Singular Boundary Research Articles

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

Forecast analysis and sliding mode control on a stochastic epidemic model with alertness and vaccination

In this paper, a stochastic SEIR epidemic model is studied with alertness and vaccination. The goal is to stabilize the infectious disease system quickly. The dynamic behavior of the model is analyzed and an integral sliding mode controller with distributed compensation is designed. By using Lyapunov function method, the sufficient conditions for the existence and uniqueness of global positive solutions and the existence of ergodic stationary distributions are obtained. The stochastic center manifold and stochastic average method are used to simplify the system into a one-dimensional Markov diffusion process. The stochastic stability and Hopf bifurcation are analyzed using singular boundary theory. An integral sliding mode controller with non-parallel distributed compensation is designed by linear matrix inequality (LMI) method, which realizes the stability of system and prevents the outbreak of epidemic disease. The correction of theoretical analysis and the effectiveness of controller are validated using numerical simulation performed in MATLAB/Simulink.

NEW APPROACH BASED ON COLLOCATION AND SHIFTED CHEBYSHEV POLYNOMIALS FOR A CLASS OF THREE-POINT SINGULAR BVPS

In the recent decades, variety of real-life problems arises in astrophysics have been mimic using the class of three-point singular boundary value problems (BVPs). Finding an effective and accurate approach for a class of three-point BVPs is still a difficult problem, though. The goal of this paper is to design a numerical strategy for approximating a class of three-point singular boundary value problems using the collocation technique and shifted Chebyshev polynomials. Utilizing shifted Chebyshev polynomials, the problem is reduced to a matrix form, which is then converted into a system of nonlinear algebraic equations by employing the collocation points. The key advantages of the new approach are (a) it is a straightforward mathematical formulation, which makes it effortless to code, and (b) it is easily adaptable to solve various classes of three-point singular boundary value problems. The convergence analysis is carried out to ensure the viability of the proposed scheme. Various examples are considered and tested in order to illustrate its applicability and efficiency. The results show excellent accuracy and efficiency compared to the other existing methods.

A fixed point iterative scheme based on Green's function for numerical solutions of singular BVPs

<abstract><p>We suggest a novel iterative scheme for solutions of singular boundary value problems (SBVPs) that is obtained by embedding Green's function into the Picard-Mann Hybrid (PMH) iterative scheme. This new scheme we call PMH-Green's iterative scheme and prove its convergence towards a sought solution of certain SBVPs. We impose possible mild conditions on the operator or on the parameters involved in our scheme to obtain our main outcome. After this, we prove that this new iterative scheme is weak $ w^{2} $-stable. Eventually, using two different numerical examples of SBVPs, we show that our new approach suggests highly accurate numerical solutions as compared the corresponding Picard-Green's and Mann-Green's iterative schemes.</p></abstract>

An ACA-BM-SBM for 2D acoustic sensitivity analysis

<abstract> <p>In this paper, we present a novel computational approach (named ACA-BM-SBM) for the calculation of 2D acoustic sensitivity by combining the Burton-Miller-type singular boundary method (BM-SBM) with the adaptive cross-approximation (ACA) algorithm. The BM-SBM circumvents the source singularities of the fundamental solutions by introducing the origin intensity factors, and it eliminates the fictitious frequency problem in external acoustic fields by introducing the Burton-Miller formula. As a semi-analysis meshless method, the BM-SBM can accurately solve the external acoustic problem governed by the Helmholtz equation. Nevertheless, the computational inefficiency introduced by the dense coefficient matrix renders this method suboptimal, particularly for large-scale simulations. As the number of nodes increases, the computation time and store memory increase dramatically. ACA is a purely algebraic method based on hierarchical matrices which can be used to partition the coefficient matrix step by step. By employing ACA, the BM-SBM can be effectively accelerated, and this results in less computation time, as well as fewer memory requirements. Numerical experiments, including Dirichlet and Neumann boundary conditions, illustrate that the proposed approach is an accurate, efficient and fast numerical method for acoustic sensitivity analysis.</p> </abstract>

Trace and boundary singularities of positive solutions of a class of quasilinear equations

