Nonlinear Sturm Liouville Research Articles

Solvability of a Class of Fractional Advection–Dispersion Coupled Systems

The purpose of this study is to provide some criteria for the existence and multiplicity of solutions for a class of fractional advection–dispersion coupled systems with nonlinear Sturm–Liouville conditions and instantaneous and non-instantaneous impulses. Specifically, the existence is derived through the Nehari manifold method, and the proof of multiplicity is based on Bonanno and Bisci’s critical point theorem, which does not require proof that the functional satisfies the Palais–Smale condition. Finally, to illustrate the obtained results, an example is provided.

Maximal regularity and two‐sided estimates of the approximation numbers of the nonlinear Sturm–Liouville equation solutions with rapidly oscillating coefficients in L2(ℝ)

A theorem on the maximum regularity of solutions of the nonlinear Sturm–Liouville equation with greatly growing and rapidly oscillating potential in the space is proved in this paper. Two‐sided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear Sturm–Liouville equation are also obtained. As is known, the obtained estimates give the opportunity to choose approximation apparatus that guarantees the minimum possible error.

Multiple solutions for nonlinear Sturm–Liouville differential equations with possibly negative variable coefficients

The aim of this paper is to study a parameter-dependent nonlinear differential problem with Sturm–Liouville type equation and Dirichlet boundary conditions. In particular, by requiring a suitable behavior on the nonlinearity, we determine an interval of parameter λ for which the problem admits three or infinitely many solutions.

Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels

<abstract><p>This article presents the Laplace-Adomian decomposition method (LADM), which produces a fast convergence series solution, for two types of nonlinear fractional Sturm-Liouville (SL) problems. The fractional derivatives are defined in the Caputo, conformable, Caputo-Fabrizio in the sense of Caputo (CFC), Caputo type Atangana-Baleanu (ABC) senses. With the help of this method, approximate solutions of the investigated problems were obtained. The solutions generated from the Caputo and ABC derivatives are represented by the Mittag-Leffler function, which is intrinsic to fractional derivatives, and the solution obtained using the conformable and CFC derivatives generate the hyperbolic sine and cosine functions. Thus, we derive some novel solutions for fractional-order versions of nonlinear SL equations. The fractional calculus provides more data than classical calculus and has been widely used in mathematical modeling with memory effect. Finally, we analyzed and compared these novel solutions of the considered problems by graphs under different values of $ p $, $ \lambda $ and different orders of $ \alpha $.</p></abstract>

On nonlinear discrete Sturm–Liouville boundary value problems with unbounded nonlinearities

This paper is devoted to the study of nonlinear Sturm–Liouville problems in the discrete setting. We consider problems for which nonlinearities in both the dynamics and boundary conditions may be unbounded. For such problems, we provide a criteria for the existence of solutions.

Bifurcation from zero or infinity in nonlinearizable Sturm–Liouville problems with indefinite weight

In this paper, we consider bifurcation from zero or infinity of nontrivial solutions of the nonlinear Sturm–Liouville problem with indefinite weight. This problem is mainly important because of it is related with a selection-migration model in genetic population. We show the existence of four families of unbounded continua of nontrivial solutions to this problem bifurcating from intervals of the line of trivial solutions or the line R × {∞} (these intervals are called bifurcation intervals). Moreover, these global continua have the usual nodal properties in some neighborhoods of bifurcation intervals.

Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation

It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L^{−1} have the following property (*) λn → 0, when n → ∞, moreover, than the faster convergence to zero so the operator L^{−1} is best approximated by finite rank operators. The following question: - Is it possible for a given nonlinear operator to indicate a decreasing numerical sequence characterized by the property (*)? naturally arises for nonlinear operators. In this paper, we study the above question for the nonlinear Sturm-Liouville operator. To solve the above problem the theorem on the maximum regularity of the solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space L2(R) (R = (−∞, ∞)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear SturmLiouville equation are also obtained. As is known, the obtained estimates of Kolmogorov widths give the opportunity to choose approximation apparatus that guarantees the minimum possible error.

Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to a non-linear system of equations, that we were able to solve it via the use of some iterative techniques, like Newton’s method. The other method under consideration is the VIM, where the VIM has been modified through the use of the Laplace transform, and another effective modification has also been made to the VIM by replacing the non-linear term in the integral equation resulting from the use of the well-known VIM with the Adomian’s polynomials. In order to explain the advantages of each method over the other, several issues have been studied, including one that has an application in the field of spectral theory. The results in solutions to these problems, which were included in tables, showed that the improved VIM is better than the Sinc method, while the Sinc method addresses some advantages over the VIM when dealing with singular problems.

Qualitative properties and global bifurcation of solutions for a singular boundary value problem

This paper deals with a singular, nonlinear Sturm–Liouville problem of the form {A(x)u 0 (x)} 0 + λu(x) = f(x, u(x), u 0 (x)) on (0, 1) where A is positive on (0, 1] but decays quadratically to zero as x approaches zero. This is the lowest level of degeneracy for which the problem exhibits behaviour radically different from the regular case. In this paper earlier results on the existence of bifurcation points are extended to yield global information about connected components of solutions.

Solving a nonlinear inverse Sturm–Liouville problem with nonlinear convective term using a boundary functional method

This article is concerned with solving nonlinear inverse problems to recover the second-order nonlinear Sturm–Liouville operators, probably including a nonlinear convective term, which refers to over-specified boundary data. Utilising a homogenization approach with the boundary data, a series of boundary functions are obtained and consist of a linear space combined with the zero element. The energetic functional both qualifying the homogeneous boundary conditions and maintaining the energy is presented in the linear space based on the energetic boundary functions. The multiplier can be decided by solving a nonlinear equation. The linear systems utilized in rebuilding the uncertain leading coefficient function as well as the potential function in the nonlinear Sturm–Liouville operator are improved, so that the iterative algorithms converge rapidly. Numerical examples provided indicate that the presented method is well capable of recovering the nonlinear Sturm–Liouville operators with a nonlinear convective term.

Finite Element Approximation of the Minimal Eigenvalue and the Corresponding Positive Eigenfunction of a Nonlinear Sturm—Liouville Problem

The problem of finding the minimal eigenvalue and the corresponding positive eigenfunction of the nonlinear Sturm—Liouville problem for the ordinary differential equation with coefficients nonlinear depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. A sufficient condition for the existence of a minimal eigenvalue and the corresponding positive eigenfunction of the nonlinear Sturm— Liouville problem is established. The original differential eigenvalue problem is approximated by the finite element method with Lagrangian finite elements of arbitrary order on a uniform grid. The error estimates of the approximate eigenvalue and the approximate positive eigenfunction to exact ones are proved. Investigations of this paper generalize well known results for the Sturm—Liouville problem with linear entrance on the spectral parameter.

On the existence of infinitely many eigenvalues in a nonlinear Sturm–Liouville problem arising in the theory of waveguides

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type on an interval with boundary conditions of the first kind. The problem describes the propagation of polarized electromagnetic waves in a plane two-layer dielectric waveguide. The cases of a homogeneous and an inhomogeneous medium are studied. The existence of infinitely many positive and negative eigenvalues is proved.

Positive solutions of sub-superlinear Sturm–Liouville problems

In this paper, we introduce the notion of a strict lower/upper solution to nonlinear Sturm–Liouville boundary value problems. Based on the maximum principles, we establish a result of Leray–Schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions. Applying the result and the fixed point index theory in cones, we obtain the global existence results of positive solutions for sub-superlinear Sturm–Liouville problems.

Bifurcation analysis of a singular nonlinear Sturm–Liouville equation

In this paper we study existence of positive solutions to the following singular nonlinear Sturm–Liouville equation [Formula: see text] where α > 0, p > 1 and λ are real constants. We prove that when 0 < α ≤ ½ and p > 1 or when ½ < α < 1 and [Formula: see text], there exists a branch of continuous positive solutions bifurcating to the left of the first eigenvalue of the operator ℒαu = -(x2αu′)′ under the boundary condition limx→0x2αu′(x) = 0. The projection of this branch onto its λ component is unbounded in two cases: when 0 < α ≤ ½ and p > 1, and when ½ < α < 1 and [Formula: see text]. On the other hand, when ½ < α < 1 and [Formula: see text], the projection of the branch has a positive lower bound below which no positive solution exists. When 0 < α < ½ and p > 1, we show that a second branch of continuous positive solution can be found to the left of the first eigenvalue of the operator ℒαunder the boundary condition limx→0u(x) = 0. Finally, when α ≥ 1, the operator ℒαhas no eigenvalues under its canonical boundary condition at the origin, and we prove that in fact there are no positive solutions to the equation, regardless of λ ∈ ℝ and p > 1.

Nonlinear Sturm-Liouville dynamic equation with a measure of noncompactness in Banach spaces

This paper is devoted to prove the existence of solutions of the nonlinear Sturm-Liouville boundary value problem on time scales in Banach spaces. We obtain the sufficient conditions for the existence of solutions in terms of Kuratowski measure of noncompactness. Mönch's fixed point theorem is used to prove the main result. By the unification property of time scales, our result is valid for Sturm-Liouville differential equations and difference equations, but more interestingly by the extension property, it is also valid for Sturm-Liouville $q$-difference equation.

Global bifurcation and nodal solutions for a Sturm–Liouville problem with a nonsmooth nonlinearity

In this paper, we shall establish the unilateral global bifurcation which bifurcates from the trivial solutions axis or from infinity of a class of nonlinear Sturm–Liouville problems with nondifferentiable nonlinearity, respectively. As applications of the above results, we shall determine the interval of r, in which there exist nodal solutions for the following problem{u″(t)+ra(t)f(u)=0,t∈(0,1),u(0)=u(1)=0, where a(t) is a positive continuous function on [0,1], f∈C(R,R) but is not necessarily differentiable at the origin and infinity. Moreover, as a special case of the above problem, we shall prove more details about the existence of nodal solutions for a class of half-linear eigenvalue problems.

Comparison of iterative methods by solving nonlinear Sturm-Liouville, Burgers and Navier-Stokes equations

Abstract In this manuscript the homotopy perturbation method, the new iterative method, and the variational iterative method have been successively used to obtain approximate analytical solutions of nonlinear Sturm-Liouville, Navier-Stokes and Burgers’ equations. It is shown that the homotopy perturbation method gives approximate analytical solution near to the exact one. We have illustrated the obtained results by sketching the graph of the solutions.

On the solvability of nonlinear Sturm–Liouville problems

The focus of this paper is the study of nonlinear differential equations subject to general non-local boundary conditions. To establish sufficient conditions for the existence of solutions, we use properties of the nonlinearities and their relationship with the eigenvalues of an associated linear Sturm–Liouville problem.

AN INVERTIBLE TRANSFORMATION AND SOME OF ITS APPLICATIONS

Several applications of an explicitly invertible transformation are reported. This transformation is elementary and therefore all the results obtained via it might be considered trivial; yet the findings highlighted in this paper are generally far from appearing trivial until the way they are obtained is revealed. Various contexts are considered: algebraic and Diophantine equations, nonlinear Sturm–Liouville problems, dynamical systems (with continuous and with discrete time), nonlinear partial differential equations, analytical geometry, functional equations. While this transformation, in one or another context, is certainly known to many, it does not seem to be as universally known as it deserves to be, for instance it is not routinely taught in basic University courses (to the best of our knowledge). The main purpose of this paper is to bring about a change in this respect; but we also hope that some of the findings reported herein — and the multitude of analogous findings easily obtainable via this technique — will be considered remarkable by the relevant experts, in spite of their elementary origin.

Global bifurcation and multiple results for Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville boundary value problem { ( L u ) ( t ) = λ a ( t ) f ( u ( t ) ) , 0 < t < 1 , R 1 ( u ) = α 1 u ( 0 ) + β 1 u ′ ( 0 ) = 0 , R 2 ( u ) = α 2 u ( 1 ) + β 2 u ′ ( 1 ) = 0 , where L is the linear Sturm–Liouville operator ( L u ) ( t ) = − ( p ( t ) u ′ ( t ) ) ′ + q ( t ) u ( t ) . We obtain a global bifurcation result for a related bifurcation problem. We then use this to obtain multiple (at least eight) solutions of the Sturm–Liouville problem having specified nodal properties.