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The paper focuses on the analysis of the Euler projection Galerkin finite element method (FEM) for the dynamics of magnetization in ferromagnetic materials, described by the Landau–Lifshitz equation with the point-wise constraint $|{\textbf{m}}|=1$. The method is based on a simple sphere projection that projects the numerical solution onto a unit sphere at each time step, and the method has been used in many areas in the past several decades. However, error analysis for the commonly used method has not been done since the classical energy approach cannot be applied directly. In this paper we present an optimal $\textbf{L}^2$ error analysis of the backward Euler sphere projection method by using quadratic or higher order finite elements under a time step condition $\tau =O(\epsilon _0 h)$ with some small $\epsilon _0>0$. The analysis is based on more precise estimates of the extra error caused by the sphere projection in both $\textbf{L}^2$ and $\textbf{H}^1$ norms, and the classical estimate of dual norm. Numerical experiment is provided to confirm our theoretical analysis.

In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

In this paper, we propose two low-order nonconforming finite element methods (FEMs) for the three-dimensional Stokes flow that generalize the nonconforming FEM of Kouhia & Stenberg (1995, A linear nonconforming finite element method for nearly incompressible elasticity and Stokes flow. Comput. Methods Appl. Mech. Eng, 124, 195–212). The finite element spaces proposed in this paper consist of two globally continuous components (one piecewise affine and one enriched component) and one component that is continuous at the midpoints of interior faces. We prove that the discrete Korn inequality and a discrete inf–sup condition hold uniformly in the mesh size and also for a nonempty Neumann boundary. Based on these two results, we show the well-posedness of the discrete problem. Two counterexamples prove that there is no direct generalization of the Kouhia–Stenberg FEM to three space dimensions: the finite element space with one nonconforming and two conforming piecewise affine components does not satisfy a discrete inf–sup condition with piecewise constant pressure approximations, while finite element functions with two nonconforming and one conforming component do not satisfy a discrete Korn inequality.

In this paper we suggest a method for transforming a vector subdivision scheme generating $C^{\ell}$ limits to another such scheme of the same dimension, generating $C^{\ell+1}$ limits. In scalar subdivision, it is well known that a scheme generating $C^{\ell}$ limit curves can be transformed to a new scheme producing $C^{\ell+1}$ limit curves by multiplying the scheme's symbol with the smoothing factor $\tfrac{z+1}{2}$. We extend this approach to vector and Hermite subdivision schemes, by manipulating symbols. The algorithms presented in this paper allow to construct vector (Hermite) subdivision schemes of arbitrarily high regularity from a convergent vector scheme (from a Hermite scheme whose Taylor scheme is convergent with limit functions of vanishing first component)., 28 pages, 4 figures. Corrected typos, updated contact information

This paper is a follow-up to the recent paper of Pillichshammer and Sonnleitner (2020) [12] . We show that the isotropic discrepancy of a lattice point set is at most d 2 2 ( d + 1 ) times its spectral test, thereby correcting the dependence on the dimension d and an inaccuracy in the proof of the upper bound in Theorem 2 of the mentioned paper. The major task is to bound the volume of the neighbourhood of the boundary of a convex set contained in the unit cube. Further, we characterize averages of the distance to a lattice point set in terms of the spectral test. As an application, we infer that the spectral test – and with it the isotropic discrepancy – is crucial for the suitability of the lattice point set for the approximation of Sobolev functions.

In a previous paper by the authors, the existence of Haar projections with growing norms in Sobolev–Triebel–Lizorkin spaces has been shown via a probabilistic argument. This existence was sufficient to determine the precise range of Triebel–Lizorkin spaces for which the Haar system is an unconditional basis. The aim of the present paper is to give simple deterministic examples of Haar projections that show this growth behavior in the respective range of parameters.

In this paper, we generalize the notion of Kantorovich-type sampling operators using the Fejer-type singular integral. By means of these operators we are able to reconstruct signals (functions) which are not necessarily continuous. Moreover, our generalization allows us to take the measurement error into account. The goal of this paper is to estimate the rate of approximation by the above operators via high-order modulus of smoothness.

