Sturm-Liouville Boundary Value Problems Research Articles

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.

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.

Variational approach to a symmetric boundary value problem generated by a system of equations and separated boundary conditions

This work provides some information on the eigenvalues and eigenfunctions of a problem which is constructed by a system of equations and symmetric boundary conditions that includes the ordinary second‐order Sturm–Liouville boundary value problem. In particular, we show that the problem has an infinite number of discrete eigenvalues with a greatest lower bound and the corresponding eigenfunctions are complete in mean and energy. We introduce the results using the variational approach that enables us to consider only continuous pair functions instead of absolutely continuous pair functions.

Spectral Expansion for Conformable Fractional Sturm- Liouville Problem on the Whole Line

In this article, we discuss a conformable fractional Sturm-Liouville boundary-value problem on the whole line. We prove the existence of a spectral function for the singular conformable fractional Sturm-Lioville problem. Further, we establish a Parseval equality and spectral expansion formula by terms of the spectral function for conformable fractional Sturm-Liouville problem on the whole line.

Spectral Properties of Discontinuous Boundary Value Problem with Transmission Conditions and Manypoint Singularities

In this study, we investigate a discontinuous Sturm-Liouville boundary value problem on three intervals
 with manypoint-transmission conditions in direct sum of Sobolev space. We establish such spectral properties as Fredholmness and coreciveness with respect to the eigenvalue parameter

Дополнительные граничные условия в задачах теплопроводности с переменным по координате начальным условием

It is exceedingly difficult to obtain mathematically accurate analytical solutions of heat conduction problems with a variable initial condition. Known solutions of these problems are expressed by cumbersome functional series that converge poorly in the range of small values of time and space variables. Thus, to obtain simpler and more effective solutions of these problems is an urgent issue. The authors have used an additional required function and additional boundary conditions to obtain solutions of the problem. Application of the additional required function allows us to reduce the original partial differential equation to the integration of an ordinary differential equation. Additional boundary conditions are in such a form that their fulfillment using the resulting solution is equivalent to the fulfillment of the equation at the boundary points. The authors have developed a technique to obtain an analytical solution of the heat conduction problem under a linear change of the initial condition, based on an additional required function and additional boundary conditions. Solution of an ordinary differential equation with respect to the additional required function determines the eigenvalues. In classical methods these eigenvalues are found in the solution of the Sturm–Liouville boundary value problem. The authors have proposed another, simpler solution to determine eigenvalues. An accurate analytical solution of the heat conduction problem for an unbounded plate with a coordinate-variable initial condition is obtained. The scientific and practical value of the proposed analytical solution is the development of an innovative approach to determine eigenvalues, as well as elimination of complex integrals when we solve the equation and initial conditions of the boundary value problem. It makes possible to simplify the use of the solution obtained in engineering applications.

Orthonormal eigenfunction expansions for sixth-order boundary value problems

Sixth-order boundary value problems (BVPs) arise in thin-film flows with a surface that has elastic bending resistance. To solve such problems, we first derive a complete set of odd and even orthonormal eigenfunctions — resembling trigonometric sines and cosines, as well as the so-called “beam” functions. These functions intrinsically satisfy boundary conditions (BCs) of relevance to thin-film flows, since they are the solutions of a self-adjoint sixth-order Sturm–Liouville BVP with the same BCs. Next, we propose a Galerkin spectral approach for sixth-order problems; namely the sought function as well as all its derivatives and terms appearing in the differential equation are expanded into an infinite series with respect to the derived complete orthonormal (CON) set of eigenfunctions. The unknown coefficients in the series expansion are determined by solving the algebraic system derived by taking successive inner products with each member of the CON set of eigenfunctions. The proposed method and its convergence are demonstrated by solving two model sixth-order BVPs.

Method for determining eigen numbers in heat conduction problems for a cylinder

RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution.THE PURPOSE. Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axial and central symmetry (cylinder, ball) has been developed.METHODS. The method is based on the use of orthogonal systems of coordinate functions and additional boundary conditions. Latter are in such a form that their fulfillment by the desired solution is equivalent to the fulfillment of the differential equation of the boundary value problem at the boundary points of the region, leading to its fulfillment inside the considered region. Moreover, the accuracy of the equation depends on the number of approximations, which, in turn, depends on the number of additional boundary conditions used. Using the orthogonality property of trigonometric coordinate functions included in a series representing eigenfunctions makes it possible to increase the accuracy of the fulfillment of the differential equation of the Sturm-Liouville boundary value problem and the accuracy of determining the eigenvalues. To satisfy the initial condition, its residual is compiled and the condition of its orthogonality to all coordinate functions is required. The orthogonality of trigonometric systems of coordinate functions with respect to the unknown constants of integration leads to a system of algebraic linear equations, the number of which is equal to the number of approximations. As a result, the fulfillment of the initial condition is simplified and the accuracy of its fulfillment is increased.RESULTS. The advantage of the method is that the resulting solution contains only simple algebraic expressions, excluding special functions (Bessel function, Legendre function, gamma function).CONCLUSION. Thus, bypassing direct integration over a spatial variable, the use of additional boundary conditions makes it possible to find a solution of any complexity of the equations of the Sturm-Liouville boundary value problem, which reduces to the definition of simple integrals.

