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We examine in this work one of the exact solutions of the conservative transport equation for isotropic scattering in spherical geometry, specifically the solution that is singular at the origin and vanishes at infinity. Two representations are known for that solution: one expressed as an infinite divergent series that is derived from the spherical harmonics method and another given by an integral that results from the technique of integration along the particle path and is confirmed here by the method of characteristics. We establish a connection between these representations by showing that the Borel sum of the first reproduces the latter. We also examine computational aspects of the solution expressed in various forms and discuss some standing issues related to it. [ABSTRACT FROM AUTHOR]

The Levenberg–Marquardt method is a fundamental regularization technique for the Newton method applied to nonlinear equations, possibly constrained, and possibly with singular or even nonisolated solutions. We review the literature on the subject, in particular relating to each other various convergence frameworks and results. In this process, the analysis is performed from a unified perspective, and some new results are obtained as well. We discuss smooth and piecewise smooth equations, inexact solution of subproblems, and globalization techniques. Attention is also paid to the LP-Newton method, because of its relations to the Levenberg–Marquardt method. [ABSTRACT FROM AUTHOR]

Efficient finite difference/spectral approximation is proposed for the time-fractional diffusion equation with an inverse square potential on the unit ball in this paper. The efficiency of the method lies in the use of ball functions to mimic the singular behavior of solutions in space and L1 scheme on graded mesh to compensate for the initial weak singularity of solutions in time. The stability and convergence of the fully discrete scheme in both L 2 -norm and H 1 -norm are analyzed. Numerical examples are presented to illustrate the accuracy and efficiency of our proposed discrete scheme. [ABSTRACT FROM AUTHOR]

We are interested in the global bifurcation diagram of radial solutions for the Gelfand problem with the exponential nonlinearity and a positive radially symmetric weight in the unit ball. When the weight is constant, it is known that the bifurcation curve has infinitely many turning points if the dimension 3 ≤ N ≤ 9 , and it has no turning points if N ≥ 10. In this paper, we show that the perturbation of the weight does not affect the bifurcation structure when 3 ≤ N ≤ 9. Moreover, we find a one-parameter family of radial singular solutions for a parametrized weight and study the Morse index of the singular solution. As a result, we prove that the perturbation affects the bifurcation structure in the critical dimension N = 10. Moreover, we give a classification of the bifurcation diagrams in the critical dimension. [ABSTRACT FROM AUTHOR]

Based on the bifurcation theory, the exact solutions of fractional Date–Jimbo–Kashiwara–Miwa equation are constructed. In terms of β -derivative, a nonlinear integer-order ODE is obtained, the equation is transformed into a plane dynamical system. The phase portraits are obtained under different bifurcation conditions. According to the orbits of the phase portraits, new bright and dark solitary solutions, periodic solutions, periodic-singular solutions, and singular solutions are established, enriching the diversity of solutions. In addition, the dynamical behaviors of various types of solutions are shown by selecting appropriate parameters, which is useful for understanding the fluctuation behavior in physical models. Compared with previous literature, this paper focuses on the combination of qualitative analyses and qualitative calculations, which makes the paper more systematic and comprehensive, and fills the gap of using bifurcation theory to solve the FDJKM equation. The dynamical behavior of new solutions obtained will provide more and more attractive and complex physical phenomena to explore more mysteries. [ABSTRACT FROM AUTHOR]

This paper is devoted to describing a linear diffusion problem involving fractional-in-time derivatives and self-adjoint integro-differential space operators posed in bounded domains. One main concern of our paper is to deal with singular boundary data which are typical of fractional diffusion operators in space, and the other one is the consideration of the fractional-in-time Caputo and Riemann–Liouville derivatives in a unified way. We first construct classical solutions of our problems using the spectral theory and discussing the corresponding fractional-in-time ordinary differential equations. We take advantage of the duality between these fractional-in-time derivatives to introduce the notion of weak-dual solution for weighted-integrable data. As the main result of the paper, we prove the well-posedness of the initial and boundary-value problems in this sense. [ABSTRACT FROM AUTHOR]