This paper considers the problem of approximating the spectral factor of continuous spectral densities with finite Dirichlet energy based on finitely many samples of these spectral densities. Although there exists a closed form expression for the spectral factor, this formula shows a very complicated behavior because of the non-linear dependency of the spectral factor from spectral density and because of a singular integral in this expression. Therefore approximation methods are usually applied to calculate the spectral factor. It is shown that there exists no sampling-based method which depends continuously on the samples and which is able to approximate the spectral factor for all densities in this set. Instead, to any sampling-based approximation method there exists a large set of spectral densities so that the approximation method does not converge to the spectral factor for every spectral density in this set as the number of available sampling points is increased. The paper will also show that the same results hold for sampling-based algorithms for the calculation of the outer function in the theory of Hardy spaces.

International audience; We work on a low-order finite element approximation of the vorticity, velocity and pressure formulation of the bidimensional Stokes problem. In a previous paper, we have introduced the adequate space in which to look for the vorticity in order to have a well-posed problem. In this paper, we deal with the numerical approximation of this space, prove optimal convergence of the scheme and show numerical experiments in good accordance with the theory. We remark that despite one supplementary unknown(the vorticity), results of the scheme are much better than the ones obtained with the P1 plus bubble−P1 element in the velocity–pressure formulation.

The work of de Boor and Fix on spline approximation by quasiinterpolants has had far-reaching influence in approximation theory since publication of their paper in 1973. In this paper, we further develop their idea and investigate quasi-projection operators. We give sharp estimates in terms of moduli of smoothness for approximation with scaled shift-invariant spaces by means of quasi-projection operators. In particular, we provide error analysis for approximation of quasi-projection operators with Lipschitz spaces. The study of quasi-projection operators has many applications to various areas related to approximation theory and wavelet analysis.

We consider the diffraction of an electromagnetic plane wave by a biperiodic structure. This paper is concerned with a numerical solution of the diffraction grating problem for three-dimensional Maxwell’s equations. Based on the Dirichlet-to-Neumann (DtN) operator, an equivalent boundary value problem is formulated in a bounded domain by using a transparent boundary condition. An a posteriori error estimate-based adaptive edge finite element method is developed for the variational problem with the truncated DtN operator. The estimate takes account of both the finite element approximation error and the truncation error of the DtN operator, where the former is used for local mesh refinements and the latter is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to demonstrate the competitive behaviour of the proposed method.

The present paper introduces an efficient and accurate numerical scheme for the solution of a highly anisotropic elliptic equation, the anisotropy direction being given by a variable vector field. This scheme is based on an asymptotic preserving reformulation of the original system, permitting an accurate resolution independently of the anisotropy strength and without the need of a mesh adapted to this anisotropy. The counterpart of this original procedure is the larger system size, enlarged by adding auxiliary variables and Lagrange multipliers. This Asymptotic-Preserving method generalizes the method investigated in a previous paper [arXiv:0903.4984v2] to the case of an arbitrary anisotropy direction field.

Summarization: In this paper, we extend reduced-basis output bound methods developed earlier for elliptic problems, to problems described by ‘parameterized parabolic’ partial differential equations. The essential new ingredient and the novelty of this paper consist in the presence of time in the formulation and solution of the problem. First, without assuming a time discretization, a reduced-basis procedure is presented to ‘efficiently’ compute accurate approximations to the solution of the parabolic problem and ‘relevant’ outputs of interest. In addition, we develop an error estimation procedure to ‘a posteriori validate’ the accuracy of our output predictions. Second, using the discontinuous Galerkin method for the temporal discretization, the reduced-basis method and the output bound procedure are analysed for the semi-discrete case. In both cases the reduced-basis is constructed by taking ‘snapshots’ of the solution both in time and in the parameters: in that sense the method is close to Proper Orthogonal Decomposition (POD). Παρουσιάστηκε στο: journal of numerical analysis

This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames $\mathcal{F} = \{f_i\}_{i \in I}$ and $\mathcal{E} = \{e_j\}_{j \in G}$ ($G$ a discrete abelian group), relating the decay of the expansion of the elements of $\mathcal{F}$ in terms of the elements of $\mathcal{E}$ via a map $a \colon I \to G$. A fundamental set of equalities are shown between three seemingly unrelated quantities: the relative measure of $\mathcal{F}$, the relative measure of $\mathcal{E}$ - both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame elements - and the density of the set $a(I)$ in $G$. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds and density, and on the structure of the dual frame of a localized frame. In a subsequent paper, these results are applied to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The notion of localization and related approximation properties introduced in this paper are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of the interrelations among these localization and approximation concepts is presented., 37 pages, 1 figure

