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Assume [Formula: see text] is a planar domain, and [Formula: see text] is a locally bounded distributional solution to the elliptic equation [Formula: see text] where [Formula: see text] is a constant, [Formula: see text] and [Formula: see text] are real analytic functions defined on [Formula: see text] and the real line [Formula: see text], respectively. We establish asymptotic expansions of [Formula: see text] to arbitrary orders near [Formula: see text], which complements the recent results of Han–Li–Li on the Yamabe equation, Guo–Li–Wanon the weighted Yamabe equation, and partly extends that of Guo–Wan–Yang on the Liouville equation in a punctured disc. Our method is a combination of a priori estimate and mathematical induction.

We study asymptotic behavior of singular solutions of $$\Delta u+e^u=0$$ in $$B \backslash \{0\}$$ , where $$B=\{x \in {\mathbb {R}}^2: \; |x|-2$$ such that $$\begin{aligned} u(x)=\alpha _0 \ln |x|+O(1) \;\; \text{ as } |x| \rightarrow 0. \end{aligned}$$ We will establish asymptotic expansions up to arbitrary orders of u(x) near the isolated singular point $$x=0$$ . This provides the “sharp” results for the asymptotic behavior of u(x) in some sense.

Rigid plastic material models are suitable for modeling metal forming processes at large strains where elastic effects are negligible. A distinguished feature of many models of this class is that the velocity field is describable by non-differentiable functions in the vicinity of certain friction surfaces. Such solution behavior causes difficulty with numerical solutions. On the other hand, it is useful for describing some material behavior near the friction surfaces. The exact asymptotic representation of singular solution behavior near the friction surface depends on constitutive equations and certain conditions at the friction surface. The present paper focuses on a particular boundary value problem for anisotropic material obeying Hill&rsquo, s quadratic yield criterion under axial symmetry. This boundary value problem represents the deformation mode that appears in the vicinity of frictional interfaces in a class of problems. In this respect, the applied aspect of the boundary value problem is not essential, but the exact mathematical analysis can occur without relaxing the original system of equations and boundary conditions. We show that some strain rate and spin components follow an inverse square rule near the friction surface. An essential difference from the available analysis under plane strain conditions is that the system of equations is not hyperbolic.

We show using the Beylkin-Coifman-Rokhlin algorithm in the Haar basis that any singular integral operator can be written as the sum of a bounded operator on $L^p$, $1, Comment: Change of title. New abstract and new introduction

In the last decade, pseudospectral methods have become popular for solving optimal control problems. Pseudospectral methods do not need prior knowledge about the optimal control structure and are thus very flexible for problems with complex path constraints, which are common in optimal train control, or train trajectory optimization. Practical optimal train control problems are nonsmooth with discontinuities in the dynamic equations and path constraints corresponding to gradients and speed limits varying along the track. Moreover, optimal train control problems typically include singular solutions with a vanishing Hessian of the associated Hamiltonian. These characteristics make these problems hard to solve and also lead to convergence issues in pseudospectral methods. We propose a computational framework that connects pseudospectral methods with Pontryagin's Maximum Principle allowing flexible computations, verification and validation of the numerical approximations, and improvements of the continuous solution accuracy. We apply the framework to two basic problems in optimal train control: minimum-time train control and energy-efficient train control, and consider cases with short-distance regional trains and long-distance intercity trains for various scenarios including varying gradients, speed limits, and scheduled running time supplements. The framework confirms the flexibility of the pseudospectral method with regards to state, control and mixed algebraic inequality path constraints, and is able to identify conditions that lead to inconsistencies between the necessary optimality conditions and the numerical approximations of the states, costates, and controls. A new approach is proposed to correct the discrete approximations by incorporating implicit equations from the optimality conditions. In particular, the issue of oscillations in the singular solution for energy-efficient driving as computed by the pseudospectral method has been solved.