Forecasting supercritical behavior of drill strings in horizontal wells

The determining factor in the current changes in the world energy balance and its structure, as well as the global development of the oil and gas industry, is currently the technological factor. As a result, fundamentally new categories of horizontal and inclined wells appeared and developed. They give a multiple increase in flow rates, as they increase the production area many times over. They are used in the development of offshore projects, in swampy and very difficult areas, to increase oil recovery, restoration of inactive wells.
 New technologies based on the method of horizontal drilling have revolutionized the practice and theory of world oil production, as they have allowed the destruction of offshore oil and gas fields without the construction of expensive offshore foundations and platforms.
 When drilling horizontal wells, as a rule, the main cause of emergencies is the loss of stability of the drill string, its bifurcation protrusion and the maximum frictional interaction with the well wall. The issues of theoretical modeling of the phenomenon of unstable protrusion of columns are associated with significant difficulties, the main of which is due to the need to set the Sturm-Liouville boundary value problem on a large length of drill string. Because for deep wells, the drill string becomes geometrically similar to a human hair, many traditional mathematical methods used to integrate solving equations become poorly convergent in these cases.
 Based on the theory of curvilinear flexible rods, the problem of theoretical modeling of supercritical states of drill strings taking into account their contact interaction with the walls of oil and gas wells is set. Analytical solutions of the problem are constructed, which determine the critical values of external tensile or compressive longitudinal force and torque at a given difference in the diameters of the cavity and the drill string, the zones of supercritical states of drill strings are established.

Radial oscillations in neutron stars with delta baryons

We investigate the effect of $\mathrm{\ensuremath{\Delta}}$ baryons on the radial oscillations of neutron and hyperon stars, employing a density-dependent relativistic mean-field model. The spin-$3/2$ baryons are described by the Rarita-Schwinger Lagrangian density. The baryon-meson coupling constants for the spin-$3/2$ decuplet and the spin-$1/2$ baryonic octet are calculated using a unified approach relying on the fact that the Yukawa couplings present in the Lagrangian density of the mean-field models must be invariant under the SU(3) and SU(6) group transformations. We calculate the 20 lowest eigenfrequencies and corresponding oscillation functions of $\mathrm{\ensuremath{\Delta}}$-inclusive nuclear ($\mathrm{N}+\mathrm{\ensuremath{\Delta}}$) and hyperonic matter ($\mathrm{N}+\mathrm{H}+\mathrm{\ensuremath{\Delta}}$) by solving the Sturm-Liouville boundary value problem and also verifying its validity. We see that the lowest mode frequencies for $\mathrm{N}+\mathrm{\ensuremath{\Delta}}$ and $\mathrm{N}+\mathrm{H}$ EoSs are higher as compared to the pure nucleonic matter because of the deltas and hyperons present. Furthermore, the separation between consecutive modes increases with the addition of hyperons and $\mathrm{\ensuremath{\Delta}}$s.

Radial oscillations of dark matter admixed neutron stars

Within the relativistic mean-field model, we investigate the properties of dark matter (DM) admixed neutron stars, considering non-rotating objects made of isotropic matter. We adopt the IOPB-I hadronic equation of state (EOS) by assuming that the fermionic DM within super-symmetric models has already been accreted inside the neutron star (NS). The impact of DM on the mass-radius relationships and the radial oscillations of pulsating DM admixed neutron stars (with and without the crust) are explored. It is observed that the presence of DM softens the EOS, which in turn lowers the maximum mass and its corresponding radius. Moreover, adding DM results in higher frequencies of pulsating objects and hence we show the linearity of fundamental mode frequency of canonical NS with DM Fermi momentum. We also investigate the profile of eigenfunctions solving the Sturm-Liouville boundary value problem, and verify its validity. Further, we study the stability of NSs considering the fundamental mode frequency variation with the mass of the star, and verify the stability criterion $\partial M/\partial\rho_c > 0$. Finally, the effect of the crust on the large frequency separation for different DM Fermi momenta is shown as well.

On the evolution of beaches of finite length

This article provides closed form solutions of the Diffusion Equation (DE) for beaches of finite length bounded by outcrops; the method employed relies on the sum of two Sturm-Liouville Boundary Value Problems, which allow accounting for the effects of wave angle. From the general solution, three situations of engineering interest were extrapolated and discussed in detail. The first corresponds to a stretch of coast “pinned” at the extremes; the second refers to an “insulated” beach bounded by groins of infinite length and the third corresponds to a groin compartment bypassed at one end. All those solutions were previously known in the field of thermodynamics, but they are here derived systematically from a general Boundary Value Problem (BVP). A detailed comparison with the software GENESIS is carried out to establish the limits of application with increasing wave angles. The effects of the presence of outcrops is studied comparing the solutions for finite beaches with those proposed in the literature under the hypothesis of infinite shoreline (open coast). It is shown that the new approach can be particularly useful in the case of beach nourishments placed asymmetrically to the cell centre. For application purposes, approximate solutions are given, based on curve fitting, to calculate the remaining value of sediments in a beach fill of rectangular shape.