This work developes a quantitative framework for describing the overcompleteness of a large class of frames. A previous paper introduced notions of localization and approximation between two frames $\mathcal{F} = \{f_i\}_{i \in I}$ and $\mathcal{E} = \{e_j\}_{j \in G}$ ($G$ a discrete abelian group), relating the decay of the expansion of the elements of $\mathcal{F}$ in terms of the elements of $\mathcal{E}$ via a map $a \colon I \to G$. This paper shows that those abstract results yield an array of new implications for irregular Gabor frames. Additionally, various Nyquist density results for Gabor frames are recovered as special cases, and in the process both their meaning and implications are clarified. New results are obtained on the excess and overcompleteness of Gabor frames, on the relationship between frame bounds and density, and on the structure of the dual frame of an irregular Gabor frame. More generally, these results apply both to Gabor frames and to systems of Gabor molecules, whose elements share only a common envelope of concentration in the time-frequency plane. The notions of localization and related approximation properties are a spectrum of ideas that quantify the degree to which elements of one frame can be approximated by elements of another frame. In this paper, a comprehensive examination of the interrelations among these localization and approximation concepts is made, with most implications shown to be sharp., 34 pages, 1 figure

In this paper we show that a spherical convex body $C$ is of constant diameter $\tau$ if and only if $C$ is of constant width $\tau$, for $0, Comment: 6 pages

Non-inf-sup-stable finite element approximations to the incompressible Navier–Stokes equations based on equal-order spaces for velocity and pressure are studied in this paper. To account for the violation of the discrete inf-sup condition, different types of symmetric pressure stabilization terms are considered. It is shown in the numerical analysis that these terms also improve stabilization of dominating convection in the following sense: error bounds with constants independent of inverse powers of the viscosity are derived. For proving the bound for the $L^2$ error of the pressure the choice of a suitable initial approximation for the velocity is essential.

We analyse a new notion of total anisotropic higher-order variation which, differently from the Total Generalized Variation by Bredies et al., quantifies for possibly non-symmetric tensor fields their variations at arbitrary order weighted by possibly inhomogeneous, smooth elliptic anisotropies. We prove some properties of this total variation and of the associated spaces of tensors with finite variations. We show the existence of solutions to a related regularity-fidelity optimisation problem. We also prove a decomposition formula which appears to be helpful for the design of numerical schemes, as shown in a companion paper, where several applications to image processing are studied., Comment: 23 pages

The first part of this paper enfolds a medius analysis for mixed finite element methods (FEMs) and proves a best-approximation result in L 2 for the stress variable independent of the error of the Lagrange multiplier under the abstract conditions (LBB), condition (C) and eciency (E). The second part applies the general result to the FEM of Arnold and Winther for linear elasticity: The stress error in L 2 is controlled by the L 2 best-approximation error of the true stress by any discrete function plus data oscillations. The analysis is valid without any extra regularity assumptions on the exact solution and also covers coarse meshes and Neumann boundary conditions. Further applications include RaviartThomas finite elements for the Poisson and the Stokes problem. The result has consequences for nonlinear approximation classes related to adaptive mixed finite element methods.