We establish the nonexistence of nontrivial ancient solutions to the nonlinear heat equation $u_t=\Delta u+|u|^{p-1}u$ which are smaller in absolute value than the self-similar radial singular steady state, provided that the exponent $p$ is strictly between Serrin's exponent and that of Joseph and Lundgren. This result was previously established by Fila and Yanagida [Tohoku Math. J. (2011)] by using forward self-similar solutions as barriers. In contrast, we apply a sweeping argument with a family of time independent weak supersolutions. Our approach naturally lends itself to yield an analogous Liouville type result for the steady state problem in higher dimensions. In fact, in the case of the critical Sobolev exponent we show the validity of our results for solutions that are smaller in absolute value than a 'Delaunay'-type singular solution., Comment: In this third version, we clarified that the approach of Fila and Yanagida [Tohoku Math. J. (2011)] also works in the subcritical regime

We study the well posedness of the nonlinear Schrodinger (NLS) equation with a point interaction and power nonlinearity in dimension two and three. Behind the autonomous interest of the problem, this is a model of the evolution of so called singular solutions that are well known in the analysis of semilinear elliptic equations. We show that the Cauchy problem for the NLS considered enjoys local existence and uniqueness of strong (operator domain) solutions, and that the solutions depend continuously from initial data. In dimension two well posedness holds for any power nonlinearity and global existence is proved for powers below the cubic. In dimension three local and global well posedness are restricted to low powers.

One can associate several residue-type indices to a singular point of a two-dimensional holomorphic vector field. Some of these indices depend also on the choice of a separatrix at the singular point. We establish some relations between them, especially when the singular point is a generalized curve and the separatrix is the maximal one. These local results have global consequences, for example concerning the construction of logarithmic forms defining a given holomorphic foliation.

In this article, we are concerned with radial solutions for the best constant of the Cafarelli-Kohn-Nirenberg inequality. Firstly, we classify the radial solutions according to its asymptotic behavior as $r \to 0$ and $r \to \infty$. Secondly, we investigate the structure of radial singular solutions. Lastly, we briefly discuss the Neumann problem related to the Cafarelli-Kohn-Nirenberg inequality.

We consider the natural time-dependent fractional $p$-Laplacian equation posed in the whole Euclidean space, with parameter $12$ has been dealt with in a recent paper. We study here the "fast" regime $1, Comment: 71 pages, 3 figures. Improvement of version 1: Subsection 2.6 added, rest of paper corrected for typos, small changes performed, reference added. arXiv admin note: text overlap with arXiv:2004.05799

We consider the exact singular solution of chordal Loewner equation and investigate a sequence of vertical slits $$\gamma ^p$$ , where $$p=4n+1$$ , $$n\in \mathbb {N}$$ . Our main result is to give an exact expression of the driving function $$\lambda $$ , and its Holder exponent near 0 in terms of p, which lies in (1/2, 1] and has a natural connection with the known results.

We study singular radial solutions of the semilinear elliptic equation Δu+f(u)=0 on finite balls in RN with N≥3. We assume that f satisfies either f(u)=up+o(up) with p>(N+2)/(N−2) or f(u)=eu+o(eu) as u→∞. We provide the existence and uniqueness of the singular radial solution, and show the convergence of regular radial solutions to the singular solution. Some applications to the bifurcation diagram of an elliptic Dirichlet problem are also given. Our results generalize and improve some known results in the literature., The first author was supported by JSPS KAKENHI Grant Number JP16K05225 and the second author was supported by JSPS KAKENHI Grant Number 17K05333. This work was also supported by Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.

We study singular radially symmetric solution to the Lin-Ni-Takagi equation for a supercritical power non-linearity in dimension $N\geq 3$. It is shown that for any ball and any $k \geq 0$, there is a singular solution that satisfies Neumann boundary condition and oscillates at least $k$ times around the constant equilibrium. Moreover, we show that the Morse index of the singular solution is finite or infinite if the exponent is respectively larger or smaller than the Joseph--Lundgren exponent., 16 pages

In this paper, we explore the DNA dynamic equation arising in the oscillator-chain named as Peyrard–Bishop model for abundant solitary wave solutions. The aforesaid model is studied for the first time via a novel fractional derivative operator. A modified traveling wave transformation with the aforementioned operator is used to transform the space-time fractional Peyrard–Bishop model into an ODE. Then, the three prolific schemes namely the simplest equation method, the Kudryashov method and the modified $$(G'/G^2)$$ -expansion method are employed. These methods contribute a variety of exact soliton solutions including the bright, dark and singular solution. Also the obtained solutions are verified through symbolic soft computations. Furthermore, some results are also explained through numerical simulations that show the novelty of our work as compared to the existing literature about the classical Peyrard–Bishop model.

