPERSYMMETRY APPROACHES TO THE BOUND STATES OF THE GENERALIZED WOODS–SAXON POTENTIAL

Using the associated Jacobi differential equation, we obtain exactly bound states of the generalization of Woods–Saxon potential with the negative energy levels based on the analytic approach. According to the supersymmetry approaches in quantum mechanics, we show that these bound states by four pairs of the first-order differential operators, represent four types of the laddering equations. Two types of these supersymmetry structures, suggest the derivation of algebraic solutions by two different approaches for the bound states.

The asymptotics of a second solution to the jacobi differential equation

An asymptotic expansion is derived for a second solution to the Jacobi differential equation, which is linearly independent of the Jacobi polynomial in the interval (-1,1). This expansion is valid uniformly with respect to x in the interval . A two-term asymptotic approximation is also constructed for the zeros , as n → ∞, where x = cosΘ. This approximation holds uniformly for k = 1,2,⋯,[γn], where γ can be any fixed number in (0,1).