Estimates on the eigenvalues of complex nonlocal Sturm-Liouville problems

The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems. The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.

Higher Order Finite Element Methods for Some One-dimensional Boundary Value Problems

In this paper, third-order compact and fourth-order finite element methods (FEMs) based on simple modifications of traditional FEMs are proposed for solving one-dimensional Sturm-Liouville boundary value problems (BVPs). The key idea is based on interpolation error estimates. A simple posterior error analysis of the original piecewise linear finite element space leads to a third-order accurate solution in the L2 norm, second-order in the H1, and the energy norm. The novel idea is also applied to obtain a fourth-order FEM based on the quadratic finite element space. The basis functions in the new fourth-order FEM are more compact compared with that of the classic cubic basis functions. Numerical examples presented in this paper have confirmed the convergence order and analysis. A generalization to a class of nonlinear two-point BVPs is also discussed and tested.

VARIATIONAL FORMULATION FOR THE STURM-LIOUVILLE PROBLEM OF FRACTIONAL DIFFERENTIAL EQUATION WITH GENERALIZED (<i>P</i>, <i>Q</i>)-LAPLACIAN OPERATOR

In this paper, the Sturm-Liouville boundary value problem is studied for fractional differential equation with generalized (<i>p</i>, <i>q</i>)-Laplacian operator. By imposing mild assumptions on nonlinearity <i>f</i>, several new existence results of at least one or two nontrivial weak solutions are established through variational methods and critical point theorems. Furthermore, the criteria is also investigated for the nonexistence result.

Rich dynamics in planar systems with heterogeneous nonnegative weights

This paper studies the global structure of the set of nodal solutions of a generalized Sturm–Liouville boundary value problem associated to the quasilinear equation$ -(\phi(u'))' = \lambda u + a(t)g(u), \quad \lambda\in {\mathbb R}, $where $ a(t) $ is non-negative with some positive humps separated away by intervals of degeneracy where $ a\equiv 0 $. When $ \phi(s) = s $ this equation includes a generalized prototype of a classical model going back to Moore and Nehari [35], 1959. This is the first paper where the general case when $ {\lambda}\in\mathbb{R} $ has been addressed when $ a\gneq 0 $. The semilinear case with $ a\lneq 0 $ has been recently treated by López-Gómez and Rabinowitz [28,29,30].

Sturm-Liouville boundary value problems for fractional differential equations with p-Laplacian operator via Riesz-Caputo fractional derivatives

Sturm-Liouville boundary value problems for fractional differential equations with p-Laplacian operator via Riesz-Caputo fractional derivatives

Fractional Sturm–Liouville Eigenvalue Problems, II

We continue the study of a non-self-adjoint fractional three-term Sturm–Liouville boundary value problem (with a potential term) formed by the composition of a left Caputo and left Riemann–Liouville fractional integral under Dirichlet type boundary conditions. We study the existence and asymptotic behavior of the real eigenvalues and show that for certain values of the fractional differentiation parameter α, 0&lt;α&lt;1, there is a finite set of real eigenvalues and that, for α near 1/2, there may be none at all. As α→1− we show that their number becomes infinite and that the problem then approaches a standard Dirichlet Sturm–Liouville problem with the composition of the operators becoming the operator of second order differentiation.

Study on Infinitely Many Solutions for a Class of Fredholm Fractional Integro-Differential System

This paper deals with a class of nonlinear fractional Sturm–Liouville boundary value problems. Each sub equation in the system is a fractional partial equation including the second kinds of Fredholm integral equation and the p-Laplacian operator, simultaneously. Infinitely many solutions are derived due to perfect involvements of fractional calculus theory and variational methods with some simpler and more easily verified assumptions.

Perturbed potential temperature field in the atmospheric boundary layer

AbstractThis article discusses the modeling of perturbed potential temperature field in an atmospheric boundary layer (ABL). We adopt a convection–diffusion model with specified initial and boundary conditions that resulted from simplifying the linearized equation of the standard continuity equation for potential temperature field in the state of weak turbulent fluxes. By implementing the method of separation of variables to the nonsteady‐state perturbed potential temperature, we obtain a regular Sturm–Liouville boundary value problem (BVP) for the spatial‐dependent, vertical distribution component of the perturbed potential temperature. By transforming the problem in the canonical form into the Liouville normal form, we provide asymptotic solutions for the corresponding second‐order BVP using the Wentzel–Kramers–Brillouin (WKB) theory. We further observe a remarkable qualitative agreement between the asymptotic solutions and numerical simulations. As other convection–diffusion models typically perform, the perturbed potential temperature diminishes and approaches the steady‐state condition over an extended period of time.