The stability and axisymmetric deformation of two immiscible, viscous, perfect or leaky dielectric fluids confined in the annulus between two concentric cylinders are studied in the presence of radial electric fields. The fields are set up by imposing a constant voltage potential difference between the inner and outer cylinders. We derive a set of equations for the interface in the long-wavelength approximation which retains the essential physics of the system and allows for interfacial deformations to be as large as the annular gap hence accounting for possible touchdown at the inner or outer electrode. The effects of the electric parameters are evaluated initially by performing a linear stability analysis which shows excellent agreement with the linear theory of the full axisymmetric problem in the appropriate long wavelength regime. The nonlinear interfacial dynamics are investigated by carrying out direct numerical simulations of the derived long wave models, both in the absence and presence of electric fields. For non-electrified thin layer flows (i.e. one of the layers thin relative to the other) the long-time dynamics agree with the lubrication approximation results found in literature. When the liquid layers have comparable thickness our results demonstrate the existence of both finite time and infinite time singularities (asymptotic touching solutions) in the system. It is shown that a two-side touching solution is possible for both the non-electrified and perfect dielectric cases, while only one-side touching is found in the case of leaky dielectric liquids, where the flattened interface shape resembles the pattern solutions found in literature. Meanwhile the finite-time singular solution agrees qualitatively with the experiments of~\cite{Reynolds1965}., 30 pages, 8 figures

We construct singular solutions to the Yamabe equation using a reduction of the problem in an equivariant setting. This provides a non-trivial geometric example for which the analysis is simpler than in Mazzeo-Pacard program. Our construction provides also a non-trivial example of a weak solution to the Yamabe problem involving an equation with (smooth) coefficients., arXiv admin note: text overlap with arXiv:1911.11891

Our Methodology is to construct using a “trivial” solution to massive gravitons, and a nonsingular start for expansion of the universe. Our methodology has many unintended consequences, not the least is a relationship between a small time step, t, the minimum scale factor and even the tension or property values of the initial space-time wall, and that is a consequence of a “trivial” solution taking into account “massive” gravitons. I.e. this solution has a mass term times the partial derivative with respect to time of an expression in brackets. The expression in brackets is the cube of a scale factor minus the square of the scale factor. Bonus that this equation is set to zero. It is deemed trivial due to the insistence of having a singular solution. If that is dropped, we have a different venue. In addition, the Friedman equation for nonsingular cosmology can have a quadratic dependence upon a density (of space-time), leading to a way to incorporate right at the surface of the initial “space-time” bubble an uncertainty principle. From there we suggest a first principle Schrodinger equation, with the caveat that time does not exist, within the space-time nonsingular bubble, but is formed right afterwards. From there we again form solutions for strength of GW signals and suggestions as to polarization states. Our quest is motivated by our last articles question, where “We conclude by stating the following question. Can extra dimensions come from a Multiverse feed into Pre-Planckian space-time? See Theorem at the end of this publication. Our answer is in the affirmative, and it has intellectual similarities to George Chapline’s work with Black hole physics”. From there we next will in future articles postulate conditions for experimental detectors for subsequent data sets to obtain falsifiable data sets.

We consider the non-stationary problem on perturbation of a plane liquid crystal layer by electric field created by charges on the capacitor plates. The plates are disposed periodically on upper and lower sides along the liquid crystal layer. When solving the Laplace equation for electric potential in the exterior of the liquid crystal (in vacuum), the method of straight lines is applied. The equation for the potential describing the non-uniform distribution of the dielectric permittivity tensor inside the liquid crystal layer is solved iteratively using the fast Fourier transform. To take into account the singularities of the electric potential caused by a sharp change in boundary conditions at the ends of the capacitor plates, a special solution of the problem with homogeneous boundary conditions is constructed. Gluing the singular solution with the numerical solution, we improve the accuracy of calculating the forces and couple force which are included in the right-hand sides of the governing equations of motion for the liquid crystal.

The Scaled Boundary Finite Element Method (SBFEM) is a technique in which approximation spaces are constructed using a semi-analytical approach. They are based on partitions of the computational domain by polygonal/polyhedral subregions, where the shape functions approximate local Dirichlet problems with piecewise polynomial trace data. Using this operator adaptation approach, and by imposing a starlike scaling requirement on the subregions, the representation of local SBFEM shape functions in radial and surface directions is obtained from eigenvalues and eigenfunctions of an ODE system, whose coefficients are determined by the element geometry and the trace polynomial spaces. The aim of this paper is to derive a priori error estimates for SBFEM’s solutions of harmonic test problems. For that, the SBFEM spaces are characterized in the context of Duffy’s approximations for which a gradient-orthogonality constraint is imposed. As a consequence, the scaled boundary functions are gradient-orthogonal to any function in Duffy’s spaces vanishing at the mesh skeleton, a mimetic version of a well-known property valid for harmonic functions. This orthogonality property is applied to provide a priori SBFEM error estimates in terms of known finite element interpolant errors of the exact solution. Similarities with virtual harmonic approximations are also explored for the understanding of SBFEM convergence properties. Numerical experiments with 2D and 3D polytopal meshes confirm optimal SBFEM convergence rates for two test problems with smooth solutions. Attention is also paid to the approximation of a point singular solution by using SBFEM close to the singularity and finite element approximations elsewhere, revealing optimal accuracy rates of standard regular contexts.

