Sturm-Liouville Research Articles

An analytical fast field solver for surface permanent magnet co‐axial magnetic gears

AbstractA mathematical model of co‐axial magnetic gears is given for the prediction of magnetic field distribution and the magnetic forces. The modelling approach is based on the separation of variables method (SVM), Sturm–Liouville problem, and spectral analysis. This method is computationally more efficient and has the ability to account for finite permeability of magnetic saliency. An finite element method (FEM) validation is performed for a case study, and the results are observed to be almost matched. The computational time of the analytical model has been almost in the same order of that of the FEM for a specified rotor position (both in a matter of a second or less); however, when 100 number of rotor positions were accounted in the simulation, the computational time of FEM was increased to 82 s, while the computational complexity of SVM was almost unaffected.

A High-Resolution Exponential Spline Method and Its Convergence Analysis for Two-Point Mixed Boundary Value Problems

In this paper, we present a new numerical method for effectively solving general second-order ordinary differential equations with mixed boundary conditions. Our approach utilizes a quasi-variable mesh enabling us the flexibility to adapt the mesh density according to different boundary layer problems. By employing a discretization technique that incorporates the construction of exponential spline, we achieve third-order accuracy at internal grid points, while boundary points exhibit fourth-order accuracy. Since the method is based on off-step grid points, it eliminates the need to modify the method while employing it to singular problems. We provide computational results to various numerical problems which arise in different physical phenomena like Burger’s equation and Sturm–Liouville equation. A comparison with recent findings underscores the superior performance of our method.

Optimizing the noise control in a two-layer conduit

Abstract In the modern world, noise pollution is a major concern due to its prevalence. This work focuses on optimizing noise control in a two-layer conduit. A conduit comprises an inlet and double-outlet zone (DOZ). The upper walls of the DOZ are lined with perforated absorbent material while the lower walls are layered with fibrous. Mathematically, the physical problem is formulated with a field wave equation together with rigid and impedance boundary conditions in the respective zones. Such governing boundary value problem (BVP) leads to the Sturm-Liouville (SL) category in which standard orthogonality relations (OR) are indispensable. The system of the linear equations of the BVP is acquired with a semi-analytical Mode-Matching (MM) approach by implementing the continuity conditions of sound pressures and velocities at the matching junction with the aid of OR. These systems are truncated and solved numerically with computation learning to obtain the reflected and transmitted modal amplitudes in respective zones. Due to the lining’s perforated upper walls, fibrous lower walls, and reversal in the DOZ, the analysis of the reflected and transmitted powers versus frequencies is significantly observed and is shown in graphical findings. The algebraic derivation is validated by satisfying the conservation law of power flux and matching continuity conditions for impedance, perforated, and fibrous lined boundaries.

Unraveling the Complexity of Inverting the Sturm–Liouville Boundary Value Problem to Its Canonical Form

The Sturm–Liouville boundary value problem (SLBVP) stands as a fundamental cornerstone in the realm of mathematical analysis and physical modeling. Also known as the Sturm–Liouville problem (SLP), this paper explores the intricacies of this classical problem, particularly the relationship between its canonical and Liouville normal (Schrödinger) forms. While the conversion from the canonical to Schrödinger form using Liouville’s transformation is well known in the literature, the inverse transformation seems to be neglected. Our study attempts to fill this gap by investigating the inverse of Liouville’s transformation, that is, given any SLP in the Schrödinger form with some invariant function, we seek the SLP in its canonical form. By closely examining the second Paine–de Hoog–Anderson (PdHA) problem, we argue that retrieving the SLP in its canonical form can be notoriously difficult and can even be impossible to achieve in its exact form. Finding the inverse relationship between the two independent variables seems to be the main obstacle. We confirm this claim by considering four different scenarios, depending on the potential and density functions that appear in the corresponding invariant function. In the second PdHA problem, this invariant function takes a reciprocal quadratic binomial form. In some cases, the inverse Liouville transformation produces an exact expression for the SLP in its canonical form. In other situations, however, while an exact canonical form is not possible to obtain, we successfully derived the SLP in its canonical form asymptotically.