A two-point boundary value problem whose highest-order derivative is a Riemann–Liouville–Caputo derivative of order α ∈ (1 , 2) is considered. A similar problem was considered in Gracia et al. (BIT 60:411–439, 2020) but under a simplifying assumption that excluded singular solutions. In the present paper, this assumption is not imposed; furthermore, the finite difference method of the BIT paper, which was proved to attain 1st-order convergence under a sign restriction on the convective term, is replaced by a piecewise polynomial collocation method which can give any desired integer order of convergence on a suitably graded mesh. An error analysis of the collocation method is given which removes the above sign restriction and numerical results are presented to support our theoretical conclusions. The tools devised for this analysis include new comparison principles for Caputo initial-value problems and weakly singular Volterra integral equations that are of independent interest. Numerical experiments demonstrate the sharpness of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we shall discuss singular solutions of semilinear elliptic equations with general supercritical growth on spherically symmetric Riemannian manifolds. More precisely, we shall prove the existence, uniqueness and asymptotic behavior of the singular radial solution, and also show that regular radial solutions converges to the singular solution. In particular, we shall provide these properties on spherically symmetric Riemannian manifolds including the hyperbolic space as well as the sphere. [ABSTRACT FROM AUTHOR]

In this paper, we first construct some explicit solutions to the b-family of equations, which will become unbounded in a finite time. Then, we investigate the asymptotic stability of the aforementioned singular solutions of the b-family of equations in the Sobolev space H s with s > 7 2 . It is also interesting to point out that this stability highly depends on the values of parameter b, that is, b ∈ (- 1 , 2 ] . The proof is based on the detailed analysis on the estimates of the perturbed solutions and the properties of the corresponding linear operators. [ABSTRACT FROM AUTHOR]

Let u be a positive singular solution to boundary semilinear elliptic problems with a gradient term and a possibly singular nonlinearity. We prove the symmetry and monotonicity of u via the moving plane procedure. [ABSTRACT FROM AUTHOR]

The existence of positive singular solutions of 1 - Δ q u = (1 + g (x)) | ∇ u | p in B 1 , u = 0 on ∂ B 1 , is proved, where B 1 is the unit ball in R N , N ≥ 3 , 2 < q < N , N (q - 1) N - 1 < p < q and g ≥ 0 is a Hölder continuous function with g (0) = 0 . Also, the existence of positive singular solutions of 2 - Δ q u = | ∇ u | p in Ω , u = 0 on ∂ Ω. is proved, where Ω is a bounded smooth domain in R N , N ≥ 3 , 2 < q < N and N (q - 1) N - 1 < p < q . Finally, the existence of a bounded positive classical solution of (2) with the additional property that ∇ u (x) · x > 0 for large |x| is proved, in the case of Ω an exterior domain R N , N ≥ 3 and p > N (q - 1) N - 1 . [ABSTRACT FROM AUTHOR]

We study general problems modeling electrostatic microelectromechanical systems devices (Pλ )φ(r,−u′(r))=λ∫0rf(s)g(u(s))ds,r∈(0,1),00\lambda \gt 0 is a parameter. We obtain results on the existence and regularity of a touchdown solution to (Pλ{P}_{\lambda }) and find upper and lower bounds on the respective pull-in voltage. In the particular case, when φ(r,v)=rα∣v∣βv\varphi \left(r,v)={r}^{\alpha }{| v| }^{\beta }v, i.e., when the associated differential equation involves the operator r−γ(rα∣u′∣βu′)′{r}^{-\gamma }\left({r}^{\alpha }{| u^{\prime} | }^{\beta }u^{\prime} )^{\prime} , we obtain an exact asymptotic behavior of the touchdown solution in a neighborhood of the origin.

This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained.

Among the most successful techniques for solving nonlinear complementarity problems is the one consisting of reformulation of the problem in question as a system of nonlinear equations, by means of the so-called complementarity functions. Different complementarity functions lead to nonlinear systems with different smoothness and regularity properties, thus allowing for application of different classed of numerical methods. In this paper we compare the Newton method for the smooth reformulation with the semismooth Newton method for the reformulation relying on the nonsmooth Fischer--Burmeister complementarity function, with a special emphasis on the cases when the solution in question violates the strict complementarity condition. [ABSTRACT FROM AUTHOR]

This work is concerned with conditions ensuring stability of a given solution of a system of nonlinear equations with respect to large (not asymptotically thin) classes of right-hand side perturbations. Our main focus is on those solutions that are in a sense singular, and hence, their stability properties are not guaranteed by "standard" inverse function-type theorems. In the twice differentiable case, these issues have received some attention in the existing literature. Moreover, a few results in this direction are known in the case when the first derivative is merely B-differentiable. Here, we further elaborate on a similar setting, but the main attention is paid to the case of piecewise smooth equations. Specifically, we study the effect of singularity of a solution for some active smooth selection on the overall stability properties, and we provide sufficient conditions ensuring the needed stability properties in the cases when such smooth selections may exist. Finally, an application to a piecewise smooth reformulation of complementarity problems is given. [ABSTRACT FROM AUTHOR]