This paper shows that, given a doubling weight w w on the unit sphere S d − 1 \mathbb {S}^{d-1} of R d \mathbb {R}^d , there exists a positive constant K w , d K_{w,d} such that, for each positive integer n n and each integer N ≥ max x ∈ S d − 1 K w , d w ( B ( x , n − 1 ) ) N\ge \max _{x\in {\mathbb {S}^{d-1}}} \frac {K_{w,d}} {w(B(x, n^{-1}))} , there exists a set of N N distinct nodes z 1 z_1 , …, z N z_N on S d − 1 \mathbb {S}^{d-1} for which ( ∗ ) 1 w ( S d − 1 ) ∫ S d − 1 f ( x ) w ( x ) d σ d ( x ) = 1 N ∑ j = 1 N f ( z j ) , ∀ f ∈ Π n d , \begin{equation} \tag {$\ast $} \frac {1}{w({\mathbb {S}^{d-1}})} \int _{{\mathbb {S}^{d-1}}} f(x) w(x)\, d\sigma _d(x)=\frac 1N \sum _{j=1}^N f(z_j),\qquad \forall f\in \Pi _n^d, \end{equation} where d σ d d\sigma _d , B ( x , r ) B(x,r) , and Π n d \Pi _n^d denote the surface Lebesgue measure on S d − 1 {\mathbb {S}^{d-1}} , the spherical cap with center x ∈ S d − 1 x\in \mathbb {S}^{d-1} and radius r > 0 r>0 , and the space of all spherical polynomials of degree at most n n on S d − 1 {\mathbb {S}^{d-1}} , respectively, and w ( E ) = ∫ E w ( x ) d σ d ( x ) w(E)=\int _E w(x) \, d\sigma _d(x) for E ⊂ S d − 1 E\subset {\mathbb {S}^{d-1}} . If, in addition, w ∈ L ∞ ( S d − 1 ) w\in L^\infty ({\mathbb {S}^{d-1}}) , then the above set of nodes can be chosen to be well separated: \[ min 1 ≤ i ≠ j ≤ N arccos ⁡ ( z i ⋅ z j ) ≥ c w , d N − 1 d − 1 > 0. \min _{1\leq i\neq j\leq N}\arccos (z_i\cdot z_j)\geq c_{w,d} N^{-\frac 1{d-1}}>0. \] It is further proved that the minimal number of nodes N n ( w d σ d ) \mathcal {N}_{n} (wd\sigma _d) required in ( ∗ \ast ) for a doubling weight w w on S d − 1 {\mathbb {S}^{d-1}} satisfies \[ N n ( w d σ d ) ∼ max x ∈ S d − 1 1 w ( B ( x , n − 1 ) ) , n = 1 , 2 , … . \mathcal {N}_n (wd\sigma _d) \sim \max _{x\in {\mathbb {S}^{d-1}}} \frac 1 {w(B(x, n^{-1}))},\qquad n=1,2,\ldots . \] Proofs of these results rely on new convex partitions of S d − 1 {\mathbb {S}^{d-1}} that are regular with respect to a given weight w w and integer N N . Similar results are also established on the unit ball and the standard simplex of R d \mathbb {R}^d . Our results extend the recent results of Bondarenko, Radchenko, and Viazovska on spherical designs.

This paper is devoted to study the global well-posedness of solutions for the Cauchy problem of the fractional Cahn–Hilliard equation in ℝ N {\mathbb{R}^{N}} ( N ∈ ℕ + {N\in\mathbb{N}^{+}} ), provided that the initial datum is sufficiently small. In addition, the L p {L^{p}} -norm ( 1 ≤ p ≤ ∞ {1\leq p\leq\infty} ) temporal decay rate for weak solutions and the higher-order derivative of solutions are also studied.

This paper aims to develop a simple and unified technique in finding asymptotic expansion of orthogonal polynomials from their difference equations. By preserving the symmetry in the difference equation, we are able to express the higher-order terms in the asymptotic expansion as an integral whose integrand can be explicitly obtained by a recurrence relation, while the integration constant is to be determined by a matching condition that relates to the initial conditions and coefficients in the difference equation.

The approximation of a general d-variate function f by the shifts �(� − �), � ∈ � ⊂ R d , of a fixed functionoccurs in many applications such as data fit- ting, neural networks, and learning theory. When � = hZ d is a dilate of the integer lattice, there is a rather complete understanding of the approximation problem (6, 18) using Fourier techniques. However, in most applications the center set � is either given, or can be chosen with complete freedom. In both of these cases, the shift-invariant setting is too restrictive. This paper studies the approximation problem in the caseis arbitrary. It establishes approximation theorems whose error bounds reflect the local density of the points in �. Two different settings are analyzed. The first is when the setis prescribed in advance. In this case, the theorems of this paper show that, in analogy with the classical univariate spline ap- proximation, improved approximation occurs in regions where the density is high. The second setting corresponds to the problem of non-linear approximation. In that setting the setcan be chosen using information about the target function f. We discuss how to 'best' make these choices and give estimates for the approximation error.

