Some singular third-order boundary value problems

In this paper, we consider some singular formally symmetric (self-adjoint) boundary value problems generated by a singular third-order differential expression and separated and coupled boundary conditions. In particular, we consider that the minimal symmetric operator generated by the third-order differential expression has the deficiency indices (3,3). We investigate same spectral properties related with these problems, and we introduce a method to find the resolvent operator.     

Uniformly third-order accurate TVNE interpolations

In this paper, we develop an approximation procedure for functionsof bounded variations. Our procedure cannot increase the variationof the function being approximated, and also avoids discontinuityof the interpolation function.     

Factorization strategies for third-order tensors

Operations with tensors, or multiway arrays, have become increasingly prevalent in recent years. Traditionally, tensors are represented or decomposed as a sum of rank-1 outer products using either the CANDECOMP/PARAFAC (CP) or the Tucker models, or some variation thereof. Such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array. The specifics of the approximate representation (such as how many terms to use in the sum, orthogonality constraints, etc.) depend on the application.In this paper, we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem. Reminiscent of matrix factorizations, we present a new factorization of a tensor as a product of tensors. To derive the new factorization, we define a closed multiplication operation between tensors. A major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications.Specifically, this new multiplication allows us to introduce concepts such as tensor transpose, inverse, and identity, which lead to the notion of an orthogonal tensor. The multiplication also gives rise to a linear operator, and the null space of the resulting operator is identified. We extend the concept of outer products of vectors to outer products of matrices. All derivations are presented for third-order tensors. However, they can be easily extended to the order-p(p>3) case. We conclude with an application in image deblurring.     

Nonlinear three-point third-order boundary value problems

In this paper we consider three-point boundary value problems for third-order differential equations. We provide simple sufficient conditions in order to obtain a priori bounds on solutions. We then rely on the topological transversality theorem of Granas or on Schaefer’s theorem to prove existence of solutions.     

On new third-order convergent iterative formulas

In this paper, we present new iteration methods with cubic convergence for solving nonlinear equations. The main advantage of the new methods are free from second derivatives and it permit that the first derivative is zero in some points. Analysis of efficiency shows that the new methods can compete with Newton’s method and the classical third-order methods. Numerical results indicate that the new methods are effective and have definite practical utility.      

Periodic solutions of a resonant third-order equation

We study the existence of periodic solutions for a third-order equation of resonant type. Under suitable conditions we prove the existence of at least one periodic solution of the problem applying Mawhin coincidence degree theory.     

Analysis of third-order methods for secular equations

Third-order numerical methods are analyzed for secular equations. These equations arise in several matrix problems and numerical linear algebra applications. A closer look at an existing method shows that it can be considered as a classical method for an equivalent problem. This not only leads to other third-order methods, it also provides the means for a unifying convergence analysis of these methods and for their comparisons. Finally, we consider approximated versions of the aforementioned methods.

     

Chaotic dynamics of a third-order Newton-type method

The dynamics of a classical third-order Newton-type iterative method is studied when it is applied to degrees two and three polynomials. The method is free of second derivatives which is the main limitation of the classical third-order iterative schemes for systems. Moreover, each iteration consists only in two steps of Newton's method having the same derivative. With these two properties the scheme becomes a real alternative to the classical Newton method. Affine conjugacy class of the method when is applied to a differentiable function is given. Chaotic dynamics have been investigated in several examples. Applying the root-finding method to a family of degree three polynomials, we have find a bifurcation diagram as those that appear in the bifurcation of the logistic map in the interval.     

A variant of Newton's method with accelerated third-order convergence

In the given method, we suggest an improvement to the iteration of Newton's method. Derivation of Newton's method involves an indefinite integral of the derivative of the function, and the relevant area is approximated by a rectangle. In the proposed scheme, we approximate this indefinite integral by a trapezoid instead of a rectangle, thereby reducing the error in the approximation. It is shown that the order of convergence of the new method is three, and computed results support this theory. Even though we have shown that the order of convergence is three, in several cases, computational order of convergence is even higher. For most of the functions we tested, the order of convergence in Newton's method was less than two and for our method, it was always close to three.     

