Weak Singularity Research Articles

Local H1-norm error analysis of a mixed finite element method for a time-fractional biharmonic equation

In this work, a time-fractional biharmonic equation with a Caputo derivative of fractional order α∈(0,1) is considered, whose solutions exhibit a weak singularity at initial time t=0. For this problem, a system of two second-order differential equations is derived by introducing a intermediate variable p=−Δu, then discretised the system using the standard finite element method in space together with the L1 discretisation of Caputo derivative on graded mesh in time. The H1-norm stability result of the method is established, and then a sharp H1-norm local convergent result is presented. Finally, numerical experiments are provided to further verify our theoretical analysis for each fixed value of α.

BEM analysis of multilayer thin structures using a composite transformation method for boundary integrals

In this paper, a composite transformation method is developed to evaluate the singular and nearly singular boundary integrals in the boundary element analysis for multilayer thin structures. The composite transformation method is implemented as follows: Firstly, a sinh transformation method is developed to eliminate the influence of nearly singular integrals in the multilayer thin-structural problem; and then a complex transformation method is improved to deal with the weakly singular integrals. The complex transformation formulations of weakly singular integrals are derived from the composition of the (α, β) coordinate transformation and the sinh transformation in β direction. With this method for weakly singular integrals, the weak singularity in the radial direction and the potentially near singularity in the circumferential direction can be removed. Finally, the proposed transformation formulations of singular and nearly singular integrals are used to analyze multi-domain elasticity problem for multilayer thin structures. The numerical results for several examples are presented to demonstrate the computational accuracy of the proposed method.

Weak Singularities of the Isothermal Entropy Change as the Smoking Gun Evidence of Phase Transitions of Mixed-Spin Ising Model on a Decorated Square Lattice in Transverse Field.

The magnetocaloric response of the mixed spin-1/2 and spin-S () Ising model on a decorated square lattice is thoroughly examined in presence of the transverse magnetic field within the generalized decoration-iteration transformation, which provides an exact mapping relation with an effective spin-1/2 Ising model on a square lattice in a zero magnetic field. Temperature dependencies of the entropy and isothermal entropy change exhibit an outstanding singular behavior in a close neighborhood of temperature-driven continuous phase transitions, which can be additionally tuned by the applied transverse magnetic field. While temperature variations of the entropy display in proximity of the critical temperature a striking energy-type singularity , two analogous weak singularities can be encountered in the temperature dependence of the isothermal entropy change. The basic magnetocaloric measurement of the isothermal entropy change may accordingly afford the smoking gun evidence of continuous phase transitions. It is shown that the investigated model predominantly displays the conventional magnetocaloric effect with exception of a small range of moderate temperatures, which contrarily promotes the inverse magnetocaloric effect. It turns out that the temperature range inherent to the inverse magnetocaloric effect is gradually suppressed upon increasing of the spin magnitude S.

Numerical Solution of Integral-Algebraic Equations with a Weak Boundary Singularity by k-step Methods

Numerical Solution of Integral-Algebraic Equations with a Weak Boundary Singularity by k-step Methods

Formula omitted]-robust [formula omitted]-norm error estimate of nonuniform Alikhanov scheme for fractional sub-diffusion equation

The sharp H1-norm error estimate of a finite difference method for two-dimensional time-fractional diffusion equation is established, where the Caputo time-fractional derivative term is approximated by Alikhanov scheme on graded mesh. Under reasonable assumption that the exact solution has typical weak singularity at initial time, the temporal convergence order of the fully scheme is O(N−min{rα,2}) and the error bound does not blow up when α approximates 1−. The theoretical analysis utilizes an improved discrete fractional Grönwall inequality, and the correctness of the theoretical result is verified by numerical experiments.

Periodic bouncing solutions of the Lazer–Solimini equation with weak repulsive singularity

We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be cataloged according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincaré–Birkhoff Theorem.

Second-Order and Nonuniform Time-Stepping Schemes for Time Fractional Evolution Equations with Time–Space Dependent Coefficients

The numerical analysis of time fractional evolution equations with the second-order elliptic operator including general time–space dependent variable coefficients is challenging, especially when the classical weak initial singularities are taken into account. In this paper, we introduce a concise technique to construct efficient time-stepping schemes with variable time step sizes for two-dimensional time fractional sub-diffusion and diffusion-wave equations with general time–space dependent variable coefficients. By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems. For the diffusion-wave problem, our scheme is constructed by employing the recently established symmetric fractional-order reduction method. The unconditional stability of proposed schemes is rigorously discussed under mild assumptions on variable coefficients and, based on reasonable regularity assumptions and weak time mesh restrictions, the second-order convergence is obtained with respect to discrete \(H^1\)-norm. Numerical experiments are given to demonstrate the theoretical statements.

