Strong Singularity Research Articles

A fundamental solution for radiofrequency heating due to a point electrical current in an infinite spatial domain: Numerical proofs with MATHEMATICA and ANSYS

Radiofrequency ablation (RFA) is a minimally invasive therapy widely used for the treatment of tumours. Mathematical modelling of RFA allows medical practitioners to understand the involved electro-thermal transport, thereby helping to improve the effectiveness of prescribed treatments. To this end, we aimed to deduce and numerically prove an analytical transient-time solution for temperature due to a point source of electrical current in an infinite medium. The solution, deduced with the Fourier transform, involves double integrals. The outer integral is a time integral, which was calculated with the trapezoidal rule. The inner integral is a generalised spatial integral with defects, which was computed using the MATHEMATICA command ‘NIntegrate’. For the radius r ≥ 0.088841 m, the computation with NIntegrate was straightforward and the results conformed well to those obtained with ANSYS. For r < 0.088841 m, NIntegrate diverged owing to the strong singularity at the defect. An algorithm was thus implemented to allow approximate calculation of the temperature at any distance from the current source with high precision. The analytical solution and the associated Green's function may be used as fundamental solutions in BEM for RFA modelling. MATHEMATICA notebooks used for calculations are provided as proofs.

Positive solutions for a class of elliptic equations

This paper is concerned with a class of semilinear elliptic equations with a potential function−Δu=λ|x|−αu−|x|σupinΩ∖{0}, where λ,σ∈R, α>0, p>1, and Ω⊂RN(N≥3) is a bounded smooth domain with 0∈Ω. We establish the existence, nonexistence and asymptotic behavior of positive solutions when the potential function |x|−α has strong singularity at the origin. When the potential function |x|−α has weak singularity at the origin, we obtain some qualitative properties of positive solutions.

Proposed Model of the Giant Thermal Hall Effect in Two-Dimensional Superconductors: An Extension to the Superconducting Fluctuation Regime.

We extend the thermodynamic approach for the description of the thermal Hall effect in two-dimensional superconductors above the critical temperature, where fluctuation Cooper pairs contribute to the conductivity, as well as in disordered normal metals where the particle-particle channel is important. We express the Hall heat conductivity in terms of the product of temperature derivatives of the chemical potential and of the magnetization of the system. Based on this general expression, we derive the analytical formalism that qualitatively reproduces the superlinear increase of the thermal Hall conductivity with the decrease of temperature observed in a large variety of experimentally studied systems [Grissonnanche etal., Nature (London) 571, 376 (2019)NATUAS0028-083610.1038/s41586-019-1375-0]. We also predict a nonmonotonic behavior of the thermal Hall conductivity in the regime of quantum fluctuations, in the vicinity of the second critical field and at very low temperatures.

On the Cutoff Approximation for the Boltzmann Equation with Long-Range Interaction

The Boltzmann collision operator for long-range interactions is usually employed in its “weak form” in the literature. However the weak form utilizes the symmetry property of the spherical integral and thus should be understood more or less in the principle value sense especially for strong angular singularity. To study the integral in the Lebesgue sense, it is natural to define the collision operator via the cutoff approximation. In this way, we give a rigorous proof to the local well-posedness of the Boltzmann equation with the long-range interactions. The result has the following main features and innovations: (1). The initial data is not necessarily a small perturbation around equilibrium but satisfies compatible conditions. (2). A quasi-linear method instead of the standard linearization method is used to prove existence and non-negativity of the solution in a suitably designed energy space depending heavily on the initial data. In such space, we derive the first uniqueness result for the equation in particular for hard potential case.

Dirichlet Problem for a Generalized Cauchy–Riemann Equation with a Supersingular Point on a Half-Plane

For equations with a Cauchy–Riemann operator involving a strong point singularity in the lower coefficient on a half-plane, an integral representation of the solution is obtained in the class of bounded functions and a Dirichlet-type problem is studied. The calculation of the Vekua–Pompeiu integral is examined in the case when the density of the integral has strong singularities in a set of points or lines.