We provide the first significant extension of the unified transform (also known as the Fokas method) applied to elliptic boundary value problems, namely, we extend the method to curvilinear polygons and partial differential equations (PDEs) with variable coefficients. This is used to solve the generalized Dirichlet-to-Neumann map. The central component of the unified transform is the coupling of certain integral transforms of the given boundary data and of the unknown boundary values. This has become known as the global relation and, in the case of constant coefficient PDEs, simply links the Fourier transforms of the Dirichlet and Neumann boundary values. We extend the global relation to PDEs with variable coefficients and to domains with curved boundaries. Furthermore, we provide a natural choice of global relations for separable PDEs. These generalizations are numerically implemented using a method based on Chebyshev interpolation for efficient and accurate computation of the integral transforms that appear in the global relation. Extensive numerical examples are provided, demonstrating that the method presented in this paper is both accurate and fast, yielding exponential convergence for sufficiently smooth solutions. Furthermore, the method is straightforward to use, involving just the construction of a simple linear system from the integral transforms, and does not require knowledge of Green’s functions of the PDE. Details on the implementation are discussed at length.

In this paper we prove a convergence result for a discretization of the three-dimensional stationary compressible Navier–Stokes equations assuming an ideal gas pressure law $p(\rho )=a \rho ^{\gamma }$ with $\gamma> \frac{3}{2}$. It is the first convergence result for a numerical method with adiabatic exponents $\gamma $ less than $3$ when the space dimension is 3. The considered numerical scheme combines finite volume techniques for the convection with the Crouzeix–Raviart finite element for the diffusion. A linearized version of the scheme is implemented in the industrial software CALIF3S developed by the French Institut de Radioprotection et de Sûreté Nucléaire.

The paper is devoted to a reaction-diffusion problem describing diffusion and consumption of nutrients in a biological tissue consisting of small cells periodically arranged in an extracellular matrix. Cells consume nutrients with a rate proportional to cell area and to nutrient concentration. The dependence on the nutrient concentration can be linear or nonlinear. The cells are modeled by a potential approximating the Dirac’s delta-function. The potential has a periodically distributed support of small measure. The problem contains two small parameters: the diameter of a cell and the distance between the cells (in comparison with the characteristic macroscopic size). In the multi-dimensional formulation assuming some restriction on the relation of parameters, we prove convergence of solution of this problem to the solution of a limiting homogenized problem. We show that the problem is non-homogenizable in classical sense if this restriction fails.

This paper gives a direct proof of localization of dual norms of bounded linear functionals on the Sobolev space ${W^{1,p}_0(\varOmega )}$, $1 \leq p \leq \infty $. The basic condition is that the functional in question vanishes over locally supported test functions from ${W^{1,p}_0(\varOmega )}$ which form a partition of unity in $\varOmega $, apart from close to the boundary $\partial \varOmega $. We also study how to weaken this condition. The results allow in particular to establish local efficiency and robustness with respect to the exponent $p$ of a posteriori estimates for nonlinear partial differential equations in divergence form, including the case of inexact solvers. Numerical illustrations support the theory.

International audience; This paper is devoted to the study of the construction of a viscous approximation of the nonconservative bitemperature Euler system. Starting from a BGK model coupled with Ampère and Poisson equations proposed in [1], we perform a Chapman-Enskog expansion up to order 1 leading to a Navier-Stokes system. Next, we prove that this system is compatible with the entropy of the bitemperature Euler system.

In this paper we study estimation, continuous dependence and Hyers–Ulam stability for continuous solutions of a second order iterative equation. First we give an estimate for a bound of its continuous solutions. Then we give a Lipschitz estimation with the Lipschitz conditions (i.e., Holder $$\hbox {exponent} =1$$ ) to given functions, which implies a continuous dependence of solutions and encourages a further discussion on the Hyers–Ulam stability. We similarly give a Holder estimation with the Holder conditions (i.e., Holder $$\hbox {exponent}

In this paper we derive a priori $L^{\infty }(L^{2})$ and L2(L2) error estimates for a linear advection–reaction equation with inlet and outlet boundary conditions. The goal is to derive error estimates for the discontinuous Galerkin method that do not blow up exponentially with respect to time, unlike the usual case when Gronwall’s inequality is used. While this is possible in special cases, such as divergence-free advection fields, we take a more general approach using exponential scaling of the exact and discrete solutions. Here we use a special scaling function, which corresponds to time taken along individual pathlines of the flow. For advection fields, where the time that massless particles carried by the flow spend inside the spatial domain is uniformly bounded from above by some $\widehat{T}$, we derive $\mathcal{O}$(hp+1/2) error estimates where the constant factor depends only on $\widehat{T}$, but not on the final time T. This can be interpreted as applying Gronwall’s inequality in the error analysis along individual pathlines (Lagrangian setting), instead of physical time (Eulerian setting).

An optimal convergence condition for Newton iteration is presented which is at least as weak as the one obtained by Traub and Woźniakowski leading also to an at least as precise complexity. The novelty of the paper is the introduction of a restricted convergence domain. That is we find a more precise location where the Newton iterates lie than in earlier studies. Consequently the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the local convergence as well as the complexity of Newton iteration. Numerical examples further justify the theoretical results.

Recently a new fractional differentiation was introduced to get rid of the singularity in the Riemann-Liouville and Caputo fractional derivative. The new fractional derivative has then generate a new class of ordinary differential equations. These class of fractional ordinary differential equations cannot be solved using conventional Adams–Bashforth numerical scheme, thus, in this paper a new three-step fractional Adams–Bashforth scheme with the Caputo–Fabrizio derivative is formulated for the solution linear and nonlinear fractional differential equations. Stability analysis result shows that the proposed scheme is conditionally stable. Applicability and suitability of the scheme is justified when applied to solve some novel chaotic systems with fractional order α ∈ (0, 1).

In a series of papers Barron, Goebel, and Jensen studied Partial Differential Equations (PDE)s for quasiconvex (QC) functions \cite{barron2012functions, barron2012quasiconvex,barron2013quasiconvex,barron2013uniqueness}. To overcome the lack of uniqueness for the QC PDE, they introduced a regularization: a PDE for $\e$-robust QC functions, which is well-posed. Building on this work, we introduce a stronger regularization which is amenable to numerical approximation. We build convergent finite difference approximations, comparing the QC envelope and the two regularization. Solutions of this PDE are strictly convex, and smoother than the robust-QC functions.

This paper aims at approximation of functions by linear integral operators on variable exponent spaces associated with a general exponent function on a domain of a Euclidean space. Under a log-Holder continuity assumption of the exponent function, we present quantitative estimates for the approximation and solve an open problem raised in our earlier work. As applications of our key estimates, we provide high orders of approximation by quasi-interpolation type and linear combinations of Bernstein type integral operators on variable exponent spaces. We also introduce K -functionals and moduli of smoothness on variable exponent spaces and discuss their relationships and applications.

For fixed real numbers $$c>0,$$ $$\alpha >-\frac{1}{2},$$ the finite Hankel transform operator, denoted by $$\mathcal {H}_c^{\alpha }$$ is given by the integral operator defined on $$L^2(0,1)$$ with kernel $$K_{\alpha }(x,y)= \sqrt{c xy} J_{\alpha }(cxy).$$ To the operator $$\mathcal {H}_c^{\alpha },$$ we associate a positive, self-adjoint compact integral operator $$\mathcal Q_c^{\alpha }=c\, \mathcal {H}_c^{\alpha }\, \mathcal {H}_c^{\alpha }.$$ Note that the integral operators $$\mathcal {H}_c^{\alpha }$$ and $$\mathcal Q_c^{\alpha }$$ commute with a Sturm-Liouville differential operator $$\mathcal D_c^{\alpha }.$$ In this paper, we first give some useful estimates and bounds of the eigenfunctions $$\varphi ^{(\alpha )}_{n,c}$$ of $$\mathcal H_c^{\alpha }$$ or $$\mathcal Q_c^{\alpha }.$$ These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator $$\mathcal D_c^{\alpha }.$$ If $$(\mu _{n,\alpha }(c))_n$$ and $$\lambda _{n,\alpha }(c)=c\, |\mu _{n,\alpha }(c)|^2$$ denote the infinite and countable sequence of the eigenvalues of the operators $$\mathcal {H}_c^{(\alpha )}$$ and $$\mathcal Q_c^{\alpha },$$ arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer $$n\ge 0,$$ $$\lambda _{n,\alpha }(c)$$ is decreasing with respect to the parameter $$\alpha .$$ As a consequence, we show that for $$\alpha \ge \frac{1}{2},$$ the $$\lambda _{n,\alpha }(c)$$ and the $$\mu _{n,\alpha }(c)$$ have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.

