334 results
Search Results
2. Regularity results for the 2D critical Oldroyd-B model in the corotational case
- Author
-
Zhuan Ye
- Subjects
Logarithm ,Cauchy stress tensor ,Computer Science::Information Retrieval ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Dissipation ,Vorticity ,01 natural sciences ,010101 applied mathematics ,A priori and a posteriori ,Oldroyd-B model ,Gravitational singularity ,0101 mathematics ,Laplace operator ,Mathematics - Abstract
This paper studies the regularity results of classical solutions to the two-dimensional critical Oldroyd-B model in the corotational case. The critical case refers to the full Laplacian dissipation in the velocity or the full Laplacian dissipation in the non-Newtonian part of the stress tensor. Whether or not their classical solutions develop finite time singularities is a difficult problem and remains open. The object of this paper is two-fold. Firstly, we establish the global regularity result to the case when the critical case occurs in the velocity and a logarithmic dissipation occurs in the non-Newtonian part of the stress tensor. Secondly, when the critical case occurs in the non-Newtonian part of the stress tensor, we first present many interesting global a priori bounds, then establish a conditional global regularity in terms of the non-Newtonian part of the stress tensor. This criterion comes naturally from our approach to obtain a global L∞-bound for the vorticity ω.
- Published
- 2019
3. The stability index of hypersurfaces with constant scalar curvature in spheres
- Author
-
Qing-Ming Cheng, Haizhong Li, and Guoxin Wei
- Subjects
Mean curvature flow ,Mean curvature ,Hypersurface ,Geodesic ,General Mathematics ,Prescribed scalar curvature problem ,Mathematical analysis ,Constant (mathematics) ,Curvature ,Scalar curvature ,Mathematics ,Mathematical physics - Abstract
The totally umbilical and non-totally geodesic hypersurfaces in the (n + 1)-dimensional spheres are characterized as the only hypersurfaces with weak stability index 0. In our 2010 paper we proved that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in an (n + 1)-dimensional sphere Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H and H3 are constant. In this paper, we prove the same results, without the assumption that H3 is constant. In fact, we show that the weak stability index of a compact hypersurface M with constant scalar curvature n(n − 1)r, r> 1, in Sn + 1(1), which is not a totally umbilical hypersurface, is greater than or equal to n + 2 if the mean curvature H is constant.
- Published
- 2014
4. On travelling wavefronts of Nicholson's blowflies equation with diffusion
- Author
-
Ming Mei and Chi-Kun Lin
- Subjects
Wavefront ,General Mathematics ,Mathematical analysis ,Reaction–diffusion system ,Perturbation (astronomy) ,Wave speed ,Mathematics - Abstract
This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.
- Published
- 2010
5. Canard cycles in the presence of slow dynamics with singularities
- Author
-
Freddy Dumortier and P. De Maesschalck
- Subjects
General Mathematics ,Mathematical analysis ,Tangent ,Periodic orbits ,Perturbation (astronomy) ,Gravitational singularity ,Turning point ,Critical curve ,Eigenvalues and eigenvectors ,Mathematics ,Stability change - Abstract
We study the cyclicity of limit periodic sets that occur in families of vector flelds of slow-fast type. The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point. At this turning point a stability change takes place: on one side of the turning point, the dynamics point strongly towards the curve of singularities, on the other side the dynamics point away from the curve of singularities. The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point, i.e. orbits that stay close to the curve of singularities despite the exponentially-strong repulsion near this curve. All existing results deal with a non-zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities. In this paper we study what happens when the slow dynamics exhibit singularities. In particular our study includes the cyclicity of the slow-fast 2-saddle cycle, formed by a regular saddle-connection (the fast part) and a part of the curve of singularities (the slow part). We see that the relevant information is no longer merely contained in the slow divergence integral. This paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow-fast vector flelds on a 2-manifold M. We are interested in families of vector flelds X" (possibly depending on other parameters as well) where the unperturbed \fast" vector fleld X0 has a curve of singular points ∞, called a critical curve. We call a point p on ∞ normally attracting (resp. normally repelling) when DX0(p) has a strictly negative (resp. strictly positive) eigenvalue corresponding to an eigendirection not tangent to ∞. When ∞ has both normally attracting points and normally repelling points it may occur that X0 has orbits connecting two such points. Let F be such a fast orbit so that the !-limit and fi-limit lie on ∞. We study the limit periodic set LF formed by F and the piece of ∞ going from the !-limit of F to the fi-limit of F. Of course the nonzero eigenvalue of DX0(p) must bifurcate along this piece of ∞; assume that this happens in a unique point p⁄, called a turning point. (One also says normal hyperbolicity of X0 is lost in p⁄.) In such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits; such orbits are called canard cycles. ‡ @’
- Published
- 2008
6. Singular solutions for the Uehling–Uhlenbeck equation
- Author
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Juan J. L. Velázquez, Stéphane Mischler, and Miguel Escobedo
- Subjects
Physical point ,Regular singular point ,Singular function ,Semigroup ,Singular solution ,General Mathematics ,Mathematical analysis ,Mathematics - Abstract
In this paper we prove the existence of solutions of the Uehling–Uhlenbeck equation that behave like k −7/6 as k → 0. From the physical point of view, such solutions can be thought as particle distributions in the space of momentum having a sink (or a source) of particles with zero momentum. Our construction is based on the precise estimates of the semigroup for the linearized equation around the singular function k −7/6 that we obtained in an earlier paper.
