Sturm-Liouville Eigenvalue Problem Research Articles

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.

Optimal inverse problems of potentials for two given eigenvalues of Sturm–Liouville problems

The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.

Higher-dimensional holographic superconductors in Born–Infeld electrodynamics and f(R) gravity

In this paper, the properties of higher dimensional holographic superconductors are studied in the background of f(R) gravity and Born–Infeld electrodynamics. A specific model of f(R) gravity is considered, allowing a perturbative approach to the problem. The Sturm–Liouville eigenvalue problem is used to analytically calculate the critical temperature and the condensation operator. An expression for the critical temperature in terms of the charge density including the correction from modified gravity is derived. It is seen that the higher values of the Born–Infeld coupling parameter make the condensation harder to form. In addition, the limiting values of this parameter, above which Born–Infeld electrodynamics cannot be applied, are found for different dimensions. Another interesting property is that the increasing modifications of f(R) gravity lead to larger values of the critical temperature and a decrease in the condensation gap, which means that the condensation is easier to form.

Explicit scheme based on integral transforms for estimation of source terms in diffusion problems in heterogeneous media

The estimation of source terms present in differential equations has various applications, ranging from structural assessment, industrial process monitoring, equipment failure detection, environmental pollution source detection to identification applications in medicine. Significant progress has been made in recent years in methodologies capable of estimating this parameter. This work employs a methodology based on an explicit formulation of the integral transformation to characterize the unknown source term, reconstructing it through the expansion in known eigenfunctions of the Sturm-Liouville eigenvalue problem. To achieve this, a linear model is considered in a heterogeneous medium with known and spatially varying physical properties and two heat sources, with both temporal and spatial dependencies, and only spatial dependence. The eigenvalue problem contains information about the heterogeneous properties and is solved using the generalized integral transformation technique. Additionally, an initial interpolation of the sensor data is proposed for each observation time, making the inverse problem computationally lighter. The solutions of the inverse problem exhibit optimal performance, even with noisy input data and sources with abrupt discontinuities. The temperatures recovered by the direct problem considering the recovered source closely match synthetic experimental data, showing errors less than 1%, ensuring the robustness and reliability of the technique for the proposed application.

Application of machine learning regression models to inverse eigenvalue problems

In this work, we study the numerical solution of inverse eigenvalue problems from a machine learning perspective. Two different problems are considered: the inverse Sturm-Liouville eigenvalue problem for symmetric potentials and the inverse transmission eigenvalue problem for spherically symmetric refractive indices. Firstly, we solve the corresponding direct problems to produce the required eigenvalues datasets in order to train the machine learning algorithms. Next, we consider several examples of inverse problems and compare the performance of each model to predict the unknown potentials and refractive indices respectively, from a given small set of the lowest eigenvalues. The supervised regression models we use are k-Nearest Neighbours, Random Forests and Multi-Layer Perceptron. Our experiments show that these machine learning methods, under appropriate tuning on their parameters, can numerically solve the examined inverse eigenvalue problems.

Modeling of silencer by using various porous materials in a trifurcated waveguide structure

Researchers in applied mathematics, physics, and engineering are investigating how to reduce unwanted or ravage noise to an acceptable level. This article describes a trifurcated waveguide structure that has soft upper boundary surfaces and rigid lower boundary surfaces with different porosity effects. There is an asymmetrical arrangement of ducts around the x-axis. There is a major problem in determining how scattered fields behave in different duct areas that are subject to a variety of material attributes for walled ducts. In standard practice, the mode-matching method can be used to address governing system arising from the Sturm–Liouville eigenvalue problem. The scattering configuration for A-glass, Basalt Wool, E-glass, and Steel Wool types of porous materials is examined for different settings of the underlying structure in order to analyze noise effects. It is pertinent to predict the bulk acoustic properties at frequencies below the range in which standard measurement methods are capable of measuring in many acoustic modeling applications. A semi-analytical method for porous media is presented here that provides physically feasible predictions of bulk properties at arbitrarily low frequencies, as well as reasonable predictions of bulk properties at higher frequencies. A close relationship exists between silencer geometries and underlying structures from a practical perspective.

Numerical Computation of Spectral Solutions for Sturm-Liouville Eigenvalue Problems

This paper focuses on the study of Sturm-Liouville eigenvalue problems. In the classical Chebyshev collocation method, the Sturm-Liouville problem is discretized to a generalized eigenvalue problem where the functions represent interpolants in suitably rescaled Chebyshev points. We are concerned with the computation of high-order eigenvalues of Sturm-Liouville problems using an effective method of discretization based on the Chebfun software algorithms with domain truncation. We solve some numerical Sturm-Liouville eigenvalue problems and demonstrate the efficiency of computations.

