Order Sturm-Liouville Problems Research Articles

Isospectral sixth order Sturm-Liouville eigenvalue problems

‎In this research, we introduce an approach to find a family of sixth order SturmLiouville problems having the same spectrum. Using Darboux Lemma and the fact that any second order Sturm-Liouville problem with the Dirichlet boundary conditions is equivalent to a sixth order Sturm-Liouville problem, the considered problems are formulated.

Exponentially convergent symbolic algorithm of the functional-discrete method for the fourth order Sturm–Liouville problems with polynomial coefficients

A new symbolic algorithmic implementation of the functional-discrete (FD-) method is developed and justified for the solution of fourth order Sturm--Liouville problem on a finite interval in the Hilbert space. The eigenvalue problem for the fourth order ordinary differential equation with polynomial coefficients is investigated. The sufficient conditions of an exponential convergence rate of the proposed approach are received. The obtained estimates of the absolute errors of FD-method significantly improve the accuracy of the estimates obtained earlier by I.P~Gavrilyuk, V.L.~Makarov and A.M.~Popov in 2010. Our algorithm is symbolic and operates with the decomposition coefficients of the eigenfunction corrections in some basis. The number of summands in these decompositions depends on the degree of the potential coefficients and the correction number. Our method uses only the algebraic operations and basic operations on $(2\times 1)$ column vectors and $(2\times 2)$ matrices. The proposed approach does not require solving any boundary value problems and computations of any integrals, unlike the previous variants of FD-method by I.P.~Gavrilyuk, V.L.~Makarov, A.M.~Popov and N.M.~Romaniuk in 2010 and 2017. The corrections to eigenpairs are computed exactly as analytical expressions, and there are no rounding errors. The numerical examples illustrate the theoretical results. The numerical results obtained with the FD-method are compared with the numerical test results obtained with other existing numerical techniques.

Spline functions, the biharmonic operator and approximate eigenvalues

The biharmonic operator plays a central role in a wide array of physical models, such as elasticity theory and the streamfunction formulation of the Navier–Stokes equations. Its spectral theory has been extensively studied. In particular the one-dimensional case (over an interval) constitutes the basic model of a high order Sturm–Liouville problem. The need for corresponding numerical simulations has led to numerous works. The present paper relies on a discrete biharmonic calculus. The primary object of this calculus is a high-order compact discrete biharmonic operator (DBO). The DBO is constructed in terms of the discrete Hermitian derivative. However, the underlying reason for its accuracy remained unclear. This paper is a contribution in this direction, expounding the strong connection between cubic spline functions (on an interval) and the DBO. The first observation is that the (scaled) fourth-order distributional derivative of the cubic spline is identical to the action of the DBO on grid functions. It is shown that the kernel of the inverse of the discrete operator is (up to scaling) equal to the grid evaluation of the kernel of $$\left[ \left( \frac{d}{dx}\right) ^4\right] ^{-1}$$ , and explicit expressions are presented for both kernels. As an important application, the relation between the (infinite) set of eigenvalues of the fourth-order Sturm–Liouville problem and the finite set of eigenvalues of the discrete biharmonic operator is studied. The discrete eigenvalues are proved to converge (at an “optimal” $$O(h^4)$$ rate) to the continuous ones. Another consequence is the validity of a comparison principle. It is well known that there is no maximum principle for the fourth-order equation. However, a positivity result is derived, both for the continuous and the discrete biharmonic equation, showing that in both cases the kernels are order preserving.

Adaptive finite element method for eigensolutions of regular second and fourth order Sturm-Liouville problems via the element energy projection technique

PurposeThe purpose of this paper is to present a numerically adaptive finite element (FE) method for accurate, efficient and reliable eigensolutions of regular second- and fourth-order Sturm–Liouville (SL) problems with variable coefficients.Design/methodology/approachAfter the conventional FE solution for an eigenpair (i.e. eigenvalue and eigenfunction) of a particular order has been obtained on a given mesh, a novel strategy is introduced, in which the FE solution of the eigenproblem is equivalently viewed as the FE solution of an associated linear problem. This strategy allows the element energy projection (EEP) technique for linear problems to calculate the super-convergent FE solutions for eigenfunctions anywhere on any element. These EEP super-convergent solutions are used to estimate the FE solution errors and to guide mesh refinements, until the accuracy matches user-preset error tolerance on both eigenvalues and eigenfunctions.FindingsNumerical results for a number of representative and challenging SL problems are presented to demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.Research limitations/implicationsThe method is limited to regular SL problems, but it can also solve some singular SL problems in an indirect way.Originality/valueComprehensive utilization of the EEP technique yields a simple, efficient and reliable adaptive FE procedure that finds sufficiently fine meshes for preset error tolerances on eigenvalues and eigenfunctions to be achieved, even on problems which proved troublesome to competing methods. The method can readily be extended to vector SL problems.

Some results on the fractional order Sturm-Liouville problems

In this work, we introduce some new results on the Lyapunov inequality, uniqueness and multiplicity results of nontrivial solutions of the nonlinear fractional Sturm-Liouville problems \t\t\t{D0+q(p(t)u′(t))+Λ(t)f(u(t))=0,1<q≤2,t∈(0,1),αu(0)−βp(0)u′(0)=0,γu(1)+δp(1)u′(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} D_{0^{+}}^{q} (p(t)u'(t))+\\Lambda(t)f(u(t))=0,\\quad1 < q\\leq2, t\\in (0,1), \\\\ \\alpha u(0)-\\beta p(0)u'(0)=0,\\qquad\\gamma u(1)+\\delta p(1)u'(1)=0, \\end{cases} $$\\end{document} where α, β, γ, δ are constants satisfying 0neq vertbetagamma+alphagammaint_{0}^{1}frac{1}{p(tau)},dtau +alpha deltavert<+infty, p(cdot) is positive and continuous on [0,1]. In addition, some existence results are given for the problem \t\t\t{D0+q(p(t)u′(t))+Λ(t)f(u(t),λ)=0,1<q≤2,t∈(0,1),αu(0)−βp(0)u′(0)=0,γu(1)+δp(1)u′(1)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\textstyle\\begin{cases} D_{0^{+}}^{q} (p(t)u'(t))+\\Lambda(t)f(u(t),\\lambda)=0,\\quad1 < q\\leq2, t\\in (0,1), \\\\ \\alpha u(0)-\\beta p(0)u'(0)=0,\\qquad\\gamma u(1)+\\delta p(1)u'(1)=0, \\end{cases} $$\\end{document} where lambdageq0 is a parameter. The proof is based on the fixed point theorems and the Leray-Schauder nonlinear alternative for single-valued maps.

Some criteria for discreteness of spectrum of half-linear fourth order Sturm–Liouville problem

We prove a necessary and su±cient conditions for discreteness of the set of all eigenvalues of half–linear eigenvalue problem with locally integrable weights. Our conditions appear to be equivalent to the compact embedding of certain weighted Sobolev and Lebesgue spaces.

A Numerical method for solving a class of fractional Sturm-Liouville eigenvalue problems

This article is devoted to both theoretical and numerical studies of eigenvalues of regular fractional $2\alpha $-order Sturm-Liouville problem where $\frac{1}{2}< \alpha \leq 1$. In this paper, we implement the reproducing kernel method RKM) to approximate the eigenvalues. To find the eigenvalues, we force the approximate solution produced by the RKM satisfy the boundary condition at $x=1$. The fractional derivative is described in the Caputo sense. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the eigenfunctions of the proposed problem. Uniformly convergence of the approximate eigenfunctions produced by the RKM to the exact eigenfunctions is proven.

Matrix methods for computing eigenvalues of Sturm–Liouville problems of order four

This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm–Liouville problem subjected to a kind of fixed boundary conditions. Furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov’s methods as well as boundary value methods for second order regular Sturm–Liouville problems. Moreover, the determination of the correction term formulas of the matrix methods is investigated in order to obtain better approximations of the problem with fixed boundary conditions since the exact eigenvalues for q=0 are known in this case. Finally, some numerical examples are illustrated.

Historic, Recent and Ongoing Applications of the Wittrick-Williams Algorithm

Eigenvalues (i.e. natural frequencies or buckling load factors) of structures are usually found by the finite element method, which involves a linear eigenproblem for which many solution methods exist. This paper covers the much harder transcendental eigenproblem which occurs when finite element discretisation errors are avoided by using the exact member equations yielded by solution of their governing differential equations. It outlines the Wittrick-Williams (W-W) algorithm, which was developed to obtain the eigenvalues of such transcendental eigenproblems, summarising its development history, its many applications and the development of associated mode finding methods. Key underlying concepts are indicated. Working at a trial value of the eigenparameter, the W-W algorithm computes the value of J, the number of eigenvalues of the structure between zero and the trial value, by summing the 'sign count' property of the transcendental overall stiffness matrix of the structure with Jm for all members of the structure, where Jm is the number of clamped/clamped member eigenvalues between zero and the trial value. An overview is given of member stiffness matrices, and associated values of Jm, for a wide range of member types. These include: Bernoulli-Euler beams and beam-columns; Timoshenko beams and beam-columns; and plates within prismatic plate assemblies which are isotropic or anisotropic (to cover both metal and laminated composite plates) and which may include out-of-plane shear deformation. Extensions of the algorithm include: cases for which the transcendental overall stiffness matrix is complex and Hermitian instead of being real and symmetric; the use of Lagrangian multipliers; very fast solutions for structures which are repetitive along one, two or three Cartesian directions; rotationally repetitive structures with members along the axis of rotational periodicity; multi-level substructuring; modes computed to the near machine accuracy to which the eigenvalues have always been found; design optimisation of stiffened wing panels using the W-W algorithm to obtain sensitivities to design changes; modal density plots; and locating pass bands and stop bands. Some non-structural areas of application of the W-W algorithm are indicated, including using a structural analogy in mathematics both to enable second and fourth order Sturm-Liouville problems to be solved competitively and also to obtain unique results for effectively infinite level trees of Sturm-Liouville equations.