<p style='text-indent:20px;'>We study positive functions satisfying (E)<inline-formula><tex-math id="M1">\begin{document}$ \;-\Delta u+m|\nabla u|^q-u^p = 0 $\end{document}</tex-math></inline-formula> in a domain <inline-formula><tex-math id="M2">\begin{document}$ {\Omega} $\end{document}</tex-math></inline-formula> or in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^{_N}_+ $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M4">\begin{document}$ p>1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ 1<q<2 $\end{document}</tex-math></inline-formula>. We give sufficient conditions for the existence of a solution to (E) with a nonnegative measure <inline-formula><tex-math id="M6">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> as boundary data; these conditions are expressed in terms of Bessel capacities on the boundary. We also study removability of boundary singular sets, and solutions with an isolated singularity on <inline-formula><tex-math id="M7">\begin{document}$ \partial\Omega $\end{document}</tex-math></inline-formula>. The different results depend on two critical exponents for <inline-formula><tex-math id="M8">\begin{document}$ p = p_c: = \frac{N+1}{N-1} $\end{document}</tex-math></inline-formula> and for <inline-formula><tex-math id="M9">\begin{document}$ q = q_c: = \frac{N+1}{N} $\end{document}</tex-math></inline-formula>, and on the sign of <inline-formula><tex-math id="M10">\begin{document}$ q-\frac{2p}{p+1} $\end{document}</tex-math></inline-formula></p>

A reliable algorithm for a class of singular nonlinear two-point boundary value problems arising in physiology

In this paper, we present a reliable numerical algorithm to determine approximate solutions of the two-point boundary value problems having Robin boundary conditions that naturally occur in the investigation of distinct tumor growth issues, the dispersal of heat sources in the person head and steady state oxygen diffusion in spherical cell possessing Michaelis–Menten uptake kinetics. This approach is based on a modified concept of Adomian polynomials (AP), and the two-step Adomian decomposition method (TSADM) merged with Padé approximants. Furthermore a Maple package ADMP is applied to solve various problems, which is very easy to use and efficient and needed to input the system of equations with initial or boundary conditions and diverse essential parameters to deliver the analytic approximate solutions within a few seconds. The suggested scheme does not require linearization, perturbations, guessing the initial terms, a set of basis function or other limiting presumptions, which yields the solutions in closed form. Many examples are examined to make clear the scope and validity of the package ADMP.

Quenching estimates for a non-Newtonian filtration equation with singular boundary conditions

This study concerns with the quenching features of solutions of the non-Newtonian filtration equation. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined are certain cases. Numerical experiments are provided to illustrate and provide additional validation of the theoretical predictions to the quenching rates and times.

Existence and nonexistence results for a class of non‐self‐adjoint fourth‐order singular boundary value problems arising in real life

In this work, we propose the following fourth‐order non‐self‐adjoint SBVPs to investigate where is the parameter, and with respect to three different homogeneous boundary conditions. By using monotone iterative technique we show the existence of at least one solution, which depends on the parameter . We report presence of two solutions computationally. We show the sign of the both solutions and derive the estimations of the parameter which indicates the region of nonexistence. To validate the estimations of we propose an iterative technique with the help of Greens function. We place the numerical results as an evidence of our derived theoretical results.

Singular boundary behaviour and large solutions for fractional elliptic equations.

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equationsposed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equationwith zero boundarydata. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equationsdriven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Greenfunction.

Numerical approximation of higher order singular boundary value problem by using Haar functions

Numerical approximation of higher order singular boundary value problem by using Haar functions

Numerical Estimation of a Mode III Fracture Mechanics Parameter for a Three-Dimensional V-Notch on a Steel Bold

In this work a couple of three-dimensional problems in the domain of linear elastic Fracture Mechanics are examined. These are problems of solid bodies (they could be steel bolds or rivets) with a surface crack singularity (V-notch). They are reduced to Laplace equation problems by considering a Lamé potential. The boundary singularity is numerically treated as per the singular function boundary integral method (SFBIM), which in the literature is known as one of the so-called Trefftz methods. Thus, the general solution of the governing equation, in the vicinity of the surface crack, is expressed as an asymptotic expansion, the coefficients of which are approximated by polynomials. The remaining numerical steps are followed according to this method with which very fast convergence and very high accuracy are observed. In fact, the CPU time and the numerical error recorded with this numerical technique are significantly smaller than those achieved with the finite element method (FEM) which was also used to solve the same problems. The calculated value of Mode III Fracture Mechanics parameter (FMP) indicates that there is no danger of crack propagation. Thus, the extension of the method to this category of problems is considered as a novel application of this algorithm in Fracture Mechanics. Keywords: Mode III Fracture Mechanics parameter, crack singularity, governing equation, singular

Triple-Positive Solutions for a Nonlinear Singular Fractional q-Difference Equation at Resonance

Fractional q-calculus plays an extremely important role in mathematics and physics. In this paper, we aim to investigate the existence of triple-positive solutions for nonlinear singular fractional q-difference equation boundary value problems at resonance by means of the fixed-point index theorem and the q-Laplace transform, where the nonlinearity f(t,u,v) permits singularities at t=0,1 and u=v=0. The obtained theorem is well illustrated with the aid of an example.