Analysis of a Sturm–Liouville Problem Arising in Atmosphere

We present recent results in study of a mathematical model of the sea-breeze flow, arising from a general model of the ’morning glory’ phenomena. Based on analysis of the Dirichlet spectrum of the corresponding Sturm–Liouville problem and application of the Fredholm alternative, we establish conditions of existence/uniqueness of solutions to the given problem.

Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions

For the generalized Sturm–Liouville problem (GSLP), a new formulation is undertaken to reduce the number of unknowns from two to one in the target equation for the determination of eigenvalue. The eigenparameter-dependent shape functions are derived for using in a variable transformation, such that the GSLP becomes an initial value problem for a new variable. For the uniqueness of eigenfunction an extra condition is imposed, which renders the right-end value of the new variable available; a derived implicit nonlinear equation is solved by an iterative method without using the differential; we can achieve highly precise eigenvalues. For the nonlocal Sturm–Liouville problem (NSLP), we consider two types of integral boundary conditions on the right end. For the first type of NSLP we can prove sufficient conditions for the positiveness of the eigenvalue. Negative eigenvalues and multiple solutions may exist for the second type of NSLP. We propose a boundary shape function method, a two-dimensional fixed-quasi-Newton method and a combination of them to solve the NSLP with fast convergence and high accuracy. From the aspect of numerical analysis the initial value problem of ordinary differential equations and scalar nonlinear equations are more easily treated than the original GSLP and NSLP, which is the main novelty of the paper to provide the mathematically equivalent and simpler mediums to determine the eigenvalues and eigenfunctions.

Reconstruction of the solution of inverse Sturm–Liouville problem

In this paper we are concerned with an inverse problem with Robin boundary conditions, which states that, when the potential on [0,1/2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$[0,1/2]$\\end{document} and the coefficient at the left end point are known a priori, a full spectrum uniquely determines its potential on the whole interval and the coefficient at the right end point. We shall give a new method for reconstructing the potential for this problem in terms of the Mittag-Leffler decomposition of entire functions associated with this problem. The new reconstructing method also deduces a necessary and sufficient condition for the existence issue.

Spectral and Oscillation Theory for an Unconventional Fractional Sturm–Liouville Problem

Spectral and Oscillation Theory for an Unconventional Fractional Sturm–Liouville Problem

Double-Step Shape Invariance of Radial Jacobi-Reference Potential and Breakdown of Conventional Rules of Supersymmetric Quantum Mechanics

The paper reveals some remarkable form-invariance features of the ‘Jacobi-reference’ canonical Sturm–Liouville equation (CSLE) in the particular case of the density function with the simple pole at the origin. It is proven that the CSLE under consideration preserves its form under the two second-order Darboux–Crum transformations (DCTs) with the seed functions represented by specially chosen pairs of ‘basic’ quasi-rational solutions (q-RSs), i.e., such that their analytical continuations do not have zeros in the complex plane. It is proven that both transformations generally either increase or decrease by 2 the exponent difference (ExpDiff) for the mentioned pole while keeping two other parameters unchanged. The change is more complicated in the latter case if the ExpDiff for the pole of the original CSLE at the origin is smaller than 2. It was observed that the DCTs in question do not preserve bound energy levels according to the conventional supersymmetry (SUSY) rules. To understand this anomaly, we split the DCT in question into the two sequential Darboux deformations of the Liouville potentials associated with the CSLEs of our interest. We found that the first Darboux transformation turns the initial CSLE into the Heun equation written in the canonical form while the second transformation brings us back to the canonical form of the hypergeometric equation. It is shown that the first of these transformations necessarily places the mentioned ExpDiff into the limit-circle (LC) range and then the second transformation keeps the pole within the LC region, violating the conventional prescriptions of SUSY quantum mechanics.