For $n\ge 3$, $00$. As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation $u_t=\frac{n-1}{m}\Delta u^m$ in $(\mathbb{R}^n\setminus\{0\})\times (0,\infty)$ with initial value $u_0$ satisfying $f_{\lambda_1}(x)\le u_0(x)\le f_{\lambda_2}(x)$, $\forall x\in\mathbb{R}^n\setminus\{0\}$, which satisfies $U_{\lambda_1}(x,t)\le u(x,t)\le U_{\lambda_2}(x,t)$, $\forall x\in \mathbb{R}^n\setminus\{0\}, t\ge 0$, for some constants $\lambda_1>\lambda_2>0$. We also prove the asymptotic behaviour of such singular solution $u$ of the fast diffusion equation as $t\to\infty$ when $n=3,4$ and $\frac{n-2}{n+2}\le m0$ under appropriate conditions on the initial value $u_0$., Comment: 34 pages

We answer affirmatively a question of Aviles posed in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted $L^\infty$ spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an \emph{Ansatz} for the Yamabe problem with a prescribed singular set of maximal dimension $(n-2)/2$, for which, using the same machinery, we provide an alternative construction to the one given by Pacard. His linear theory uses $L^p$-theory on manifolds, while our approach studies the equations in the ambient space and is therefore suitable for generalization to nonlocal problems. In a forthcoming paper, we will prove analogous results in the fractional setting., 24 pages

The analysis of singular solutions in optimal control problems is addressed. The case of systems with two inputs is investigated characterising all the possible combination of singular arcs and constant boundary values. It is described the extension to a two input system of a previously proposed procedure for computing the control along the singular arcs in a state feedback form for one input dynamics. The procedure makes use of the possibility of computation in an analytical form of the costate as a function of the state. The example of a SIRC epidemic model is used to verify the effectiveness of the result.

A singular perturbation analysis for the Rolie-Poly model is presented for the limit of large relaxation time. The model describes entangled linear polymer melts and time scales for a Rouse time and reptation time. First, the case of the Rouse time approaching zero is named the nonstretching model, and the addition of a solvent model gives the small parameter to be the ratio of the retardation time to the slow reptation time. This analysis is instructive in formulating the perturbation analysis for the full Rolie-Poly model without the solvent, where the small parameter is the ratio of Rouse time to reptation time. The resulting formulae elucidate the characteristic features of thixotropic yield stress fluids, including yield stress hysteresis, delayed yielding and long term persistence of a decreased viscosity after cessation of flow. Two fundamental initial value problems are solved: startup of shear flow and cessation of shear flow. The scalings lead naturally to simplified dynamical systems that describe regimes of fast, slow and yielded dynamics, and show how the combination of these regimes can be used to describe the entire flow evolution.

We study positive singular solutions to the Dirichlet problem for the semilinear elliptic equation Δ u + λ f ( u ) = 0 in the unit ball on R N . We assume that f has the form f ( u ) = u p + g ( u ) , where p > ( N + 2 ) / ( N − 2 ) and g ( u ) is a lower order term. We first show the uniqueness of the singular solution to the problem, and then study the existence of the singular extremal solution. In particular, we show a necessary and sufficient condition for the existence of the singular extremal solution in terms of the weak eigenvalue of the linearized problem.

Multi-linear functionals or tensors are useful in study and analysis multi-dimensional signal and system. Tensor approximation, which has various applications in signal processing and system theory, can be achieved by generalizing the notion of singular values and singular vectors of matrices to tensor. In this paper, we showed local convergence of a parallelizable numerical method (based on the Jacobi iteration) for obtaining the singular values and singular vectors of a tensor.