Existence-uniqueness and iterative methods for third-order boundary value problems

In this paper we shall provide necessary and sufficient conditions for the existence and uniqueness of solutions of third-order nonlinear differential equations satisfying three-point boundary conditions. For the linear case, we propose a constructive method which is a variation of the method of chasing. For the nonlinear problems sufficient conditions are provided to ensure the convergence of a general class of iterative methods. Several examples are also included.     

Positive solutions for singular third-order nonhomogeneous boundary value problems

In this paper, we investigate the existence of positive solutions for singular third-order nonhomogeneous boundary value problems. By using a fixed point theorem of cone expansion-compression type due to Krasnosel??skii, we establish various results on the existence or nonexistence of single and multiple positive solutions to the singular boundary problems in the explicit intervals for the nonhomogeneous term. An example is also given to illustrate some of the main results.     

On a third-order multi-point boundary value problem at resonance

In this paper, we prove some existence results for a third order multi-point boundary value problem at resonance. Our method is based upon the coincidence degree theory of Mawhin.     

A simply constructed third-order modifications of Newton's method

In this paper, we present a simple and easily applicable approach to construct some third-order modifications of Newton's method for solving nonlinear equations. It is shown by way of illustration that existing third-order methods can be employed to construct new third-order iterative methods. The proposed approach is applied to the classical Chebyshev–Halley methods to derive their second-derivative-free variants. Numerical examples are given to support that the methods thus obtained can compete with known third-order methods.     

On third-order limiter functions for finite volume methods

In this article, we propose a finite volume limiter function for a reconstruction on the three-point stencil. Compared to classical limiter functions in the MUSCL framework, which yield 2^nd-order accuracy, the new limiter is 3^rd-order accurate for smooth solution. In an earlier work, such a 3^rd-order limiter function was proposed and showed successful results [2]. However, it came with unspecified parameters. We close this gap by giving information on these parameters.     

Oscillation criteria for third-order nonlinear differential equations

In this paper, we are concerned with the oscillation properties of the third order differential equation

Oscillatory criteria for third-order nonlinear difference equation with impulses

In this paper, we study the oscillation of a third-order nonlinear difference equation with impulses. Some sufficient conditions for the oscillatory behavior of solutions of third-order impulsive nonlinear difference equation are obtained. Some known results in the literature are generalized and improved.     

Existence and uniqueness of solutions for third-order nonlinear boundary value problems

In this paper, we present some general results of existence and uniqueness of solutions of nonlinear two-point boundary value problems for third-order nonlinear differential equations by using the Shooting method. As applications we give certain concrete sufficient conditions for the existence and uniqueness.     

Second- and third-order bias reduction for one-parameter family models

In this paper we derive second- and third-order bias-corrected maximum likelihood estimates in general uniparametric models. We compare the corrected estimates and the usual maximum likelihood estimate in terms of their mean squared errors. We also obtain closed-form expressions for bias-corrected estimates in one-parameter exponential family models. Our results cover many important and commonly used distributions. Simulation results are also given.     

Solvability for a third-order three-point BVP on time scales

We are concerned with the existence of a nontrivial solution to a nonlinear third-order three-point boundary-value problem on general time scales. Using the corresponding Green function, without a non-negative or monotone-type assumption on the nonlinearity, we obtain sufficient conditions for the existence of at least one nontrivial solution, using the Leray–Schauder nonlinear alternative theorem. This paper extends recent results in differential equations to difference equations, quantum equations, and general dynamic equations on time scales.     

Generalized double Wronskian solutions of the third-order isospectral AKNS equation

The generalized double Wronskian solutions of the third-order isospectral AKNS equation are obtained. Thus we found rational solutions, Matveev solutions, complexitons and interaction solutions. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction.