Singular Boundary Research Articles

Solving singular boundary value problems using higher-order compact finite difference schemes with a novel higher-order implementation of Robin boundary conditions

This study introduces a novel and very accurate method for implementing Robin boundary conditions while solving boundary value problems with compact finite difference schemes. Most studies numerically approximate Neumann and Robin boundary conditions with a first-order approximation, however, this reduces the accuracy in general. Our implementation produces highly accurate results in solving nonlinear singular boundary value problems. Other reported results achieved utilizing various techniques are used for comparison. Tables and graphs showing the results are given.

Kinetic models for semiflexible polymers in a half-plane

Based on a general discrete model for a semiflexible polymer chain, we introduce a formal derivation of a kinetic equation for semiflexible polymers in the half-plane via a continuum limit. It turns out that the resulting equation is the kinetic Fokker-Planck-type equation with the Laplace-Beltrami operator under a non-local trapping boundary condition. We then study the well-posedness and the long-chain asymptotics of the solutions of the resulting equation. In particular, we prove that there exists a unique measure-valued solution for the corresponding boundary value problem. In addition, we prove that the equation is hypoelliptic and the solutions are locally H\"older continuous near the singular boundary. Finally, we provide the asymptotic behaviors of the solutions for large polymer chains.

A Novel Coupled Meshless Model for Simulation of Acoustic Wave Propagation in Infinite Domain Containing Multiple Heterogeneous Media

This study presents a novel coupled meshless model for simulating acoustic wave propagation in heterogeneous media, based on the singular boundary method (SBM) and Kansa’s method (KS). In the proposed approach, the SBM was used to model the homogeneous part of the propagation domain, while KS was employed to model a heterogeneity. The interface compatibility conditions associated with velocities and pressures were imposed to couple the two methods. The proposed SBM–KS coupled approach combines the respective advantages of the SBM and KS. The SBM is especially suitable for solving external sound field problems, while KS is attractive for nonlinear problems in bounded non-homogeneous media. Moreover, the new methodology completely avoids grid generation and numerical integration compared with the finite element method and boundary element method. Numerical experiments verified the accuracy and effectiveness of the proposed scheme.

Accurate Laguerre collocation solutions to a class of Emden–Fowler type BVP

We solve numerically the nonlinear and double singular boundary value problem formed by the well known Emden–Fowler equation , s > 1 along with the boundary conditions and In order to capture the exponential decrease of its solution we use the Laguerre–Gauss–Radau collocation method and infer its convergence. We show that the value of u ʹ at origin, which plays a fundamental role in these problems, definitely satisfies some rigorous accepted bounds. A particular attention is paid to the Thomas–Fermi case, i.e. We treat the problems as boundary values ones without any involvement of ones with initial values. The method is robust with respect to scaling and order of approximation.

Existence theory for multiple solutions to second-order singular Dirichlet boundary value problem modeling the Antarctic Circumpolar Current

In the article, we present multiple solutions for a second-order singular Dirichlet boundary value problem that arises when modeling the ocean flow of the Antarctic Circumpolar Current. The main tools of the proof are the Leray–Schauder nonlinear alternative principle and a well-known fixed point theorem in cones.

Numerical Approximation of the Quenching Time for One-Dimensional p-Laplacian with Singular Boundary Flux

This paper concerns the study of the numerical approximation for a discrete non-newtonian filtration system with nonlinear boundary conditions. We find some conditions under which the solution of a discrete form of above problem quenches in a finite time and estimate its discrete quenching time. We also establish the convergence of the discrete quenching time to the theoretical one when the mesh size tends to zero.

Pointwise error estimates and local superconvergence of Jacobi expansions

As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of | x − a | |x-a| (with | a | > 1 |a|>1 ) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about 95 95% range of [ − 1 , 1 ] [-1,1] except for a small neighbourhood near the singular point x = a . x=a. In this paper, we rigorously show that the Jacobi expansion for a more general class of Φ \Phi -functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired log ⁡ n \log n -factor in the pointwise error estimate for the Legendre expansion recently stated in Babus̆ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748–773] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.

Existence of positive solutions for singular p-Laplacian Hadamard fractional differential equations with the derivative term contained in the nonlinear term

In this paper, based on the properties of Green function and the eigenvalue of a corresponding linear operator, the existence of positive solutions is investigated by spectral analysis for a infinite-points singular p-Laplacian Hadamard fractional differential equation boundary value problem, and an example is given to demonstrate the validity of our main results.

Nonlinear dynamic analysis of a stochastic delay wheelset system

Considering not only the stochastic track irregularity and the possible effect of stochastic parameter excitation, but also the time delay of spring, that is, its response is in place, but the force generated is not in place, a stochastic delay wheelset system is established. The infinite dimensional system is reduced to the finite dimensional stochastic differential equation by using the center manifold, and further reduced to a one-dimensional diffusion process by using the stochastic averaging method. The stability of the wheelset system is obtained by analyzing the singular boundary theory and calculating the maximum Lyapunov exponent. The conditions and types of stochastic bifurcation in wheelset system are obtained by combining probability density function. The numerical simulation verifies the correctness of the theoretical analysis and shows that the time delay affects the critical hunting instability speed of wheelset. The stochastic term affects the lateral displacement of the stochastic delay wheelset system.

Impact of correlated Gaussian colored noises on stability and stationary probability density for the randomly forced two-species competitive Gompertz model

In this paper, random factors described by correlated Gaussian colored noises are introduced into a two-species competition model with Gompertz growth. Firstly, stochastic stability is studied with the help of singular boundary theory. After that, the stationary probability density of the limiting system is given explicitly and then the extrema of invariant measure are discussed in detail. Finally, the average time of system escaping from the attraction domain is calculated using mean first passage time, and noise enhanced stability is observed. Our results show that random factors are more likely to make the system lose its original stability. Instead, the sample trajectories will stay near the maximum of the stationary probability density for a long time. Importantly, noise intensity and correlation degree are destructive, which can expand the distribution range of population density, weaken the stability of the system, and then easily lead to ecological imbalance. Conversely, the role of correlation time and proper competition between species is constructive and conducive to coexistence.

Modified 2.5D singular boundary methods to deal with spurious eigensolutions in exterior acoustic problems

This paper studies the problem of spurious eigensolutions in the context of the singular boundary method (SBM) formulated in the two-and-a-half-dimensional (2.5D) domain and proposes two numerical schemes to overcome this numerical difficulty. The SBM can be seen as a version of the method of fundamental solutions where the source points are located at the physical boundary. Similar to other boundary-type discretization schemes, the SBM also encounters the non-uniqueness problem at the vicinity of the eigensolutions of the corresponding interior problem. In the 2.5D domain framework, the so-called fictitious eigenfrequencies appearing in 3D problems arise in the form of spurious dispersion curves associated with propagation modes of the corresponding interior problem. The two enhanced 2.5D SBM approaches proposed in this work, based on the Burton–Miller method in one case and the dual surface method in the other, are designed to filter out the spurious eigenvalues from the simulation results and deliver accurate solutions along the wavenumber-frequency spectrum. Three benchmark examples including the radiation problems of an infinitely long cylinder under Dirichlet and Neumann boundary conditions and the radiation problem of a longitudinally infinite object with a constant star-like cross section subjected to a Dirichlet boundary condition are considered to study the proposed methods. The results demonstrate the capability of the proposed numerical schemes to successfully avoid the non-uniqueness problem when the 2.5D SBM is employed.

A 2.5D automatic FEM-SBM method for the evaluation of free-field vibrations induced by underground railway infrastructures

This paper presents an efficient method to predict underground railway-induced vibrations. The method uses the finite element method (FEM) to model the railway tunnel structure and the singular boundary method (SBM) to model the wave propagation in the surrounding soil. The FEM mesh and the distribution of SBM collocation points at the tunnel/soil interface are generated using an automatic meshing strategy. The presented method is one of the main components of VIBWAY, a user-friendly prediction tool to address railway-induced vibration problems. This paper presents three calculation examples in which the soil response due to forces applied on the tunnel structure are computed in terms of transfer functions. The results obtained for each one of the calculation examples are compared with those computed using a model based on a 2.5D FEM-BEM approach. The presented comparisons show that the proposed approach is a suitable strategy for predicting underground railway-induced vibrations, both in terms of accuracy and computational efficiency. Moreover, the use of an automatic meshing strategy and the SBM formulation not only eases the implementation of the approach but it also makes it easier to use, which is one of the key features of the VIBWAY tool.