Schrödinger equation for Sturm–Liouville operator with singular propagation and potential

Schrödinger equation for Sturm–Liouville operator with singular propagation and potential

Donoghue 𝑚-functions for Singular Sturm–Liouville operators

Let A ˙ \dot {A} be a densely defined, closed, symmetric operator in the complex, separable Hilbert space H \mathcal {H} with equal deficiency indices and denote by N i = ker ⁡ ( ( A ˙ ) ∗ − i I H ) \mathcal {N}_i = \ker ((\dot {A})^* - i I_{\mathcal {H}}) , dim ⁡ ( N i ) = k ∈ N ∪ { ∞ } \dim (\mathcal {N}_i)=k\in \mathbb {N} \cup \{\infty \} , the associated deficiency subspace of A ˙ \dot {A} . If A A denotes a self-adjoint extension of A ˙ \dot {A} in H \mathcal {H} , the Donoghue m m -operator M A , N i D o ( ⋅ ) M_{A,\mathcal {N}_i}^{Do} (\,\cdot \,) in N i \mathcal {N}_i associated with the pair ( A , N i ) (A,\mathcal {N}_i) is given by M A , N i D o ( z ) = z I N i + ( z 2 + 1 ) P N i ( A − z I H ) − 1 P N i | N i M_{A,\mathcal {N}_i}^{Do}(z)=zI_{\mathcal {N}_i} + (z^2+1) P_{\mathcal {N}_i} (A - z I_{\mathcal {H}})^{-1} P_{\mathcal {N}_i} \vert _{\mathcal {N}_i} , z ∈ C ∖ R , z\in \mathbb {C}\setminus \mathbb {R}, with I N i I_{\mathcal {N}_i} the identity operator in N i \mathcal {N}_i , and P N i P_{\mathcal {N}_i} the orthogonal projection in H \mathcal {H} onto  N i \mathcal {N}_i . Assuming the standard local integrability hypotheses on the coefficients p , q , r p, q,r , we study all self-adjoint realizations corresponding to the differential expression τ = 1 r ( x ) [ − d d x p ( x ) d d x + q ( x ) ] \tau =\frac {1}{r(x)}[-\frac {d}{dx}p(x)\frac {d}{dx} + q(x)] for a.e.  x ∈ ( a , b ) ⊆ R x\in (a,b) \subseteq \mathbb {R} , in L 2 ( ( a , b ) ; r d x ) L^2((a,b); rdx) , and, as our principal aim in this paper, systematically construct the associated Donoghue m m -functions (respectively, ( 2 × 2 ) (2 \times 2) matrices) in all cases where τ \tau is in the limit circle case at least at one interval endpoint a a or b b .

On the Dependence of Acoustic Modes on Media Parameters in the Pekeris-Airy Waveguide

A waveguide consisting of an isospeed water layer overlying a halfspace with a sound speed gradient is considered. The dependence of horizontal wavenumbers of normal modes in such a waveguide on the sound frequency, water depth and geoacoustic parameters of the bottom are investigated. It is shown that these dependencies are described by ordinary differential equations that can be solved numerically at very low computational cost providing a substantial increase in the efficiency of broadband modelling of sound propagation. The explicit formulae for the derivatives of horizontal wavenumbers with respect to the bottom parameters can be also used in dispersion-based geoacoustic inversion methods. This approach can be extended to the case of a waveguide consisting of several layers of Airy-type media. We also propose a new and concise proof that the spectrum of the Sturm-Liouville problem from which the modes are found for this waveguide is purely discrete.

Analysis of Norm-Attainability and Convergence Properties of Orthogonal Polynomials in Weighted Sobolev Spaces

This paper explores norm-attainability of orthogonal polynomials in Sobolev spaces, investigating properties like existence, uniqueness, and convergence. It establishes the convergence of these polynomials in Sobolev spaces, addressing orthogonality preservation and derivative behaviors. Spectral properties, including Sturm-Liouville eigenvalue problems, are analyzed, enhancing the understanding of these polynomials. The study incorporates fundamental concepts like reproducing kernels, Riesz representations, and Bessel’s inequality. Results contribute to the theoretical understanding of orthogonal polynomials, with potential applications in diverse mathematical and computational contexts.