In this paper, we present the finite difference method for Caputo-type parabolic equation with fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0, 1) and the spatial derivative is the fractional Laplacian. The Caputo derivative is evaluated by the L 1 approximation, and the fractional Laplacian with respect to the space variable is approximated by the Caffarelli–Silvestre extension. The difference schemes are provided together with the convergence and error estimates. Finally, numerical experiments are displayed to verify the theoretical results.

A shape-memory polymer (SMP) is capable of memorizing its original shape, and can acquire a temporary shape upon deformation and returns to its permanent shape in response to an external stimulus such as a temperature change. SMPs have been widely used industrial and medical applications. Previously, differential equation models were developed to describe SMPs and their applications. However, these models are often of very complicated form, which require accurate numerical simulations. In this paper we argue that a variable-order fractional differential equation model of the shape-memory behavior is more suitable than constant-order fractional differential equation models in terms of modeling the memory behavior of SMPs. We develop a numerical method to simulate the variable-order model and, in particular, to identify the unknown variable order of the model. Numerical experiments are presented to show the utility of the method.

We introduce an efficient algorithm for computing fractional integrals and derivatives and apply it for solving problems of the calculus of variations of fractional order. The proposed approximations are particularly useful for solving fractional boundary value problems. As an application, we solve a special class of fractional Euler-Lagrange equations. The method is based on Hale and Townsend algorithm for finding the roots and weights of the fractional Gauss-Jacobi quadrature rule and the predictor-corrector method introduced by Diethelm for solving fractional differential equations. Illustrative examples show that the given method is more accurate than the one introduced in [Comput. Math. Appl. 66 (2013), no. 5, 597--607], which uses the Golub-Welsch algorithm for evaluating fractional directional integrals., Comment: This is a preprint of a paper whose final and definite form is with 'Chaos, Solitons & Fractals', ISSN: 0960-0779. Submitted 1 Dec 2016; Article revised 17 Apr 2017; Article accepted for publication 19 Apr 2017; see [http://dx.doi.org/10.1016/j.chaos.2017.04.034]

In this paper, we study a space-time nonlocal diffusion model that reduces to the classical diffusion equation in the local limit. Firstly, we show the uniqueness and existence of the weak solution of the nonlocal model, and study the local limit of the nonlocal model as horizon parameters approach zero. Particularly, it is shown that the convergence is uniform at a rate of O ( δ + σ 2 ) , under certain regularity assumptions on initial and source data. Next we propose a fully discrete scheme, by exploiting the quadrature-based finite difference method in time and the Fourier spectral method in space, and show its stability. The numerical scheme is proved to be asymptotically compatible so that it preserves the local limiting behavior at the discrete level. Numerical experiments are provided to illustrate the theoretical results.

In this paper the Polynomial Least Squares Method is applied in order to compute analytical approximate polynomial solutions for the fractional Riccati type differential equations. The accuracy of the method is tested by means of the comparison with previous results for several applications.

In this paper, we investigate isometric extension problem in general normed space. We prove that an isometry between spheres can be extended to a linear isometry between the spaces if and only if the natural positive homogeneous extension is additive on spheres. Moreover, this conclusion still holds provided that the additivity holds on a restricted domain of spheres.

While it was noted by R. Hardy and proved in a famous paper by C. A. Micchelli that radial basis function interpolants s(x)=j(xxj) exist uniquely for the multiquadric radial function (r)=r2+c2 as soon as the (at least two) centres are pairwise distinct, the error bounds for this interpolation problem always demanded an added constant to s. By using Pontryagin native spaces, we obtain error bounds that no longer require this additional constant expression.