Efficient Temporal Third/Fourth-Order Finite Element Method for a Time-Fractional Mobile/Immobile Transport Equation with Smooth and Nonsmooth Data.

In recent years, the numerical theory of fractional models has received more and more attention from researchers, due to the broad and important applications in materials and mechanics, anomalous diffusion processes and other physical phenomena. In this paper, we propose two efficient finite element schemes based on convolution quadrature for solving the time-fractional mobile/immobile transport equation with the smooth and nonsmooth data. In order to deal with the weak singularity of solution near , we choose suitable corrections for the derived schemes to restore the third/fourth-order accuracy in time. Error estimates of the two fully discrete schemes are presented with respect to data regularity. Numerical examples are given to illustrate the effectiveness of the schemes.

On the Behavior of the Spectrum of a Perturbed Steklov Boundary Value Problem with a Weak Singularity

On the Behavior of the Spectrum of a Perturbed Steklov Boundary Value Problem with a Weak Singularity

Numerical analysis of the spectrum for the highly oscillatory integral equation with weak singularity

In this paper, we focus on the spectra numerical computation for the integral equation with the absolute oscillation and power-law or logarithmic singularity. Finite section method is applied to transform the integral equation to an algebraic eigenvalue problem. The entries of the coefficient matrix appearing in the bivariate highly oscillatory singular integrals can be represented explicitly in Gamma or the exponential integral functions. The decay rate of the entries is established to construct the truncation scheme. Then the infinite algebraic eigenvalue problem can be simplified to be the finite one. The corresponding error of the infinite and finite algebraic systems is also bounded. Finally, the numerical experiments are provided to illustrate the theoretical analysis.

Quantum improved charged black holes

We consider quantum effects of gravitational and electromagnetic fields in spherically symmetric black hole spacetimes in the asymptotic safety scenario. Introducing both the running gravitational and electromagnetic couplings from the renormalization group equations and applying a physically sensible scale identification scheme based on the Kretschmann scalar, we construct a quantum mechanically corrected, or quantum improved Reissner-Nordstrom metric. We study the global structure of the quantum improved geometry and show, in particular, that the central singularity is resolved, being generally replaced with a regular Minkowski-core, where the curvature tensor vanishes. Exploring cases with more general scale identifications, we further find that the space of quantum improved geometries is divided into two regions: one for geometries with a regular Minkowski-core and the other for those with a weak singularity at the center. At the boundary of the two regions, the geometry has either Minkowski-, de Sitter-, or anti-de Sitter(AdS)-core.

Analysis and discretization of a variable-order fractional wave equation

We analyze a variable-order time-fractional wave equation, which models, e.g., the vibration of a membrane in a viscoelastic environment. We prove that the solutions to the variable-order ordinary differential equations in the spectral decomposition of the solution to the fractional wave equation exhibit power-law decaying characteristics and overcome the difficulty that its solution operator does not have an exponential decay in contrast to its variable-order fractional diffusion analogue.We prove an optimal-order error estimate of a numerical discretization of the variable-order fractional wave equation only under regularity assumptions of the data of the model but with no smoothness assumption of its solution. As the solution exhibits initial weak singularity, the local truncation error is suboptimal. A conventional analysis gives a suboptimal-order estimate. We develop a new technique to derive the desired optimal-order convergence rate. We also conduct numerical experiments to substantiate the mathematically proved findings.

Pointwise error estimate of an alternating direction implicit difference scheme for two-dimensional time-fractional diffusion equation

An alternating direction implicit (ADI) difference method is adopted to solve the two-dimensional time-fractional diffusion equation with Dirichlet boundary condition whose solution has some weak singularity at initial time. L1 scheme on uniform mesh is used to discretize the temporal Caputo fractional derivative. Pointwise-in-time error estimate is given for the fully discrete ADI scheme, where the error bound does not blowup when α (the order of fractional derivative) approaches 1−. It is shown both in theoretically and numerically that the temporal convergence order of the ADI scheme is O(τ2α+τtnα−1) at time t=tn; hence the scheme is globally O(τα) accurate in temporal direction, but it is O(τmin⁡{2α,1}) when t is away from 0.