A high-order numerical method for solving nonlinear derivative-dependent singular boundary value problems using trigonometric B-spline basis function

A high-order numerical method for solving nonlinear derivative-dependent singular boundary value problems using trigonometric B-spline basis function

Existence and Uniqueness of Positive Solutions to a Class of Singular Integral Boundary Value Problems of Fractional Order

Existence and Uniqueness of Positive Solutions to a Class of Singular Integral Boundary Value Problems of Fractional Order

Existence of positive solutions for singular fractional boundary value problems with p-Laplacian

In this paper, we obtain the existence of positive solutions for the singular fractional boundary value problem with p-Laplacian. The existence of positive solutions is established using the Avery-Peterson fixed point theorem. In addition, we include an example for the demonstration of our main result.

Existence of solutions for a delay singular high order fractional boundary value problem with sign-changing nonlinearity

This paper consider the existence of at least one positive solution of a Riemann-Liouville fractional delay singular boundary value problem with sign-changing nonlinerty. To establish sufficient conditions we use the Guo-Krasnosel?skii fixed point theorem.

A sinc-collocation approximation solution for strongly nonlinear class of weakly singular two-point boundary value problems

In this study, an efficient collocation method based on Sinc function coupled with double exponential transformation is developed. This approach is used for solving a class of strongly nonlinear regular or weekly singular two-point BVPs with homogeneous or non homogeneous boundary conditions. The properties of the Sinc-collocation scheme were used to reduce the computations of the problem to the nonlinear system of equations. To use the Newton method in solving the nonlinear system, its vectormatrix form was obtained. The convergence analysis of the method is discussed. The analysis show that the method is convergent exponential. In order to investigate the capability and accuracy of the method, it is applied to solve several existing problems chosen from the open literature. The numerical results compared with other existing methods. The obtained results indicate high capacity and rapid convergence of the proposed method.

A Modified Formulation of Singular Boundary Method for Exterior Acoustics

This paper proposes a modified formulation of the singular boundary method (SBM) by introducing the combined Helmholtz integral equation formulation (CHIEF) and the self-regularization technique to exterior acoustics. In the SBM, the concept of the origin intensity factor (OIF) is introduced to avoid the singularities of the fundamental solutions. The SBM belongs to the meshless boundary collocation methods. The additional use of the CHIEF scheme and the self-regularization technique in the SBM guarantees the unique solution of the exterior acoustics accurately and efficiently. Consequently, by using the SBM coupled with the CHIEF scheme and the self-regularization technique, the accuracy of the numerical solution can be improved, especially near the corresponding internal characteristic frequencies. Several numerical examples of two-dimensional and three-dimensional benchmark examples about exterior acoustics are used to verify the effectiveness and accuracy of the proposed method. The proposed numerical results are compared with the analytical solutions and the solutions obtained by the other numerical methods.

Bernstein operational matrix of differentiation and collocation approach for a class of three-point singular BVPs: error estimate and convergence analysis

Singular boundary value problems (BVPs) have widespread applications in the field of engineering, chemical science, astrophysics and mathematical biology. Finding an approximate solution to a problem with both singularity and non-linearity is highly challenging. The goal of the current study is to establish a numerical approach for dealing with problems involving three-point boundary conditions. The Bernstein polynomials and collocation nodes of a domain are used for developing the proposed numerical approach. The straightforward mathematical formulation and easy to code, makes the proposed numerical method accessible and adaptable for the researchers working in the field of engineering and sciences. The priori error estimate and convergence analysis are carried out to affirm the viability of the proposed method. Various examples are considered and worked out in order to illustrate its applicability and effectiveness. The results demonstrate excellent accuracy and efficiency compared to the other existing methods.

The bounds of eigenvalue for complex singular boundary value problems

The bounds of eigenvalue for complex singular boundary value problems