Singular Higher Order Research Articles

Iterative unique positive solutions for a new class of nonlinear singular higher order fractional differential equations with mixed-type boundary value conditions

In this paper, we consider a new class of singular nonlinear higher order fractional boundary value problems supplemented with sum of Riemann–Stieltjes integral type and nonlocal infinite-point discrete type boundary conditions. The fractional derivative of different orders is involved in the nonlinear terms and boundary conditions, and the nonlinear terms are allowed to be singular in regard to not only time variable but also space variables. A unique positive solution is established by using the fixed point theorem of mixed monotone operator. In addition, some significant properties of the unique solution depending on the parameter λ are stated. In the end, two examples are worked out to illustrate our main results.

Existence of Solutions of Impulsive Lidstone BVPs of Singular Higher Order Fractional Differential Equations

We present a new general method for converting an impulsive fractional differential equation to an equivalent integral equation. By using this method and employing a fixed point theorem in Banach space, we establish existence results of solutions for two boundary value problems of impulsive singular higher order fractional differential equation. An example is presented to illustrate the efficiency of the results obtained.

Positive Periodic Solutions for Singular Higher Order Delay Differential Equations

In this article, we consider the singular higher order delay differential equations. Under certain assumptions, some new criteria for guaranteeing the existence of positive periodic solutions are presented. Moreover, two examples are provided to demonstrate the effectiveness and applications of the main results.

Existence and uniqueness of positive solutions for a class of n-th-order BVPs on timescales

This paper investigates conditions on the existence and uniqueness of solutions for a class of Sturm–Liouville type singular higher–order boundary value problems on timescales by using mixed monotone method.

Spectral properties of singular discrete linear Hamiltonian systems

This paper is concerned with spectral properties of singular discrete linear Hamiltonian systems. It is shown that properties of the essential spectrum of each self-adjoint subspace extension (SSE) of the corresponding minimal subspace are independent of the values of the coefficients of the system on any finite subinterval. The analyticity of the Weyl function is studied by employing the Schwarz reflection principle for the system in the limit point case. Based on the above work, several sufficient conditions are obtained for each SSE to have no essential spectrum points in an interval of the real line in the strong limit point case, and then a sufficient condition for the essential spectrum to be bounded from below (above) and a sufficient condition for the pure discrete spectrum are presented, respectively. As a direct consequence, the related spectral properties of singular higher order symmetric vector difference expressions are given.

Singular higher order boundary value problems for ordinary differential equations

We study singular boundary value problems for differential equations with left focal boundary conditions of the form,

Existence and Uniqueness of Solutions for Singular Higher Order Continuous and Discrete Boundary Value Problems

By mixed monotone method, the existence and uniqueness are established for singular higher-order continuous and discrete boundary value problems. The theorems obtained are very general and complement previous known results.

Nonoscillatory criteria for singular higher order differential equations

A new nonoscillatory theory is presented for higher order singular nonlinear equations. In particular our nonlinearity may be singular in both the dependent and independent variables.

Sturm–Liouville and Focal Higher Order BVPs with Singularities in Phase Variables

Abstract The paper deals with the existence of solutions for singular higher order differential equations with Sturm–Liouville or (𝑝, 𝑛 – 𝑝) right focal boundary conditions or the (𝑛 – 𝑝, 𝑝) left focal boundary conditions. Right-hand sides of differential equations may be singular in the zero values of all their phase variables. The proofs are based on the regularization and sequential techniques.

On the Hamilton–Jacobi Treatment of Constrained Systems

Systems with singular-higher order Lagrangians are investigated by two methods: Dirac method and Hamilton–Jacobi method. An example is studied and it is shown that the Hamilton–Jacobi method gives the correct canonical generalized equations of motion, contrary to Dirac method, where Dirac conjecture is invalid.

Canonical global symmetry and quantal conservation laws for a system with singular higher order Lagrangian

Starting from the phase-space generating functional of the Green function for a system with singular higher order Lagrangian, the generalized canonical Ward identities under the global symmetry transformation in phase space is deduced. The local transformation connected with this global symmetry transformation is studied, and the quantal conservation laws are obtained for such a system. We give a preliminary application to higher derivative Yang-Mills theory; a generalized quantal BRS conserved quantity is found.

A Numerical Solution for Viscoelastic Half Planes

The standard singular higher order transcendental functions, appearing in all the fundamental frequency domain response quantities of viscoelastic half planes, are identified and separated from the regular parts utilizing the notion of convolution algebra. The relevant mathematical properties of such logarithmic singularities have been investigated to recommend analytical differentiations and integrations in the space of distributions. Only the regular parts of the physical quantities are operated numerically. The foundation coefficients, both impedances and compliances, for the finite element analysis of soil-structure interaction problems, corresponding to any order of interpolation polynomial can be evaluated numerically using the algorithm developed here. Numerical results for the impedance coefficients and the foundation surface stress profiles associated with the finite element linear displacement field are illustrated as an example.