Accurate solutions of fourth order Sturm–Liouville problems

Recently we introduced a new method which we call the Extended Sampling Method to compute the eigenvalues of second order Sturm–Liouville problems with eigenvalue dependent potential. We shall see in this paper how we use this method to compute the eigenvalues of fourth order Sturm–Liouville problems and present its practical use on a few examples.

Discrete Sturm-Liouville problems with nonlinear parameter in the boundary conditions

This paper deals with discrete second order Sturm-Liouville problems where the parameter that is part of the Sturm-Liouville difference equation appears nonlinearly in the boundary conditions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.

The topology of the monodromy map of a second order ODE

We consider the following question: given A ∈ SL ( 2 , R ) , which potentials q for the second order Sturm–Liouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential q ∈ L 2 ( [ 0 , 2 π ] ) to μ ( q ) = Φ ˜ ( 2 π ) , the lift to the universal cover G = SL ( 2 , R ) ˜ of SL ( 2 , R ) of the fundamental matrix map Φ : [ 0 , 2 π ] → SL ( 2 , R ) , Φ ( 0 ) = I , Φ ′ ( t ) = ( 0 1 q ( t ) 0 ) Φ ( t ) . Let H be the real infinite-dimensional separable Hilbert space: we present an explicit diffeomorphism Ψ : G 0 × H → H 0 ( [ 0 , 2 π ] ) such that the composition μ ○ Ψ is the projection on the first coordinate, where G 0 is an explicitly given open subset of G diffeomorphic to R 3 . The key ingredient is the correspondence between potentials q and the image in the plane of the first row of Φ, parametrized by polar coordinates, which we call the Kepler transform. As an application among others, let C 1 ⊂ L 2 ( [ 0 , 2 π ] ) be the set of potentials q for which the equation − u ″ + q u = 0 admits a nonzero periodic solution: C 1 is diffeomorphic to the disjoint union of a hyperplane and Cartesian products of the usual cone in R 3 with H .

Discrete Sturm-Liouville Problems With Parameter in the Boundary Conditions

This paper deals with discrete second order Sturm-Liouville problems in which the parameter that is part of the Sturm-Liouville difference equation also appears linearly in the boundary conditions. An appropriate Green's formula is developed for this problem, which leads to the fact that the eigenvalues are simple, and that they are real under appropriate restrictions. A boundary value problem can be expressed by a system of equations, and finding solutions to a boundary value problem is equivalent to finding the eigenvalues and eigenvectors of the coefficient matrix of a related linear system. Thus, the behavior of eigenvalues and eigenvectors is investigated using techniques in linear algebra, and a linear-algebraic proof is given that the eigenvalues are distinct under appropriate restrictions. The operator is extended to a self-adjoint operator and an expansion theorem is proved.

Multiplicity of Positive Solutions for Higher Order Sturm-Liouville Problems

We establish the existence of an arbitrary number of positive solutions to the 2mth order Sturm-Liouville type problem (−1)my(2m)(t) = f(t, y(t)), 0 ≤ t ≤ 1, αy(2i)(0)− βy(0) = 0, 0 ≤ i ≤ m− 1, γy(1) + δy(1) = 0, 0 ≤ i ≤ m− 1, where f : [0, 1]×[0,∞) → [0,∞) is continuous. We accomplish this by making growth assumptions on f which we state in terms which generalize assumptions in recent works regarding superlinear and/or sublinear growth in f .

Numerical methods for higher order Sturm–Liouville problems

We review some numerical methods for self-adjoint and non-self-adjoint boundary eigenvalue problems.

Eigenvalues of fourth order Sturm-Liouville problems using Fliess series

We shall extend our previous results (Chanane, 1998) on the computation of eigenvalues of second order Sturm-Liouville problems to fourth order ones. The approach is based on iterated integrals and Fliess series.