On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

The ∞ ‐Bilaplacian is a third‐order fully nonlinear PDE given by (1) In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞ ‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p‐Bilaplacian (2) We prove convergence of the numerical solution to the weak solution of and show that we are able to pass to the limit p → ∞ . We perform various tests aimed at understanding the nature of solutions of and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of ‐solutions.     

Numerical study of a conservative weighted compact difference scheme for the symmetric regularized long wave equations

In this article, a new weighted and compact conservative difference scheme for the symmetric regularized long wave (SRLW) equations is considered. The new scheme is decoupled and linearized in practical computation, that is, at each time step only two tridiagonal systems of linear algebraic equations need to be solved. It is proved by the discrete energy method that the compact scheme is uniquely solvable, the convergence and stability of the difference scheme are obtained, and its numerical convergence order is in the ‐norm. Numerical experiment results show that the scheme is efficient and reliable.     

H1-superconvergence of finite difference method based on Q1-element on quasi-uniform mesh for the 3D Poisson equation

In this paper, the finite difference (FD) method is considered for the 3D Poisson equation by using the Q₁-element on a quasi-uniform mesh. First, under the regularity assumption of , the H¹-superconvergence of the FD solution u_h based on the Q₁-element to the first-order interpolation function is obtained. Next, the H¹-superconvergence of the second-order interpolation postprocessing function based on the FD solution u_h to u is provided. Finally, numerical tests are presented to show the H¹-superconvergence result of the FD postprocessing function to u if .     

Superconvergence analysis of an energy stable scheme with three step backward differential formula-finite element method for nonlinear reaction–diffusion equation

A three step backward differential formula scheme is proposed for nonlinear reaction–diffusion equation and superconvergence results are studied with Galerkin finite element method unconditionally. Energy stability is testified for the constructed scheme with an artificial term. Splitting technique is utilized to get rid of the ratio between the time step size and the subdivision parameter . Temporal error estimate in H²-norm is derived, which leads to the boundedness of the solutions of the time-discrete equations. Unconditional spatial error estimate in L²-norm is deduced which help bound the numerical solutions in L^∞-norm. Superconvergent property of in H¹-norm with order is obtained by taking difference between two time levels of the error equations unconditionally. The global superconvergent property is deduced through the above results. Two numerical examples show the validity of the theoretical analysis.     

Numerical analysis for a system coupling curve evolution attached orthogonally to a fixed boundary,to a reaction–diffusion equation on the curve

We consider a semidiscrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain , such that the curve meets the boundary orthogonally, and the forcing is a function of the solution of a reaction–diffusion equation that holds on the evolving curve. We prove optimal order error bounds for the resulting approximation and present numerical experiments.     

Ultraconvergence of finite element method by Richardson extrapolation for elliptic problems with inhomogeneous boundary conditions

In this article, Richardson extrapolation technique is employed to investigate the local ultraconvergence properties of Lagrange finite element method using piecewise polynomials of degrees () for the second order elliptic problem with inhomogeneous boundary. A sequence of special graded partition are proposed and a new interpolation operator is introduced to achieve order local ultraconvergence for the displacement and derivative.     

Half‐inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse

In this paper, we consider the inverse spectral problem for the impulsive Sturm–Liouville differential pencils on [0, π] with the Robin boundary conditions and the jump conditions at the point . We prove that two potentials functions on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potentials given on and (ii) the potentials given on , where 0 < α < 1 , respectively. Inverse spectral problems, Sturm–Liouville operator, spectrum, uniqueness.     

A robust scheme based on novel-operational matrices for some classes of time-fractional nonlinear problems arising in mechanics and mathematical physics

In this paper, we present a novel approach based on shifted Gegenbauer wavelets to attain approximate solutions of some classed of time-fractional nonlinear problems. First, we present the approximation of a function of two variables u(x,t) with help of shifted Gegenbauer wavelets and then some novel operational matrices are proposed with the help of piecewise functions to investigate the positive integer derivative ( D _x and D _t), fractional-order derivative ( and ), fractional-order integration ( and ) and delay terms ( and ) of approximated function u(x,t). In order to transform the discussed nonlinear problem into linear problem Picard iterative scheme has been adopt. The current scheme converts the discussed highly nonlinear time-fractional problem into system of linear algebraic equation the help of developed operational matrices and Picard idea. Analysis on the error bound and convergence to authenticate the mathematical formulation of the computational algorithm. We solve various test problems, such as the van der Pol oscillator model, generalized Burger–Huxley, neutral delay parabolic differential equations, sine-Gordon, parabolic integro-differential equation and nonlinear Schrödinger equations to show the efficiency and accuracy of linearized shifted Gegenbauer wavelets method. A comprehensive comparative examination shows the credibility, accuracy, and reliability of the presently proposed computational approach. Also, this scheme can be extended conveniently to other multi-dimensional physical problems of highly nonlinear fractional or variable order of complex nature.     

Spectral statistics of random Schrödinger operator with growing potential

In this work we investigate the spectral statistics of random Schrödinger operators acting on where are i.i.d random variables distributed uniformly on [0,1].     