This paper presents a new implementation of a reliable iterative method proposed by Temimi and Ansari namely (TAM) for approximate solutions of a nonlinear problem that arises in the thin film flow of a third grade fluid on a moving belt. The solution is obtained in the form of a series that converges to the exact solution with easily computed components, without any restrictive assumptions for nonlinear terms. The results are bench-marked against a numerical solution based on the classical Runge–Kutta method (RK4) and an excellent agreement is observed. Error analysis of the approximate solution is performed using the error remainder and the maximal error remainder. An exponential rate for the convergence is achieved. A symbolic manipulator Mathematica ®10 was used to evaluate terms in the iterative process.

In this paper, we study a stochastic fractional integro-differential equation with impulsive effects in separable Hilbert space. Using a finite dimensional subspace, semigroup theory of linear operators and stochastic version of the well-known Banach fixed point theorem is applied to show the existence and uniqueness of an approximate solution. Next, these approximate solutions are shown to form a Cauchy sequence with respect to an appropriate norm, and the limit of this sequence is then a solution of the original problem. Moreover, the convergence of Faedo–Galerkin approximation of solution is shown. In the last, we have given an example to illustrate the applications of the abstract results.

The limit q-Bernstein operator Bq comes out naturally as the limit for the sequence of q-Bernstein operators in the case 0 < q < 1. Alternatively, it can be viewed as a modification of the Szasz-Mirakyan operator related to the Euler distribution. In this paper, a necessary and sufficient condition for a function g to be an image of an entire function under Bq is presented.

This paper deals with a theoretical mathematical analysis of a Cauchy problem for the time-fractional diffusion-wave equation in the upper half-plane, \(x\in \mathbb {R}\), \(t\in \mathbb {R}^+\), where the Caputo fractional derivative of order \(\alpha \in \left( 0,2\right) \) is considered. An explicit solution to this Cauchy problem is obtained via separation of variables. A first proof of the validity of the obtained results is provided for a certain kind of initial conditions. Throughout this work a new expression of the solution to this problem and its utility for carrying out rigurous proofs are presented. Finally, several new properties of the solution are obtained.

In this paper, we investigate certain input-state relations and the exponential stability for singular systems of linear difference equations. For the treatment of singular systems, we use the projector-based approach. Based on the result of decoupling, we construct admissible spaces for the inhomogeneity part of the singular systems. Three so-called Bohl–Perron type stability theorems, which are known in the literature for regular explicit difference equations, are extended to singular difference equations.

In this paper we establish a multiscale approximation for random fields on the sphere using spherical needlets—a class of spherical wavelets. We prove that the semidiscrete needlet decomposition converges in mean and pointwise senses for weakly isotropic random fields on S d , d ≥ 2 . For numerical implementation, we construct a fully discrete needlet approximation of a smooth 2 -weakly isotropic random field on S d and prove that the approximation error for fully discrete needlets has the same convergence order as that for semidiscrete needlets. Numerical examples are carried out for fully discrete needlet approximations of Gaussian random fields and compared to a discrete version of the truncated Fourier expansion.

As a first step towards the numerical analysis of the stochastic primitive equations of the atmosphere and the oceans, the time discretization of these equations by an implicit Euler scheme is studied. From the deterministic point of view, the 3D primitive equations are studied in their full form on a general domain and with physically realistic boundary conditions. From the probabilistic viewpoint, this paper deals with a wide class of nonlinear, state dependent, white noise forcings which may be interpreted in either the Ito or the Stratonovich sense. The proof of convergence of the Euler scheme, which is carried out within an abstract framework, covers the equations for the oceans, the atmosphere, the coupled oceanic-atmospheric system as well as other related geophysical equations. The authors obtain the existence of solutions which are weak in both the PDE and probabilistic sense, a result which is new by itself to the best of our knowledge.

This paper is devoted to studying an interpolation problem on the circle, which can be considered an intermediate problem between Lagrange and Hermite interpolation. The difference as well as the novelty is that we prescribe Lagrange values at the 2 n roots of a complex number with modulus one and we prescribe values for the first derivative only on half of the nodes. We obtain two types of expressions for the interpolation polynomials: the barycentric expressions and another one given in terms of an orthogonal basis of the corresponding subspace of Laurent polynomials. These expressions are very suitable for numerical computation. Moreover, we give sufficient conditions in order to obtain convergence in case of continuous functions and we obtain the rate of convergence for smooth functions. Finally we present some numerical experiments to highlight the results obtained.