On Isospectral Composite Beams

We consider a composite system consisting of two identical straight elastic beams under longitudinal vibration connected by an elastic interface capable of counteracting the relative vibration of the two beams with its shearing stiffness. We construct examples of isospectral composite beams, i.e., countable one-parameter families of beams having different shearing stiffness but exactly the same eigenvalues under a given set of boundary conditions. The construction is explicit and is based on the reduction to a one-dimensional Sturm–Liouville eigenvalue problem and the application of a Darboux’s lemma.

On the Solution of Mathematical Model Including Space-Time Fractional Diffusion Equation in Conformable Derivative, Via Weighted Inner Product

This research aims to accomplish an analytic solution to mathematical models involving space-time fractional differential equations in the conformable sense in series form through the weighted inner product and separation of variables method. The main advantage of this method is that various linear problems of any kind of differential equations can be solved by using this method. First, the corresponding eigenfunctions are established by solving the Sturm-Liouville eigenvalue problem. Secondly, the coefficients of the eigenfunctions are determined by employing weighted inner product and initial condition. Thirdly, the analytic solution to the problem is constructed in the series form. Finally, an illustrative example is presented to show how this method is implemented for fractional problems and exhibit its effectiveness and accuracy.

Optimal recovery of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions

Optimal recovery of potentials for Sturm-Liouville eigenvalue problems with separated boundary conditions

Continuous dependence of eigenvalues on potential functions for nonlocal Sturm–Liouville equations

In this paper, we discuss the continuous dependence of eigenvalues on the potential function for a nonlocal Sturm–Liouville equation with truncated Marchaud fractional derivative term by two‐parameter method. To this end, we first study the properties of eigenvalues for a two‐parameter nonlocal Sturm–Liouville eigenvalue problem. We obtain the two‐parameter nonlocal Sturm–Liouville eigenvalue problem that has countable number of simple real eigenvalues. Meanwhile, the asymptotic behavior of eigenvalues is studied by aid of the analytic perturbation theory.

Accurate Eigenvalues for the Sturm-Liouville Problems, Involving Generalized and Periodic Ones

In the paper, the eigenvalues of Sturm-Liouville problems (SLPs), generalized SLPs and periodic SLPs are solved. First, we propose a new method to transform the SLP with mixed boundary conditions to a {generalized} SLP for a transformed variable, for which the Dirichlet boundary conditions occur on two-side, but the coefficients are nonlinear functions of eigenvalue. To computing the eigenvalue and eigenfunction, we further recast the transformed system to an initial value problem for a new variable. In terms of the relative residual of two consecutive terminal values of the new variable a nonlinear equation is solved for seeking the eigenvalue by the fictitious time integration method (FTIM), which monotonically converges to the exact eigenvalue. We solve a numerically characteristic equation by the half-interval method (HIM) and a derivative-free iterative scheme LHL {(Liu, Hong $\&$ Li, 2021)} to achieve high precision eigenvalues. Next, the generalized SLP is transformed to a {new} one, so that the Dirichlet boundary condition happens on the right-end. By using the boundary shape function method and the uniqueness condition of the transformed variable, a definite initial value problem is derived for the new variable. To match the right-end Dirichlet boundary condition a numerically characteristic equation is {obtained and} solved by the HIM and LHL. Finally, new techniques for solving the periodic SLPs with three types periodic boundary conditions are proposed, which preserve the periodic boundary conditions with the aids of boundary shape functions. Three iterative algorithms are developed, which converge quickly.  All the proposed iterative algorithms are identified by testing some examples.

A note on “Analytical solution for the time-fractional telegraph equation by the method of separating variables”

Chen et al. (2008) investigated the analytical solution of the time-fractional telegraph equation with three kinds of nonhomogeneous boundary conditions by the method of separating variables. However, for the Sturm-Liouville eigenvalue problem derived under the Robin boundary conditions, the eigenvalues and eigenfunctions provided by the authors have some defects. In this note, we mainly analyze and modify the eigenvalues and eigenfunctions in detail and obtain the new analytical solution. Numerical experiments show that our new analytical solution is valid.

Ağırlıklı iç çarpım ile zaman kesirli problem

In this research, we discuss the construction of analytic solution of homogenous initial boundary value problem including PDEs of fractional order. Since homogenous initial boundary value problem involves Caputo fractional order derivative, it has classical initial and boundary conditions. By means of separation of variables method and the inner product defined on L^2 [0,l], the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in Caputo sense used in this study. We defined a new inner product with a weighted function to get coefficients in the Fourier series. Illustrative example presents the applicability and influence of separation of variables method on fractional mathematical problems.