- Published
- 2008
7. Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems
- Author
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Messoud Efendiev, Alain Miranville, and Sergey Zelik
- Subjects
Projected dynamical system ,Dynamical systems theory ,General Mathematics ,Mathematical analysis ,Attractor ,Statistical physics ,Limit set ,Dynamical system ,Random dynamical system ,Mathematics ,Linear dynamical system ,Hamiltonian system - Abstract
We suggest in this paper a new explicit algorithm allowing us to construct exponential attractors which are uniformly Hölder continuous with respect to the variation of the dynamical system in some natural large class. Moreover, we extend this construction to non-autonomous dynamical systems (dynamical processes) treating in that case the exponential attractor as a uniformly exponentially attracting, finite-dimensional and time-dependent set in the phase space. In particular, this result shows that, for a wide class of non-autonomous equations of mathematical physics, the limit dynamics remains finite dimensional no matter how complicated the dependence of the external forces on time is. We illustrate the main results of this paper on the model example of a non-autonomous reaction–diffusion system in a bounded domain.
- Published
- 2005
8. Constructing the symplectic Evans matrix using maximally analytic individual vectors
- Author
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Gianne Derks and Thomas J. Bridges
- Subjects
Determinant ,Symplectic vector space ,Pure mathematics ,General Mathematics ,Mathematical analysis ,Symplectic representation ,Symplectomorphism ,Mathematics::Symplectic Geometry ,Moment map ,Symplectic matrix ,Mathematics ,Symplectic manifold ,Symplectic geometry - Abstract
For linear systems with a multi-symplectic structure, arising from the linearization of Hamiltonian partial differential equations about a solitary wave, the Evans function can be characterized as the determinant of a matrix, and each entry of this matrix is a restricted symplectic form. This variant of the Evans function is useful for a geometric analysis of the linear stability problem. But, in general, this matrix of two-forms may have branch points at isolated points, shrinking the natural region of analyticity. In this paper, a new construction of the symplectic Evans matrix is presented, which is based on individual vectors but is analytic at the branch points—indeed, maximally analytic. In fact, this result has greater generality than just the symplectic case; it solves the following open problem in the literature: can the Evans function be constructed in a maximally analytic way when individual vectors are used? Although the non-symplectic case will be discussed in passing, the paper will concentrate on the symplectic case, where there are geometric reasons for evaluating the Evans function on individual vectors. This result simplifies and generalizes the multi-symplectic framework for the stability analysis of solitary waves, and some of the implications are discussed.
- Published
- 2003
9. Analysis of the PML equations in general convex geometry
- Author
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Erkki Somersalo and Matti Lassas
- Subjects
Convex geometry ,Scattering ,General Mathematics ,Mathematical analysis ,Boundary (topology) ,Geometry ,010103 numerical & computational mathematics ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,symbols.namesake ,Perfectly matched layer ,Bounded function ,Dirichlet boundary condition ,symbols ,0101 mathematics ,Convex function ,Mathematics - Abstract
In this work, we study a mesh termination scheme in acoustic scattering, known as the perfectly matched layer (PML) method. The main result of the paper is the following. Assume that the scatterer is contained in a bounded and strictly convex artificial domain. We surround this domain by a PML of constant thickness. On the peripheral boundary of this layer, a homogenous Dirichlet condition is imposed. We show in this paper that the resulting boundary-value problem for the scattered field is uniquely solvable for all wavenumbers and the solution within the artificial domain converges exponentially fast toward the full-space scattering solution when the layer thickness is increased. The proof is based on the idea of interpreting the PML medium as a complex stretching of the coordinates in Rn and on the use of complexified layer potential techniques.
