Singular Sturm-Liouville Boundary Value Problems Research Articles

The double exponential sinc collocation method for singular Sturm-Liouville problems

Sturm-Liouville problems are abundant in the numerical treatment of scientific and engineering problems. In the present contribution, we present an efficient and highly accurate method for computing eigenvalues of singular Sturm-Liouville boundary value problems. The proposed method uses the double exponential formula coupled with sinc collocation method. This method produces a symmetric positive-definite generalized eigenvalue system and has exponential convergence rate. Numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula.

Necessary and Sufficient Condition for the Existence of Positive Solutions to a Class of Fourth-order Singular Sturm-Liouville Boundary Value Problems

This paper studies a class of singular Sturm-Liouville boundary value problems of fourthorder differential equations.A necessary and sufficient condition for the existence of C~3[0,1]positive solutions as well as C~2[0,1]positive solutions of the nonlinear differential equations is established by means of the fixed point theorems on cones.

Small perturbation propagation in the viscoelastic fluid enclosed in a multilayer viscoelastic tube

A pulsating flow of the incompressible viscoelastic fluid enclosed in the multilayer viscoelastic tube of a variable circular section is investigated. The problem stated leads to the singular Sturm-Liouville boundary value problem which in its turn is reduced to the equivalent integral equation of the Volterra type that is solved by the successive approximation method. The influence of fluid relaxation and retardation time on the wave system characteristics is revealed.

Positive solutions for singular Sturm-Liouville boundary value problems with integral boundary conditions

In this paper, we study the second-order nonlinear singular Sturm-Liouville boundary value problems with Riemann-Stieltjes integral boundary conditions     −(p(t)u ' (t)) ' +q(t)u(t) = f(t,u(t)), 0 < t < 1, �1u(0) −�1u ' (0) = R 1 0 u(�)d�(�), where f(t,u) is allowed to be singular at t = 0,1 and u = 0. Some new results for the existence of positive solutions of the boundary value problems are obtained. Our results extend some known results from the nonsingular case to the singular case, and we also improve and extend some results of the singular cases.

Multiple positive solutions of the singular boundary value problems for second-order differential equations on the half-line

In this paper, the existence of multiple positive solutions for singular Sturm–Liouville boundary value problems on the half-line is investigated. By applying the fixed point index theorem of cone map, some existence and multiplicity results of positive solutions are derived. Our results improve and generalize many well-known results. Two examples are presented to demonstrate the application of our main results.

On the existence of positive solutions for singular boundary value problems on the half-line

On the basis of a special cone and the application of the fixed point theory, some new results for the existence of positive solutions of singular Sturm–Liouville boundary value problems on the half-line are obtained.

Positive solutions of fourth-order Sturm–Liouville boundary value problems with changing sign nonlinearity

In this paper, the existence of positive solutions is investigated for the following singular fourth-order Sturm–Liouville boundary value problem with changing sign nonlinearity: { u ( 4 ) ( t ) = a ( t ) f ( t , u ( t ) , u ″ ( t ) ) + b ( t ) , 0 < t < 1 , α 1 u ( 0 ) − β 1 u ′ ( 0 ) = δ 1 u ( 1 ) + γ 1 u ′ ( 1 ) = 0 , α 2 u ″ ( 0 ) − β 2 u ‴ ( 0 ) = δ 2 u ″ ( 1 ) + γ 2 u ‴ ( 1 ) = 0 , where a : ( 0 , 1 ) → [ 0 , + ∞ ) is continuous, f : [ 0 , 1 ] × [ 0 , + ∞ ) × ( − ∞ , 0 ] → [ 0 , + ∞ ) is continuous, b : ( 0 , 1 ) → ( − ∞ , + ∞ ) is Lebesgue integrable. Without any nonnegative assumption on nonlinearity, by choosing a proper cone and creating an available integral operator, the existence of the positive solutions for the above singular Sturm–Liouville boundary value problem is established. Some examples are worked out to indicate the application of our main result.

Positive solutions of fourth-order nonlinear singular boundary value problems

We consider the existence of positive solutions for the following fourth-order singular Sturm–Liouville boundary value problem: { 1 p ( t ) ( p ( t ) u ‴ ( t ) ) ′ − g ( t ) F ( t , u ) = 0 , 0 < t < 1 , α 1 u ( 0 ) − β 1 u ′ ( 0 ) = 0 , γ 1 u ( 1 ) + δ 1 u ′ ( 1 ) = 0 , α 2 u ″ ( 0 ) − β 2 lim t → 0 + p ( t ) u ‴ ( t ) = 0 , γ 2 u ″ ( 1 ) + δ 2 lim t → 1 − p ( t ) u ‴ ( t ) = 0 , where g , p may be singular at t = 0 and/or 1. Moreover F ( t , x ) may also have singularity at x = 0 . The existence and multiplicity theorems of positive solutions for the fourth-order singular Sturm–Liouville boundary value problem are obtained by using the first eigenvalue of the corresponding linear problems. Our results significantly extend and improve many known results including singular and nonsingular cases.

