Fourth-order Sturm Liouville Problems Research Articles

Eigenvalues and eigenfunctions of fourth-order sturm-liouville problems using Bernoulli series with Chebychev collocation points

Eigenvalues and eigenfunctions of fourth-order sturm-liouville problems using Bernoulli series with Chebychev collocation points

Finite difference approach to fourth-order linear boundary-value problems

Abstract Discrete approximations to the equation $$\begin{equation*}L_\textrm{cont}u = u^{(4)} + D(x) u^{(3)} + A(x) u^{(2)} + (A^{\prime}(x)+H(x)) u^{(1)} + B(x) u = f, \quad x\in[0,1]\end{equation*}$$are considered. This is an extension of the Sturm–Liouville case $D(x)\equiv H(x)\equiv 0$ (Ben-Artzi et al. (2018) Discrete fourth-order Sturm–Liouville problems. IMA J. Numer. Anal., 38, 1485–1522) to the non-self-adjoint setting. The ‘natural’ boundary conditions in the Sturm–Liouville case are the values of the function and its derivative. The inclusion of a third-order discrete derivative entails a revision of the underlying discrete functional calculus. This revision forces evaluations of accurate discrete approximations to the boundary values of the second-, third- and fourth-order derivatives. The resulting functional calculus provides the discrete analogs of the fundamental Sobolev properties of compactness and coercivity. It allows us to obtain a general convergence theorem of the discrete approximations to the exact solution. Some representative numerical examples are presented.

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.

Discrete fourth-order Sturm–Liouville problems

Discrete fourth-order Sturm–Liouville problems

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.

The Chebyshev Collocation Method for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems

In this work, we have suggested that the Chebyshev collocation method can be employed for detecting the eigenvalues of fourth-order Sturm-Liouville problems. Two examples are presented subsequently. Numerical eventuates indicate that the present method is accurate.

An efficient technique for finding the eigenvalues and the eigenelements of fourth-order Sturm-Liouville problems

In this paper an efficient method based on Legendre-Galerkin method for computing the eigenvalues of fourth-order Sturm-Liouville problem subject to a kind of fixed boundary conditions is developed. Properties of Legendre polynomials are first presented, these properties are then utilized to reduce the eigenvalues of fourth-order Sturm-Liouville problem to some linear algebraic equations. The method is computationally attractive, and applications are demonstrated through an illustrative example and a comparisons with other methods are made.

Solution of regular second- and fourth-order Sturm–Liouville problems by exact dynamic stiffness method analogy

This paper treats all regular second- or fourth-order Sturm–Liouville (SL) problems as generalised vibration problems of non-uniform structural members on elastic foundations. Hence such SL problems can be solved by the Wittrick–Williams (WW) algorithm and the authors’ recursive second-order exact dynamic stiffness vibration method. The coefficients of the mathematical SL problems range more widely than for structural vibration problems, and so the method must account for additional possibilities, e.g. the equivalent of large negative stiffness continuous elastic supports. The method computes exact dynamic stiffnesses and their derivatives with respect to the eigenparameter accurately by solving the associated linear boundary-value problems using a standard adaptive solver. The difficulty of calculating the number of exact fixed-end eigenvalues below any trial eigenvalue needed by the WW algorithm is overcome by dividing the whole SL problem domain into an appropriate mesh, in which each element is guaranteed to have all fixed-end eigenvalues above the current trial eigenvalue. Results for second- and fourth-order SL problems shown in the literature to be particularly challenging demonstrate the effectiveness, efficiency, accuracy and reliability of the proposed method.

Dependence of eigenvalues of a class of fourth-order Sturm–Liouville problems on the boundary

In this paper, we consider the dependence of eigenvalues of a class of fourth-order Sturm–Liouville problems on the boundary. We show that the eigenvalues depend not only continuously but smoothly on boundary points, and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher fourth-order Dirichlet eigenvalues march off to plus infinity, this is also true for the first (i.e., lowest) eigenvalue.

A simple approach for determining the eigenvalues of the fourth-order Sturm–Liouville problem with variable coefficients

In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.

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.

Chebyshev differentiation matrices for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems

In this paper, an efficient technique based on the Chebyshev spectral collocation method for computing the eigenvalues of fourth-order Sturm–Liouville boundary value problems is proposed. The excellent performance of this scheme is illustrated through four examples. Numerical results and comparison with other methods are presented.

Positive solutions for systems of a nonlinear fourth-order singular semipositone Sturm-Liouville boundary value problem

This paper concerns the existence of positive solutions for systems of a fourth-order singular semipositone Sturm-Liouville boundary value problem. By applying the fixed point index theorem, some sufficient conditions for positive solutions are established. An example is given to demonstrate the application of our main results.

Eigenvalues of a class of regular fourth-order Sturm–Liouville problems

The eigenvalues of a regular fourth-order Sturm–Liouville (SL) problems depend not only continuously but smoothly on the problem. An expression for the derivative of the eigenvalues with respect to a given parameter: an endpoint, a boundary condition, a coefficient, or the weight function, are found.

Multiple Positive Solutions of Singular Nonlinear Sturm-Liouville Problems with Carathéodory Perturbed Term

By employing a well-known fixed point theorem, we establish the existence of multiple positive solutions for the following fourth-order singular differential equationLu=p(t)f(t,u(t),u′′(t))-g(t,u(t),u′′(t)),0<t<1,α1u(0)-β1u'(0)=0,γ1u(1)+δ1u'(1)=0,α2u′′(0)-β2u′′′(0)=0,γ2u′′(1)+δ2u′′′(1)=0, withαi,βi,γi,δi≥0andβiγi+αiγi+αiδi>0, i=1,2, whereLdenotes the linear operatorLu:=(ru′′′)'-qu′′,r∈C1([0,1],(0,+∞)), andq∈C([0,1],[0,+∞)). This equation is viewed as a perturbation of the fourth-order Sturm-Liouville problem, where the perturbed termg:(0,1)×[0,+∞)×(-∞,+∞)→(-∞,+∞)only satisfies the global Carathéodory conditions, which implies that the perturbed effect ofgonfis quite large so that the nonlinearity can tend to negative infinity at some singular points.

Differential quadrature method (DQM) and Boubaker Polynomials Expansion Scheme (BPES) for efficient computation of the eigenvalues of fourth-order Sturm–Liouville problems

The differential quadrature method (DQM) and the Boubaker Polynomials Expansion Scheme (BPES) are applied in order to compute the eigenvalues of some regular fourth-order Sturm–Liouville problems. Generally, these problems include fourth-order ordinary differential equations together with four boundary conditions which are specified at two boundary points. These problems concern mainly applied-physics models like the steady-state Euler–Bernoulli beam equation and mechanicals non-linear systems identification. The approach of directly substituting the boundary conditions into the discrete governing equations is used in order to implement these boundary conditions within DQM calculations. It is demonstrated through numerical examples that accurate results for the first kth eigenvalues of the problem, where k = 1, 2, 3, … , can be obtained by using minimally 2( k + 4) mesh points in the computational domain. The results of this work are then compared with some relevant studies.

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.