Pseudospectral method for solving fractional Sturm-Liouville problem using Chebyshev cardinal functions

This work deals with the pseudospectral method to solve the Sturm-Liouville eigenvalue problems with Caputo fractional derivative using Chebyshev cardinal functions. The method is based on reducing the problem to a weakly singular Volterra integrodifferential equation. Then, using the matrices obtained from the representation of the fractional integration operator and derivative operator based on Chebyshev cardinal functions, the equation becomes an algebraic system. To get the eigenvalues, we find the roots of the characteristics polynomial of the coefficients matrix. We have proved the convergence of the proposed method. To illustrate the ability and accuracy of the method, some numerical examples are presented.

Sequential time space fractional diffusion equation including nonhomogenous initial boundary conditions

In this research, we discuss the construction of analytic solution of non-homogenous initial boundary value problem including PDEs of fractional order. By means of separation of variables method, the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem including fractional derivative in Liouville-Caputo sense.

A three-dimensional fractional solution for air contaminants dispersal in the planetary boundary layer

In this study, we investigated some closed-form solutions for solving atmospheric dispersion issues under variable atmospherical hypothesis, in a vertically positioned non-homogeneous planetary boundary-layer. In our context, a nonidentical expansion for the solution of the fractional advection-diffusion equation in a non-integer dimensional-space was examined. In a nutshell, a Sturm-Liouville eigenvalue problem with more reliable information concerning the initial value problem is discussed. The performance of the model was estimated by presenting numerical simulations against experimental data. Under these meteorological conditions, fractional-order models performed far better than those of the classical integer-order ones.

A two-index generalization of conformable operators with potential applications in engineering and physics

We developed a somewhat novel fractional-order calculus workbench as a certain generalization of the Khalil’s conformable derivative. Although every integer-order derivate can naturally be consistent with fully physical-sense problem’s quotation, this is not the standard scenario of the non-integer-order derivatives, even aiming physics systems’s modelling, solely.We revisited a particular case of the generalized conformable fractional derivative and derived a differential operator, whose properties overcome those of the integer-order derivatives, though preserving its clue advantages.Worthwhile noting, that two-fractional indexes differential operator we are dealing, departs from the single-fractional index framework, which typifies the generalized conformable fractional derivative. This distinction leads to proper mathematical tools, useful in generalizing widely accepted results, with potential applications to fundamental Physics within fractional order calculus. The later seems to be especially appropriate for exercising the Sturm-Liouville eigenvalue problem, as well as the Euler-Lagrange equation and to clarify several operator algebra matters.

A Fractional Analysis in Higher Dimensions for the Sturm-Liouville Problem

Abstract In this work, we consider the n-dimensional fractional Sturm-Liouville eigenvalue problem, by using fractional versions of the gradient operator involving left and right Riemann-Liouville fractional derivatives. We study the main properties of the eigenfunctions and the eigenvalues of the associated fractional boundary problem. More precisely, we show that the eigenfunctions are orthogonal and the eigenvalues are real and simple. Moreover, using techniques from fractional variational calculus, we prove in the main result that the eigenvalues are separated and form an infinite sequence, where the eigenvalues can be ordered according to increasing magnitude. Finally, a connection with Clifford analysis is established.

Smooth approximation of a varying refractive-index profile and its application in the computation of light waves

In this paper, the smooth approximation of light waves is studied for an open optical waveguide with a distinct refractive-index profile, which involves high-precision computation of the eigenmodes and corresponding eigenfunctions. During analysis, the refractive-index function is first approximated by a quadratic spline interpolation function. Since the quadratic spline function is a polynomial of degree two in every sub-interval (sub-layer), it is equivalent to a piecewise polynomial of degree two, based on which, the corresponding Sturm-Liouville eigenvalue problem of the Helmholtz operator in sub-layer can be solved analytically by the Kummer functions. Finally, the approximate dispersion equation is established to the TE case. Obviously, the approximate dispersion equations converge fast to the exact ones, as the maximum value of the sub-interval sizes tends to zero. Furthermore, eigenmodes may be obtained by Müller’s method with suitable initial values. To refine the accuracy, the equidistant partition and the non-equidistant partition are applied to divide the interval. Numerical simulations show that the eigenfunctions of the spline interpolation are much smoother than the ones with piecewise interpolation. In addition, the non-equidistant partition can help improve the accuracy and the order of convergence of general solutions reaches the third.