Couette Resistance Due to a Sliding Plate Over a Plate With Stripes of Nonhomogeneous Slip

Abstract Couette flow with nonhomogeneous partial-slip stripes on one plate is studied. Drag and flowrate are found by an efficient eigenfunction expansion and point match method. Longitudinal motion (parallel to the stripes) experiences lower drag than transverse motion. As the gap width between the two plates approaches zero, the drag increases to a finite value if the stripes have partial slip, as comparison to the infinite value for no slip. Analysis of the region near the junction of a perfect stick-slip boundary shows a weak stress singularity while there is no singularity for partial slip junctions.

Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

Regularization of Weak and Basic Singularities in Fractional Integro-Differential Equations

Regularization of Weak and Basic Singularities in Fractional Integro-Differential Equations

An α-robust finite difference method for a time-fractional radially symmetric diffusion problem

A time-fractional diffusion problem is considered on a spherically symmetric domain in Rd for d=1,2,3. The solution of such a problem is shown in general to have a weak singularity near the initial time t=0, and bounds on certain derivatives of this solution are derived. To solve the problem numerically, spatial polar coordinates are used; a finite difference method is constructed on a mesh that is graded in time and spherical in space. The discretisation uses the L1 scheme in time and a polar-coordinate discretisation of the diffusion term. Its convergence is analysed and error bounds are derived that are robust in α, the order of the time derivative, as α→1−. Numerical experiments show that our results are sharp.

Comparative analysis on the blow-up occurrence of solutions to Hadamard type fractional differential systems

The main intention of this paper is to deal with the weak singularity of solutions to some nonlinear Hadamard type fractional differential systems (HTFDSs) which could be viewed as the generalization of classic Hadamard fractional settings. Resorting to the lower and upper solutions technique and constructing the compatible weighted Banach space, the completely continuous operator described by Hadamard type fractional versions as well as a sufficient criterion for the existence of blow-up solutions to some nonlinear HTFDSs is established. Additionally, the comparative analysis on the blow-up rate affected by critical system parameters is also presented, and several examples clearly illustrate the effectiveness and efficiency of the proposed results.

Boundary-integral representation sans waterline integral for flows around ships steadily advancing in calm water

The boundary-integral flow-representation associated with the boundary-value problem, commonly called Neumann–Kelvin (NK) problem, that corresponds to linear potential flow around a ship steadily advancing in calm water involves an integral around the mean waterline of the ship. This ‘waterline integral’ is a notorious source of numerical difficulties and has been extensively studied. The waterline integral in the NK theory is largely – but not fully – eliminated in the modification, called Neumann–Michell (NM) theory, of the NK theory. Specifically, the NM theory includes a residual waterline-distribution of weak Rankine singularities, ignored in practical applications. A crucial element of the NM theory is a mathematical transformation that is based on a vector Green function, which is associated with the common scalar Green function used in the NK theory. This transformation is revisited in the present study. A rigorous analysis yields an answer to a fifty-year old puzzle: an exact boundary-integral flow representation that does not include a waterline integral. A remarkable feature of this new flow representation, which is a modification of the NM flow representation given previously, is that it explicitly determines the flow potential and the flow velocity at a ship hull surface in terms of the flow velocity at the hull surface, rather than in terms of the hull-surface potential as in usual boundary-integral flow representations obtained via Green’s classical identity.

A boundary weak singularity elimination method for multilayer structures

In the numerical discretization for the models of multilayer thin structures, the elements with narrow strip shape may be appeared. When using the previous methods to integrate these elements, the potentially near singularity will arise in the circumferential direction. In this paper, a weak singularity elimination method for multilayer structures of 3D boundary element method is presented to accurately calculate weakly singular integrals. Its numerical implementation is divided into the following two parts: (1) (α, β) coordinate transformation is applied to deal with the weakly singular integral arising in boundary integral equation and separate out the integral in the circumferential direction (β direction); (2) Based on the integral form of β, a corresponding sinh transformation method is developed to eliminate the near singularity in β direction. Finally, the proposed transformation formulation is extended to multi-domain potential problem and applied to the boundary analysis for multilayer structures. Several numerical examples are given to verify the computational accuracy of the proposed method. And the numerical results are promising and encouraging.