Slender body theories for rotating filaments

Slender fibres are ubiquitous in biology, physics and engineering, with prominent examples including bacterial flagella and cytoskeletal fibres. In this setting, slender body theories (SBTs), which give the resistance on the fibre asymptotically in its slenderness $\epsilon$ , are useful tools for both analysis and computations. However, a difficulty arises when accounting for twist and cross-sectional rotation: because the angular velocity of a filament can vary depending on the order of magnitude of the applied torque, asymptotic theories must give accurate results for rotational dynamics over a range of angular velocities. In this paper, we first survey the challenges in applying existing SBTs, which are based on either singularity or full boundary integral representations, to rotating filaments, showing in particular that they fail to consistently treat rotation–translation coupling in curved filaments. We then provide an alternative approach which approximates the three-dimensional dynamics via a one-dimensional line integral of Rotne–Prager–Yamakawa regularized singularities. While unable to accurately resolve the flow field near the filament, this approach gives a grand mobility with symmetric rotation–translation and translation–rotation coupling, making it applicable to a broad range of angular velocities. To restore fidelity to the three-dimensional filament geometry, we use our regularized singularity model to inform a simple empirical equation which relates the mean force and torque along the filament centreline to the translational and rotational velocity of the cross-section. The single unknown coefficient in the model is estimated numerically from three-dimensional boundary integral calculations on a rotating, curved filament.

A study on singular boundary integrals and stability of 3D time domain boundary element method

The paper presents a novel numerical scheme to effectively solve the weak, strong, and hyper-singular time-space integrals in the 3D time domain boundary element method (BEM). To deal with the singularities of P- or S-wavefront elements, the element subdivision method is used to identify non-zero valued sub-elements on the wavefronts, and the non-zero valued sub-elements containing source points are solved by the power series expansion method, which achieves the effect of obtaining satisfactory accuracy with fewer Gaussian points. In the present scheme, the non-uniform rational B-Spline (NURBS) basis functions are used to determine the physical coordinates of each sub-element with very high precision. Numerical results show that the proposed scheme can effectively implement a high-precision singular time-space integral solution under a unified framework. In addition, this paper uses the time-weighting method to improve the time integration algorithm to solve the instability problem. The validity and accuracy of the algorithm are verified by dynamics examples of the finite elastic body and infinite elastic body, and the effects of different weight points and integration schemes in the time-weighting method on the stability are discussed. The effectiveness of the proposed algorithm for large-scale elastodynamics problems is also examined using the 27 spherical cavities problem.

Transient Dynamic Response Analysis of Two-Dimensional Saturated Soil with Singular Boundary Method

In this paper, the singular boundary method (SBM) in conjunction with the exponential window method (EWM) is firstly extended to simulate the transient dynamic response of two-dimensional saturated soil. The frequency-domain (Fourier space) governing equations of Biot theory is solved by the SBM with a linear combination of the fundamental solutions. In order to avoid the perplexing fictitious boundary in the method of fundamental solution (MFS), the SBM places the source point on the physical boundary and eliminates the source singularity of the fundamental solution via the origin intensity factors (OIFs). The EWM is carried out for the inverse Fourier transform, which transforms the frequency-domain solutions into the time-domain solutions. The accuracy and feasibility of the SBM-EWM are verified by three numerical examples. The numerical comparison between the MFS and SBM indicates that the SBM takes a quarter of the time taken by the MFS.

Boundary singularities in mean curvature flow and total curvature of minimal surface boundaries

For hypersurfaces moving by standard mean curvature flow with fixed boundary, we show that if a tangent flow at a boundary singularity is given by a smoothly embedded shrinker, then the shrinker must be non-orientable. We also show that there is an initially smooth surface in \bf{R}^3 that develops a boundary singularity for which the shrinker is smoothly embedded (and therefore non-orientable). Indeed, we show that there is a non-empty open set of such initial surfaces. Let \kappa be the largest number with the following property: if M is a minimal surface in \bf{R}^3 bounded by a smooth simple closed curve of total curvature < \kappa , then M is a disk. Examples show that \kappa<4\pi . In this paper, we use mean curvature flow to show that \kappa \ge 3\pi . We get a slightly larger lower bound for orientable surfaces.

Singular boundary method for band structure calculations of in-plane waves in 2D phononic crystals

In this study, the singular boundary method, which is a boundary collocation method, is first employed to calculate the band structures of in-plane waves in two-dimensional phononic crystals. When a unit cell of a phononic crystal is considered, the whole boundaries, including the continuity and periodic boundaries are discretized by the singular boundary method. Then, a linear eigenvalue equation is derived. For a given frequency, an eigenvalue that involves the Block wave can be calculated. Therefore, by sweeping the frequencies, the band structure can be obtained. The accuracy, efficiency and convergence of the proposed method are tested by several numerical examples, and the results demonstrate that the singular boundary method can provide stable and efficient results for band structure calculations of in-plane waves in two-dimensional phononic crystals.