Numerical solution of certain Cauchy singular integral equations using a collocation scheme

The present study is devoted to developing a computational collocation technique for solving the Cauchy singular integral equation of the second kind (CSIE-2). Although, several studies have investigated the numerical approximation solution of CSIEs, the strong singularity and accuracy of the numerical methods are still two important challenges for these integral equations. In this paper, we focus on the smooth transformation and implementation of Bessel basis polynomials (BBP). The reduction of the CSIEs-2 into a system of algebraic equations with the Gauss–Legendre collocation points simplifies this technique. The technique of performing numerical approximation of the solution is well presented and illustrated in the matrix form. Also, the convergence and error bound associated with the scheme are established. Finally, several experiments show the reliability and numerical efficiency of the proposed scheme in comparison with other methods.

A new multifractal analysis-based for identifying the reservoir fluid nature

This work introduces a relevant non-linear approach to well log data processing and interpretation. The suggested method studies the behavior of fractal dimension values together with singularity spectrum attributes. It aims to delimit the hydrocarbon zones, including the detection of fluids' nature.The analysis was carried out on different datasets of 1085 well logging records related to 93 wells drilled in various geological contexts, located in different sedimentary basins in southern Algeria, both box and regularization dimensions have been calculated, and a multifractal analysis using wavelet transform allowed us to compute the singularity spectrum from which we could extract the main attributes, namely width, right height, left height, αpeak, αmin, αmax, f(αpeak), f(αmin), f(αmax) and C value. Based on the obtained values and attributes, an analytical investigation using descriptive statistics, Agglomeration Hierarchical Clustering (AHC), cross plots and scatter plots, has been performed.The obtained results illustrate that the fractal dimensions and singularity spectrum attributes can effectively characterize reservoir fluids, and are importantly affected by the fluid change, resulting in important differences and meaningful discrimination between oil and gas. It should be noted that the oil and gas widths which are significantly different from zero confirm that well logs exhibit multifractal behavior and can, therefore, be suitably investigated to explain such behavior related to physical properties and fluid qualities of reservoirs.The attained results show the oil has the broader range that manifests remarkably in all values and attributes as compared to the gas. Our findings exhibit the gas occupies high regularization dimensions, low box dimensions, low width, low right heights, low left heights which means high sets of fractal dimension, low α peak, low α max, and usually high α min as compared to oil. In addition, the widths are affected by the strength of singularity, with lower widths in gas, the gas known by strong singularities as compared to the oil which represents high widths. It is worth noting that these values and attributes should be used in combination to provide more precision in the analysis, and enhance the interpretation process of gas and oil detection.In conclusion, the obtained results demonstrate the relevance of the suggested method in detecting the nature of the fluid; it might be very interesting to examine the multiscale characteristics of the well log data and enrich the conventional analysis. However, additional investigations using multiple mathematical and statistical tools are needed to confirm these findings, obtain more results as well as improve our analysis and interpretation process.

Fast finite-time tracking control for multi-agent systems with mismatched disturbances under directed communication graphs

In this paper, the fast finite-time consensus problem on multi-agent systems is considered. Multi-agent systems are affected by mismatched disturbances and the communication graph is directed. By a sliding mode observer and a fast nonsingular terminal sliding mode control method, the feedforward-feedback controller is presented for multiple second-order systems and multiple high-order systems to achieve consensus, respectively. The proposed control method can provide short convergence time and improved control input without strong chattering and singularity phenomenon. Finally, simulations are given to verify the availability of the methods.

Positive periodic solution to an indefinite singular equation

In this paper, we prove the existence of a positive periodic solution for nonlinear differential equation with an indefinite singularity by using Green’s function and fixed point theorem in cones. The results are applicable to weak as well as strong singularities. At last, we give one example to show the application of the theorem.