Fast high order difference schemes for the time fractional telegraph equation

In this paper, a fast high order difference scheme is first proposed to solve the time fractional telegraph equation based on the ℱℒ 2-1_σ formula for the Caputo fractional derivative, which reduces the storage and computational cost for calculation. A compact scheme is then presented to improve the convergence order in space. The unconditional stability and convergence in maximum norm are proved for both schemes, with the accuracy order and , respectively. Difficulty arising from the two Caputo fractional derivatives is overcome by some detailed analysis. Finally, we carry out numerical experiments to show the efficiency and accuracy, by comparing with the ℒ 2-1_σ method.     

The Tarski irredundant basis theorem and the finite soluble groups

Denote by and , respectively, the smallest and the largest cardinality of a minimal generating set of a finite group G. The Tarski irredundant basis theorem implies that for every k with there exist a minimal generating set , an index and in G such that is again a minimal generating set of G. In this case we say that is an immediate descendant of ω. There are several examples of minimal generating sets of cardinality smaller than which have no immediate descendant and so it appears an interesting problem to investigate under which conditions an immediate descendant exists. In this paper we discuss this problem in the case of finite soluble groups.     

Existence and regularity of positive solutions of a degenerate elliptic problem

Existence and regularity of positive solutions of a degenerate elliptic Dirichlet problem of the form in Ω, on , where Ω is a bounded smooth domain in , , are obtained via new embeddings of some weighted Sobolev spaces with singular weights and . It is seen that and admit many singular points in Ω. The main embedding results in this paper provide some generalizations of the well‐known Caffarelli–Kohn–Nirenberg inequality.     

Localized peaking regimes for quasilinear parabolic equations

This paper deals with the asymptotic behavior as of all weak (energy) solutions of a class of equations with the following model representative: with prescribed global energy function Here , , , Ω is a bounded smooth domain, . Particularly, in the case it is proved that the solution u remains uniformly bounded as in an arbitrary subdomain and the sharp upper estimate of when has been obtained depending on and . In the case for all , sharp sufficient conditions on degeneration of near that guarantee the above mentioned boundedness for an arbitrary (even large) solution have been found and the sharp upper estimate of a final profile of the solution when has been obtained.     

Dimension of the space generated by Steiner systems

To every Steiner system with parameters on and blocks , we can assign its characteristic vector , which is a ‐vector whose entries are indexed by the ‐subsets of such that for each ‐subset of if and only if . In this paper, we show that the dimension of the vector space generated by all of the characteristic vectors of Steiner systems with parameters is , provided that and there is at least one such system.     

Uniformly convergent scheme for two‐parameter singularly perturbed problems with non‐smooth data

A numerical scheme is constructed for the problems in which the diffusion and convection parameters (?₁ and ?₂ , respectively) both are small, and the convection and source terms have a jump discontinuity in the domain of consideration. Depending on the magnitude of the ratios , and two different cases have been considered separately. Through rigorous analysis, the theoretical error bounds on the singular and regular components of the solution are obtained separately, which shows that in both cases the method is convergent uniformly irrespective of the size of the parameters ?₁, ?₂ . Two test problems are included to validate the theoretical results.     

On quasi‐infinitely divisible distributions with a point mass

An infinitely divisible distribution on is a probability measure μ such that the characteristic function has a Lévy–Khintchine representation with characteristic triplet , where ν is a Lévy measure, and . A natural extension of such distributions are quasi‐infinitely distributions. Instead of a Lévy measure, we assume that ν is a “signed Lévy measure”, for further information on the definition see [10]. We show that a distribution with and , where is the absolutely continuous part, is quasi‐infinitely divisible if and only if for every . We apply this to show that certain variance mixtures of mean zero normal distributions are quasi‐infinitely divisible distributions, and we give an example of a quasi‐infinitely divisible distribution that is not continuous but has infinite quasi‐Lévy measure. Furthermore, it is shown that replacing the signed Lévy measure by a seemingly more general complex Lévy measure does not lead to new distributions. Last but not least it is proven that the class of quasi‐infinitely divisible distributions is not open, but path‐connected in the space of probability measures with the Prokhorov metric.     

A note on the eigenvalues of p‐fractional Hardy–Sobolev operator with indefinite weight

In this article, we study the eigenvalues of p‐fractional Hardy operator where , , , and Ω is an unbounded domain in with Lipschitz boundary containing 0. The weight function V may change sign and may have singular points. We also show that the least positive eigenvalue is simple and it is uniquely associated to a nonnegative eigenfunction. Moreover, we proved that there exists a sequence of eigenvalues as .     

Lindelöf theorems for monotone Sobolev functions in Orlicz spaces on uniform domains

In this paper, we are concerned with Lindelöf type theorems for monotone (in the sense of Lebesgue) Sobolev functions u on a uniform domain satisfying where ? denotes the gradient, denotes the distance from z to the boundary , φ is of log‐type and ω is a weight function satisfying the doubling condition.     

On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

We construct a bounded C¹ domain Ω in for which the regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in such that the solution of in Ω and either on or on is contained in but not in for any . An analogous result holds for Sobolev spaces with .