A positive radial singular solution for Δ u + f (u) = 0 with a general supercritical growth is constructed. An exact asymptotic expansion as well as its uniqueness in the space of radial functions are also established. These results can be applied to the bifurcation problem Δ u + λ f (u) = 0 on a ball. Our method can treat a wide class of nonlinearities in a unified way, e.g., u p log u , exp (u p) and exp (⋯ exp (u) ⋯) as well as u p and e u . Main technical tools are intrinsic transformations for semilinear elliptic equations and ODE techniques. [ABSTRACT FROM AUTHOR]

In this paper, for a class of nonlinear implicit ordinary differential equation with variable coefficients, a new approach to get the analytical singular solutions are given both for first order and higher order equations. Sufficient conditions on singular solutions are strengthened into necessary and sufficient conditions in specific case. Finally, the new results are verified by practical examples and MATLAB illustrations compared with traditional methods. [ABSTRACT FROM AUTHOR]

Assume Ω ⊂ ℝ 2 is a planar domain, and u is a locally bounded distributional solution to the elliptic equation − Δ u = | x | 2 β h (x) f (u) in  Ω , where β > − 1 is a constant, h and f are real analytic functions defined on Ω and the real line ℝ , respectively. We establish asymptotic expansions of u (x) to arbitrary orders near 0 , 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. [ABSTRACT FROM AUTHOR]

The theory of singular perturbations in a unified formulation is used, for the first time, to study the propagation of two-dimensional periodic perturbations, including capillary and gravitational surface waves and accompanying ligaments in the 10 − 4 < ω < 10 3     s − 1 frequency range, in a viscous continuously stratified fluid. Dispersion relations for flow constituents are given, as well as expressions for phase and group velocities for surface waves and ligaments in physically observable variables. When the wave-length reaches values of the order of the stratification scale, the liquid behaves as homogeneous. As the wave frequency approaches the buoyancy frequency, the energy transfer rate decreases: the group velocity of surface waves tends to zero, while the phase velocity tends to infinity. In limiting cases, the expressions obtained are transformed into known wave dispersion expressions for an ideal stratified or actually homogeneous fluid. [ABSTRACT FROM AUTHOR]

We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is formulated by enriching the local polynomial spaces with appropriate singular functions. Via a detailed error analysis, the method is shown to converge optimally in both discrete and continuous energy norms. Some tests are conducted in two dimensions for singularities arising from irregular geometries in the domain. The numerical simulations illustrate the established error estimates, and show the method to be a significant improvement over a standard Hybrid High-Order method. [ABSTRACT FROM AUTHOR]

The paper is concerned with the equations of mathematical physics which describe the actual physical processes. The actual are the processes that can be, in principle, simulated experimentally. Particularly, the discussion is restricted to the singular solutions which are formally exact, but have no physical meaning at the points where the parameters of the actual processes become infinitely high. The existence of such solutions is explained in the paper by the shortcomings of the classical differential calculus based on the analysis of infinitesimal quantities. In contrast to the classical equations of mathematical physics, which are derived as the conservation equations for a media element with infinitely small dimensions, the proposed approach is based on the analysis of an element with small but finite dimensions. As a result, the obtained generalized equations retain the form of the corresponding classical equations, but include the generalized parameters studied process instead of the actual parameters. The generalized parameters are expressed of the actual parameters trough the Helmholtz operator which includes the scale parameter linked with the element dimension. For various particular problems, this parameter is found analytically or experimentally. Thus, in the proposed generalized equations of mathematical physics, the classical equations are supplemented with Helmholtz's equations for the actual parameters of the process. The right-hand sides of these equations coincide with the corresponding classical solutions of the problem. If the expression in the right-hand side is regular, the final solution reduces to the classical solution. However, if this expression is singular, the fundamental solutions of the homogeneous Helmholtz equations allow us to eliminate the singularities of the classical solution and to arrive at the regular solution. The proposed approach is demonstrated for the classical singular problem of mathematical physics — a circular membrane loaded with a concentrated force applied at the center, the contact and the crack problems of the theory of elasticity. [ABSTRACT FROM AUTHOR]

We construct some singular solutions to the four component Toda system. These solutions are almost split into two groups, each one modelled on an explicit solution to the two component Toda system (i.e. Liouivlle equation). These solutions are shown to be stable in high dimensions. This gives a sharp example on the partial regularity of stable solutions to Toda system. [ABSTRACT FROM AUTHOR]