On general-relativistic hydrogen and hydrogenic ions

This paper studies how the static non-linear electromagnetic-vacuum spacetime of a point nucleus with a negative bare mass affects the self-adjointness of the general-relativistic Dirac Hamiltonian for a test electron, without and with an anomalous magnetic moment. This study interpolates between the previously studied extreme cases of a test electron in (a) the Reissner–Weyl–Nordström spacetime (Maxwell’s electromagnetic vacuum), which sports a very strong curvature singularity with negative infinite bare mass, and (b) the Hoffmann spacetime (Born or Born–Infeld’s electromagnetic vacuum) with vanishing bare mass, which features the mildest possible curvature singularity. The main conclusion reached is: on electrostatic spacetimes of a point nucleus with a strictly negative bare mass (which may be −∞), essential self-adjointness fails unless the radial electric field diverges sufficiently fast at the nucleus and the anomalous magnetic moment of the electron is taken into account. Thus, on the Hoffmann spacetime with a (strictly) negative bare mass, the Dirac Hamiltonian of a test electron, with or without anomalous magnetic moment, is not essentially self-adjoint. All these operators have self-adjoint extensions, though, with the usual essential spectrum (−∞, −mec2] ∪ [mec2, ∞) and an infinite discrete spectrum located in the gap (−mec2, mec2).

A New Numerical Procedure for Vibration Analysis of Beam under Impulse and Multiharmonics Piezoelectric Actuators

The dynamic behavior of structures with piezoelectric patches is governed by partial differential equations with strong singularities. To directly deal with these equations, well adapted numerical procedures are required. In this work, the differential quadrature method (DQM) combined with a regularization procedure for space and implicit scheme for time discretization is used. The DQM is a simple method that can be implemented with few grid points and can give results with a good accuracy. However, the DQM presents some difficulties when applied to partial differential equations involving strong singularities. This is due to the fact that the subsidiaries of the singular functions cannot be straightforwardly discretized by the DQM. A methodological approach based on the regularization procedure is used here to overcome this difficulty and the derivatives of the Dirac-delta function are replaced by regularized smooth functions. Thanks to this regularization, the resulting differential equations can be directly discretized using the DQM. The efficiency and applicability of the proposed approach are demonstrated in the computation of the dynamic behavior of beams for various boundary conditions and excited by impulse and Multiharmonics piezoelectric actuators. The obtained numerical results are well compared to the developed analytical solution.

Weak and strong singularities problems to Li\xe9nard equation

This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation: x″+f(x(t))x′(t)+a(t)x=b(t)xα+e(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x''+f\\bigl(x(t)\\bigr)x'(t)+a(t)x= \\frac{b(t)}{x^{\\alpha }}+e(t), $$\\end{document} where the external force e(t) may change sign, α is a constant and alpha >0. The novelty of the present article is that for the first time we show that weak and strong singularities enables the achievement of a new existence criterion of positive periodic solution through an application of the Manásevich–Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, and we give the existence interval of periodic solution of this equation. At last, two examples and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.

Non-cutoff Boltzmann equation with polynomial decay perturbations

The Boltzmann equation without the angular cutoff is considered when the initial data is a small perturbation of a global Maxwellian and decays algebraically in the velocity variable. We obtain a well-posedness theory in the perturbative framework for both mild and strong angular singularities. The three main ingredients in the proof are the moment propagation, the spectral gap of the linearized operator, and the regularizing effect of the linearized operator when the initial data is in a Sobolev space with a negative index. A carefully designed pseudo-differential operator plays a central role in capturing the regularizing effect. In addition, some intrinsic symmetry with respect to the collision operator and an intrinsic functional in the coercivity estimate are essentially used in the commutator estimates for the collision operator with velocity weights.

Global visibility of a strong curvature singularity in nonmarginally bound dust collapse

We investigate here the local versus global visibility of a space-time singularity formed due to the gravitational collapse of a spherically symmetric dust cloud having a non-zero velocity function. The conditions are investigated that ensure the global visibility of the singularity, in the sense that the outgoing null geodesics leave the boundary of the matter cloud in the future, whereas, in the past, these terminate at the singularity. Explicit examples of this effect are constructed. We require that this must be a strong curvature singularity in the sense of Tipler, to ensure the physical significance of the scenario considered. This may act as a counterexample to the weak cosmic censorship hypothesis.