A problem of the solutions singularity for applied problems is discussed. It is proposed to qualify such solutions as formal mathematical results that arise from the discrepancy between the mathematical and physical models of the phenomenon or object being studied. As examples, we consider the singular solution of the Schwarzschild problem in the general theory of relativity (serving as the mathematical basis for the existence of objects called the Black Holes), the solution of the mathematical physics problem for a circular membrane loaded in the center by a concentrated force, and the solution for the problems of the theory of elasticity about a cylindrical punch and an expandable plate with a crack. A generalization of the classical definition for a function and its derivative is proposed. This generalization makes it possible to obtain regular solutions of traditional singular problems.

This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of induced regular SSEs, generated by the same difference expression on smaller finite intervals. It is shown that every SSE of the minimal subspace has a pure discrete spectrum, and the k-th eigenvalue of any given SSE is exactly the limit of the k-th eigenvalues of the induced regular SSEs; that is, spectral exactness holds, in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, in the limit point and intermediate cases, spectral inclusive holds.

This paper investigates the problem of reachable set estimation for a class of uncertain singular systems with time-varying delays from a new point of view. Our consideration is centered on the design of a proportional-derivative state feedback controller (PDSFC) such that the considered singular system is robustly normalizable and all the states of the closed-loop system can be contained by a bounded set under zero initial conditions. First, a nominal singular time-delay system is considered and sufficient conditions are obtained in terms of matrix inequalities for the existence of a PDSFC and an ellipsoid. In this case, the considered system is guaranteed to be normalizable and the reachable set of the closed-loop systems is contained by the ellipsoid. Then, the result is extended to the case of singular time-delay systems with polytopic uncertainties and relaxed conditions are derived by introducing some weighting matrix variables. Furthermore, based on the obtained results, the reachable set of the considered closed-loop singular system can be contained in a prescribed ellipsoid. Finally, the effectiveness of our results are demonstrated by two numerical examples.

In this paper, an efficient method based on the simplified reproducing kernel method (SRKM) and least squares method (LSM) is proposed for solving nonlinear singular boundary value problems. The algorithm consists of two steps. First, we apply SRKM to solve a linear equation which contains a set of parameters, second the LSM is used to find the optimal parameters. Error estimation and convergence order for the presented method are also discussed. Our numerical experiments validate our theoretical findings and demonstrate the performance of the algorithm in terms of simplicity, accuracy, and efficiency.

In this paper, we prove that the singular integral defined byis bounded on Lp() for 1 < p < ∞ and is of weak type (1,1), where Ω ∈ L log+ L() and , with a ∈ L∞() satisfying some restricted conditions.

The main purpose of this article is to propose a modified singular boundary method using the modified fundamental solution of Helmholtz equation for simulation of three-dimensional high frequency acoustic wave problems. Compared with the standard second-order discretization methods which usually need to place more than 10–12 grid points in one wavelength per direction, the newly proposed modified singular boundary method only needs 2–3 source points in one wavelength of each direction to produce the accurate solutions with relative error around 1E − 3 level which satisfies most application requirements. It is observed that the present algorithm has similar condition number to the boundary element method and can hereby be solved efficiently by the iterative solver. By adopting the range restricted generalized minimal residual algorithm iterative solver, numerical experiments with 194,400 source points have successfully been achieved on a single laptop for three-dimensional high frequency acoustic wave problems with up to wavenumber 440.

In this paper, we give the possible orders of growth of solutions to certain linear differential equations near a finite singular point. For that, an asymptotic equality of Wiman–Valiron type is proved for the derivatives of functions which are analytic in the closed complex plane except at a finite singular point.

This paper investigates the problem of finite-time dissipative control for a class of time-delayed singular Markovian jump systems (SMJSs) with time-varying transition rates by using a quantized approach. To begin with, sufficient conditions on singular stochastic finite-time admissible are obtained for the family of SMJS with time-varying transition rates. Following this, the results are extended to singular stochastic dissipative finite-time admissible of SMJS with time-varying transition rates. Then state feedback controllers are designed to ensure the singular stochastic finite-time admissible and the singular stochastic dissipative finite-time admissible of the underlying closed-loop SMJS in the forms of strict linear matrix inequalities (LMIs). Ultimately, numerical examples are used to indicate the correctness and effectiveness of the proposed methods.

By variational methods, we study the existence of multiplicity of solutions for singular quasilinear Schrodinger equations.

In this paper, we prove the existence of a ground state solution for a class of elliptic problem on R N with the nonlinearity containing both singular and exponential subcritical terms, we get some results without the A–R condition.

We study a second-order ordinary differential equation coming from the Kepler problem on . The forcing term under consideration is a piecewise constant with singular nonlinearity that changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of T-periodic solutions.