We consider the nonlinear elliptic equation Δ u + V (x) u + f (x , u (x)) = 0 on D \ { 0 } , where D is a bounded domain containing 0 in R n , n ≥ 2 , and V and f are Borel measurable functions. Under general conditions on the functions V and f, we prove the existence of positive singular solutions globally comparable to the Dirichlet Green's function of the Laplacian with pole at the origin. Our result applies to various types of semilinear equations, in particular to Δ u + W (x) u p = 0 for all real exponent p which was extensively studied for the range p > 1 . Moreover for this equation with sign-unchanging function W our condition is the optimal one. [ABSTRACT FROM AUTHOR]

A first-order differential equation of Clairaut type has a family of classical solutions, and a singular solution when the contact singular set is not empty. The projection of a singular solution of Clairaut type is an envelope of a family of fronts (Legendre immersions). In these cases, the envelopes are always fronts. We investigate singular points of envelopes for first-order ordinary differential equations, first-order partial differential equations, and systems of first-order partial differential equations of Clairaut type, respectively. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It is well-recognized that in the presence of singular (and in particular nonisolated) solutions of unconstrained or constrained smooth nonlinear equations, the existence of critical solutions has a crucial impact on the behavior of various Newton-type methods. On the one hand, it has been demonstrated that such solutions turn out to be attractors for sequences generated by these methods, for wide domains of starting points, and with a linear convergence rate estimate. On the other hand, the pattern of convergence to such solutions is quite special, and allows for a sharp characterization which serves, in particular, as a basis for some known acceleration techniques, and for the proof of an asymptotic acceptance of the unit stepsize. The latter is an essential property for the success of these techniques when combined with a linesearch strategy for globalization of convergence. This paper aims at extensions of these results to piecewise smooth equations, with applications to corresponding reformulations of nonlinear complementarity problems. [ABSTRACT FROM AUTHOR]

In order to investigate envelopes for singular surfaces, we introduce one- and two-parameter families of framed surfaces and the basic invariants, respectively. By using the basic invariants, the existence and uniqueness theorems of one- and two-parameter families of framed surfaces are given. Then we define envelopes of one- and two-parameter families of framed surfaces and give the existence conditions of envelopes which are called envelope theorems. As an application of the envelope theorems, we show that the projections of singular solutions of completely integrable first-order partial differential equations are envelopes. [ABSTRACT FROM AUTHOR]

The theory of singular perturbations in a unified formulation is used, for the first time, to study the propagation of two-dimensional periodic perturbations, including capillary and gravitational surface waves and accompanying ligaments in the 10−4<ω<103 s−1 frequency range, in a viscous continuously stratified fluid. Dispersion relations for flow constituents are given, as well as expressions for phase and group velocities for surface waves and ligaments in physically observable variables. When the wave-length reaches values of the order of the stratification scale, the liquid behaves as homogeneous. As the wave frequency approaches the buoyancy frequency, the energy transfer rate decreases: the group velocity of surface waves tends to zero, while the phase velocity tends to infinity. In limiting cases, the expressions obtained are transformed into known wave dispersion expressions for an ideal stratified or actually homogeneous fluid.

As is well known, when initialized close to a nonsingular solution of a smooth nonlinear equation, the Newton method converges to this solution superlinearly. Moreover, the common Armijo linesearch procedure used to globalize the process for convergence from arbitrary starting points, accepts the unit stepsize asymptotically and ensures fast local convergence. In the case of a singular and possibly even nonisolated solution, the situation is much more complicated. Local linear convergence (with asymptotic ratio of 1/2) of the Newton method can still be guaranteed under reasonable assumptions, from a starlike, asymptotically dense set around the solution. Moreover, convergence can be accelerated by extrapolation and overrelaxation techniques. However, nothing was previously known on how the Newton method can be coupled in these circumstances with a linesearch technique for globalization that locally accepts unit stepsize and guarantees linear convergence. It turns out that this is a rather nontrivial issue, requiring a delicate combination of the analyses on acceptance of the unit stepsize and on the iterates staying within the relevant starlike domain of convergence. In addition to these analyses, numerical illustrations and comparisons are presented for the Newton method and the use of extrapolation to accelerate convergence speed. [ABSTRACT FROM AUTHOR]