On One Integral Equation in the Theory of Transform Operators

Integral representations of solutions of one differential equation with singularities in the coefficients, containing the Bessel operator perturbed by some potential, are considered. The existence of integral representations of a certain type for such solutions is proved by the method of successive approximations using transform operators. Potentials with strong singularities at the origin are allowed. As compared with the known results, the Riemann function is expressed not via the general hypergeometric function, but, more specifically, via the Legendre function, which helps to avoid unknown constants in the estimates.

Periodic solution for prescribed mean curvature Rayleigh equation with a singularity

In this paper, we consider the existence of a periodic solution for a prescribed mean curvature Rayleigh equation with singularity (weak and strong singularities of attractive type or weak and strong singularities of repulsive type). Our proof is based on an extension of Mawhin’s continuation theorem.

Experimental evidence of a phase transition in the multifractal spectra of turbulent temperature fluctuations at a forest canopy top

Ramp–cliff patterns visible in scalar turbulent time series have long been suspected to enhance the fine-scale intermittency of scalar fluctuations compared to longitudinal velocity fluctuations. Here, we use the wavelet transform modulus maxima method to perform a multifractal analysis of air temperature time series collected at a pine forest canopy top for different atmospheric stability regimes. We show that the multifractal spectra exhibit a phase transition as the signature of the presence of strong singularities corresponding to sharp temperature drops (respectively jumps) bordering the so-called ramp (respectively inverted ramp) cliff patterns commonly observed in unstable (respectively stable) atmospheric conditions and previously suspected to contaminate and possibly enhance the internal intermittency of (scalar) temperature fluctuations. Under unstable (respectively stable) atmospheric conditions, these ‘cliff’ singularities are indeed found to be hierarchically distributed on a ‘Cantor-like’ set surrounded by singularities of weaker strength typical of intermittent temperature fluctuations observed in homogeneous and isotropic turbulence. Under near-neutral conditions, no such a phase transition is observed in the temperature multifractal spectra, which is a strong indication that the statistical contribution of the ‘cliffs’ is not important enough to account for the stronger intermittency of temperature fluctuations when compared to corresponding longitudinal velocity fluctuations.

A meshless regularized local boundary integral equation method and the selection of weight function and geometrical parameters

The local boundary integral equation (LBIE) method is a pure meshless integral method proposed by Atluri and co-workers. Since the fundamental solutions are used as the test functions, the sub-domain can be very small and the LBIE method is a promising method for solving problems of elasticity with nonhomogeneous material properties. In this paper, the subtraction technique is introduced to remove the strong singularity that results in a regularized local boundary integral equation (RLBIE) method. The formula is rather straightforward and more accurate than other treatment techniques. Compared with other meshless methods, more geometrical parameters need to be assigned in the method. If the parameters change a little bit, the solutions obtained from the method may fluctuate quickly and uncertainly. Therefore the effects of weight function, support domain, sub-domain, and monomial basis are investigated and how to select the parameters is discussed. The appropriate parameters can be determined and the meshless RLBIE method is an accurate and robust method.

Positive periodic solution for p-Laplacian Rayleigh equation with weak and strong singularities of repulsive type

In this paper, we investigate the existence of a positive periodic solution for the following p-Laplacian generalized Rayleigh equation with a singularity: $$\begin{aligned} (\phi _p(x'(t)))'+f(t,x'(t))+g(x(t))=e(t), \end{aligned}$$where g has a singularity of repulsive type at the origin. The novelty of the present article is that for the first time, we show that a weakly singularity enables the achievement of a new existence criterion of positive periodic solutions through a application of the Manasevich–Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, the result is applicable to the case of a strong singularity as well as the case of a weak singularity, and we give the existence interval of a positive periodic solution of this equation. At last, example and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.

Positive periodic solution for prescribed mean curvature generalized Li\xe9nard equation with a singularity

The main purpose of this paper is to investigate the existence of a positive periodic solution for a prescribed mean curvature generalized Liénard equation with a singularity (weak and strong singularities of attractive type, or weak and strong singularities of repulsive type). Our proof is based on an extension of Mawhin’s continuation theorem.