Existence of positive solutions for second-order semipositone differential equations on the half-line

In this paper, by means of constructing a special cone, we obtain a sufficient condition for the existence of positive solution for semipositone singular Sturm-Liouville boundary value problem on the half-line. The results obtained generalize and improve the corresponding results of Zhang and Liu in papers [X.G. Zhang, L.S. Liu, Positive solutions of superlinear semipositone singular Dirichlet boundary value problems, J. Math. Anal. Appl. 316 (2006) 525–537, Y.S. Liu, Existence and unboundedness of positive solutions for singular boundary value problems on half-line, Appl. Math. Comput. 144 (2003) 543–556].

A dissipative singular Sturm–Liouville problem with a spectral parameter in the boundary condition

In the Hilbert space L w 2 ( a , b ) , we consider nonselfadjoint singular Sturm–Liouville boundary value problem (with two singular end points a and b) in limit-circle cases at a and b, and with a spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Sturm–Liouville equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and Sturm–Liouville boundary value problem.

Waves in an Inhomogeneous Tube with a Flowing Two-Phase Fluid

Results of theoretical and mathematical justification of the problem on a pulsating flow of a two-phase barotropic bubbly fluid enclosed in an elastic semi-infinite cylindrical tube inhomogeneous along its length are presented. Linear one-dimensional equations are used. It is assumed that the tube is rigidly attached to the surrounding medium and therefore its displacement in the axial direction is absent. At infinity, the tube material is assumed to be homogeneous. To describe the pressure, flow rate, and displacement of the fluid, a pulsating pressure is given at the tube end. The problem stated is reduced to a singular Sturm-Liouville boundary-value problem, which in turn is reduced to a Volterra-type integral equation. This equation is solved by the method of successive approximations. By assuming that the corresponding potential is integrable, it is proved that these approximations converge to the exact solution of the problem. It is shown that this assumption also covers the very important practical case of piecewise inhomogeneity. For numerical realization, we consider a homogeneous tube with flowing water containing a small amount of bubbles. The effect of the volume content of bubbles on wave characteristics is revealed. In particular, it is stated that, for the oscillation regime selected, an increased bubble volume content decreases the wave velocity and considerably increases the flow speed (rate).

An inversion theorem for integral transforms related to singular Sturm-Liouville problems on the half line

We extend the classical theory of singular Sturm-Liouville boundary value problems on the half line, as developed by Titshmarsh and Levitan to generalized functions in order to obtain a general approach to handle many integral transforms, such as the sine, cosine, Weber, Hankel, and the K-transforms, in a unified way. This approach will lead to an inversion formula that holds in the sense of generalized functions. More precisely, for λ∈[0,∞) and 0≤α<∞, let ϕ(x,λ) be a solution of the Sturm-Liouville equation $$\frac{{d^2 y}}{{dx^2 }} - q(x)y = - \lambda y, y(0) = \sin \alpha , y'(0) = - \cos \alpha , \leqq x < \infty .$$ We define a test-function space ∈ A such that for each λ∈[0,∞), ϕ(.,λ)∈ A and hence for f∈ A*, we define the ϕ-transform of f by F(λ)= 〈f(x),ϕ(x,λ)〉. This paper studies properties of the ϕ-transform of f, in particular its inversion formula.

Uniqueness of positive solutions for Sturm-Liouville boundary value problems

Sufficient conditions for the uniqueness of positive solutions of singular Sturm-Liouville boundary value problems (BVP) { ( E ) ( | u ′ | m − 2 u ′ ) ′ + f ( t , u , u ′ ) = 0 , in ( θ 1 , θ 2 ) , m ≥ 2 , ( B C ) { α 1 u ( θ 1 ) − β 1 u ′ ( θ 1 ) = 0 , α 2 u ( θ 2 ) + β 2 u ′ ( θ 2 ) = 0 , \begin{equation*} \begin {cases} (\mathrm E) (|u’|^{m-2}u’)’+f(t,u,u’)=0,\quad \text {in} (\theta _1,\theta _2),m\ge 2,\ (\mathrm {BC})\begin {cases} \alpha _1u(\theta _1)-\beta _1u’(\theta _1)=0,\ \alpha _2u(\theta _2)+\beta _2u’(\theta _2)=0, \end{cases} \end{cases} \tag {BVP} \end{equation*} where α i , β i ≥ 0 \alpha _i,\beta _i\ge 0 and α i 2 + β i 2 ≠ 0 \alpha _i^2+\beta _i^2\not =0 ( i = 1 , 2 ) (i=1,2) , are established.

Sinc function computation of the eigenvalues of Sturm-Liouville problems

A collocation scheme using sine basis functions is developed to approximate the eigenvalues of regular and singular Sturm-Liouville boundary value problems. The error in the approximation of the eigenvalues is shown to converge at the rate exp(− α √ N) ( α > 0), where 2 N + 1 basis elements in the collocation scheme are used. A number of test examples are included (both finite and infinite interval boundary value problems) to indicate the accuracy and demonstrate the implementation of the method.

Regularizing Transformations for Certain Singular Sturm–Liouville Boundary Value Problems

It is shown that certain singular Sturm–Liouville boundary value problems can be transformed into regular problems by a simple transformation of the dependent variable.