Multiplicity and nonexistence of positive solutions to impulsive Sturm–Liouville boundary value problems

AbstractIn this paper, we study the existence, nonexistence, and multiplicity of positive solutions to a nonlinear impulsive Sturm–Liouville boundary value problem with a parameter. By using a variational method, we prove that the problem has at least two positive solutions for the parameter $\lambda \in (0,\Lambda )$ λ ∈ ( 0 , Λ ) , one positive solution for $\lambda =\Lambda $ λ = Λ , and no positive solution for $\lambda >\Lambda $ λ > Λ , where $\Lambda >0$ Λ > 0 is a constant.

Boundary controllability for a 1D degenerate parabolic equation with a Robin boundary condition

AbstractIn this paper, we prove the null controllability of a one-dimensional degenerate parabolic equation with a weighted Robin boundary condition at the left endpoint, where the potential has a singularity. We use some results from the singular Sturm–Liouville theory to show the well-posedness of our system. We obtain a spectral decomposition of a degenerate parabolic operator with Robin conditions at the endpoints, we use Fourier–Dini expansions and the moment method introduced by Fattorini and Russell to prove the null controllability and to obtain an upper estimate of the cost of controllability. We also get a lower estimate of the cost of controllability by using a representation theorem for analytic functions of exponential type.

Bernstein collocation technique for a class of Sturm-Liouville problems

Sturm-Liouville problems have yielded the biggest achievement in the spectral theory of ordinary differential operators. Sturm-Liouville boundary value issues appear in many key applications in natural sciences. All the eigenvalues for the standard Sturm-Liouville problem are guaranteed to be real and simple, and the related eigenfunctions form a basis in a suitable Hilbert space. This article uses the weighted residual collocation technique to numerically compute the eigenpairs of both regular and singular Strum Liouville problems. Bernstein polynomials over [0,1] has been used to develop a weighted residual collocation approach to achieve an improved accuracy. The properties of Bernstein polynomials and the differentiation formula based on the Bernstein operational matrix are used to simplify the given singular boundary value problems into a matrix-based linear algebraic system. Keeping this fact in mind such a polynomial with space defined collocation scheme has been studied for Strum Liouville problems. The main reasons to use the collocation technique are its affordability, ease of use, well-conditioned matrices, and flexibility. The weighted residual collocation method is found to be more appealing because Bernstein polynomials vanish at the two interval ends, providing better versatility. A multitude of test problems are offered along with computation errors to demonstrate how the suggested method behaves. The numerical algorithm and its applicability to particular situations are described in detail, along with the convergence behavior and precision of the current technique.

On a nonlocal integral operator commuting with the Laplacian and the Sturm–Liouville problem: Low rank perturbations of the operator

On a nonlocal integral operator commuting with the Laplacian and the Sturm–Liouville problem: Low rank perturbations of the operator

Approaching certain central-field Schrödinger problems through the harmonic oscillator green's function

We propose a method to analytically solve the Schrödinger equation with central potentials. By means of Sturm–Liouville expansion, the Green's function (GF) of the Schrödinger differential operator can be represented in the GF of the isotropic harmonic oscillator. Applying this straightforward approach, we solve the Schrödinger problem for the Coulomb potential, for the Morse potential, and for the respective inverted potentials. Using the obtained results, we formulate a method to analytically solve Schrodinger equations for multi-well potentials, demonstrating that classically forbidden regions can be treated as inverted harmonic oscillators, whereas harmonic oscillators or Morse potentials can approximate the wells.

Mixed boundary value problems involving Sturm–Liouville differential equations with possibly negative coefficients

This paper is devoted to the study of a mixed boundary value problem for a complete Sturm–Liouville equation, where the coefficients can also be negative. In particular, the existence of infinitely many distinct positive solutions to the given problem is obtained by using critical point theory.

Ambarzumyan Theorem for Conformable Type Sturm-Liouville Problem on Time Scales

In this study, we give an Ambarzumyan type theorem for a Sturm-Liouville dynamic equation which includes conformable type derivative on time scales with conformable Robin boundary conditions. Under certain conditions, we prove that potential function can be determined by using only first eigenvalue.