We consider nonnegative solutions of a semilinear heat equation u t − Δ u = u p in R N (N ≥ 3) with p = N / (N − 2) and a nonnegative initial data u 0 ∈ L N / (N − 2) (R N) which has a singularity at ξ 0 ∈ R N. We prove that there exists u 0 such that, for any ξ ∈ C α ([ 0 , ∞) ; R N) with α ∈ (1 / 2 , 1 ] and ξ (0) = ξ 0 , the problem admits a nonnegative solution u ξ ∈ C ([ 0 , T ξ ] ; L N / (N − 2) (R N)) for some T ξ with an explicit singular leading term for each t ∈ [ 0 , T ξ ] as x → ξ (t). Our result refines known counter examples for the uniqueness of the doubly critical case in view of pointwise behavior and complements known sufficient conditions on p for the existence of solutions with moving singularities. [ABSTRACT FROM AUTHOR]

In the case of singular (and possibly even nonisolated) solutions of nonlinear equations, while superlinear convergence of the Newton method cannot be guaranteed, local linear convergence from large domains of starting points still holds under certain reasonable assumptions. We consider a linesearch globalization of the Newton method, combined with extrapolation and over-relaxation accelerating techniques, aiming at a speed up of convergence to critical solutions (a certain class of singular solutions). Numerical results indicate that an acceleration is observed indeed. [ABSTRACT FROM AUTHOR]

Abstract The aim of this paper is to investigate the existence of weak solutions for a two-dimensional Schrödinger equation with a singular potential in C+ $\mathbb{C}_{+}$. Under appropriate assumptions on the nonlinearity, we introduce a new type of quantum calculus via the Morse theory and variational methods. By applying Schrödinger type inequalities and the well-known Banach fixed point theorem in conjunction with the technique of measures of weak noncompactness, the new and more accurate estimations for boundary behaviors of them are also deduced.

We study the behavior of the finite element condition numbers on a class of anisotropic meshes. These newly-developed mesh algorithms can produce numerical approximations with optimal convergence to isotropic and anisotropic singular solutions of elliptic boundary value problems in two- and three-dimensions. Despite the simplicity and fewer geometric constraints in implementation, these meshes can be highly anisotropic and do not maintain the maximum angle condition. We formulate a unified refinement principle and establish sharp estimates on the growth rate of the condition numbers of the stiffness matrix from these meshes. These results are important for effective applications of these meshes and for the design of fast numerical solvers. Numerical tests validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

We study singular radial solutions of the semilinear elliptic equation Δ u + f (u) = 0 on finite balls in R N with N ≥ 3 . We assume that f satisfies either f (u) = u p + o (u p) with p > (N + 2) / (N - 2) or f (u) = e u + o (e u) 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. [ABSTRACT FROM AUTHOR]

We study stability properties of a given solution of a constrained equation, where the constraint has the form of the inclusion into an arbitrary closed convex set. We are mostly interested in those cases when Robinson's regularity condition does not hold, and we obtain weaker conditions ensuring stability of a given solution subject to wide classes of perturbations, or, in other words, ensuring covering of a ``large"" set. Unlike previous developments of this kind, here we do not employ any necessary conicity assumptions on the constraint set, thus allowing for a much wider area of potential applications. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to develop a general theory on how the inf‐sup stable and convergent elements of the velocity Dirichlet boundary (VDB)‐Stokes problem with no‐slip VDB are still inf‐sup stable and convergent for the pressure Dirichlet boundary (PDB)‐Stokes problem with PDB in Lipschitz domain. The PDB‐Stokes problem in a Lipschitz domain usually only has a singular velocity solution which does not belong to (H1(Ω))2, sharply in contrast to the VDB‐Stokes problem whose velocity solution still belongs to (H1(Ω))2, and unexpectedly, some well‐known inf‐sup stable and convergent VDB‐Stokes elements may or may no longer correctly converge. It turns out that the inf‐sup condition of the PDB‐Stokes problem in Lipschitz domain relies on an unusual variational problem and requires adequate degrees of freedom on the domain boundary. In this paper we propose two families of staggered elements: staggered Taylor–Hood elements CPℓ+22−Pℓ with ℓ ≥ 1 (continuous in both velocity and pressure) and staggered Fortin elements CPm+22−Pmdisc with m ≥ 1 (continuous in velocity and discontinuous in pressure) on triangles, for solving the PDB‐Stokes problem in Lipschitz domain. We show that the two families are inf‐sup stable and are correctly convergent for the non‐H1 singular velocity. Numerical results illustrate the proposed elements and the theoretical results. [ABSTRACT FROM AUTHOR]