- Published
- 2001
10. Ergodic properties and Weyl M-functions for random linear Hamiltonian systems
- Author
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Russell Johnson, Sylvia Novo, and Rafael Obaya
- Subjects
Floquet theory ,Computer Science::Information Retrieval ,General Mathematics ,Mathematical analysis ,Schrödinger equation ,Hamiltonian system ,symbols.namesake ,symbols ,Ergodic theory ,Covariant Hamiltonian field theory ,Superintegrable Hamiltonian system ,Invariant (mathematics) ,Hamiltonian (control theory) ,Mathematical physics ,Mathematics - Abstract
This paper provides a topological and ergodic analysis of random linear Hamiltonian systems. We consider a class of Hamiltonian equations presenting absolutely continuous dynamics and prove the existence of the radial limits of the Weyl M-functions in the L1-topology. The proof is based on previous ergodic relations obtained for the Floquet coefficient. The second part of the paper is devoted to the qualitative description of disconjugate linear Hamiltonian equations. We show that the principal solutions at ±∞ define singular ergodic measures, and determine an invariant region in the Lagrange bundle which concentrates the essential dynamical information. We apply this theory to the study of the n-dimensional Schrödinger equation at the first point of the spectrum.
- Published
- 2000
11. Global regularity criterion for the dissipative systems modelling electrohydrodynamics involving the middle eigenvalue of the strain tensor
- Author
-
Fan Wu
- Subjects
Physics::Fluid Dynamics ,General Mathematics ,Mathematical analysis ,Dissipative system ,Infinitesimal strain theory ,Electrohydrodynamics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we study a dissipative systems modelling electrohydrodynamics in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a classical Poisson–Nernst–Planck equations. In the three-dimensional case, we establish a global regularity criteria in terms of the middle eigenvalue of the strain tensor in the framework of the anisotropic Lorentz spaces for local smooth solution. The proof relies on the identity for entropy growth introduced by Miller in the Arch. Ration. Mech. Anal. [16].
- Published
- 2021
12. Rank one plus a null-Lagrangian is an inherited property of two-dimensional compliance tensors under homogenisation
- Author
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Graeme W. Milton and Yury Grabovsky
- Subjects
Shear modulus ,Algebra ,Rank (linear algebra) ,General Mathematics ,Mathematical analysis ,Isotropy ,Tensor ,Space (mathematics) ,Representation (mathematics) ,Null (physics) ,Mathematics ,Moduli - Abstract
Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.
- Published
- 1998
13. A relationship between the periodic and the Dirichlet BVPs of singular differential equations
- Author
-
Meirong Zhang
- Subjects
Differential equation ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Infinity ,Dirichlet distribution ,symbols.namesake ,Singularity ,Ordinary differential equation ,Dirichlet's principle ,symbols ,Gravitational singularity ,Boundary value problem ,media_common ,Mathematics - Abstract
In this paper, a relationship between the periodic and the Dirichlet boundary value problems for second-order ordinary differential equations with singularities is established. This relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations. Using this relationship, we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity.
- Published
- 1998
14. Convergence of the viscosity solutions for weakly strictly hyperbolic conservation laws
- Author
-
Zhu Changjiang
- Subjects
Conservation law ,General Mathematics ,Viscosity (programming) ,Convergence (routing) ,Mathematical analysis ,Mathematics - Abstract
SynopsisThis paper is an extension of papers [14–16]. Using the theory of compensated compactness, we establish the convergence of the uniformly bounded approximate solution sequence for a class of ‘weakly strictly hyperbolic’ conservation laws.
- Published
- 1997
15. Regularity of the solutions for elliptic problems on nonsmooth domains in ℝ3, Part I: countably normed spaces on polyhedral domains
- Author
-
Benqi Guo and Ivo Babuška
- Subjects
Sobolev space ,Pure mathematics ,Continuous function ,General Mathematics ,Mathematical analysis ,Neighbourhood (graph theory) ,Structure (category theory) ,Piecewise ,Ellipse ,Mathematics ,Vector space ,Analytic function - Abstract
This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.
- Published
- 1997
16. Positivity of solutions of elliptic equations with nonlocal terms
- Author
-
A. Barabanova and W. Allegretto
- Subjects
Nonlinear system ,Partial differential equation ,Distribution (number theory) ,General Mathematics ,Mathematical analysis ,Value (mathematics) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study a nonlocal problem for a second-order partial differential equation which depends on a parametern. We prove the existence of critical values 0
such that for all≦ɳ≦and for all non-negative right-hand sides, our problem has nonnegative solutions. We obtain a formula forɳ0, the maximal possible value of, and find the exact value of ɳ for spherical ɳ. We also study the corresponding eigenvalue problem. At the end of the paper, we consider the application of our results to the nonlinear system describing the distribution of temperature and potential in a microsensor. - Published
- 1996
17. L1-approximation of Fourier series of complex-valued functions
- Author
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T. F. Xie and S. P. Zhou
- Subjects
symbols.namesake ,Fourier transform ,Fourier analysis ,Discrete-time Fourier transform ,General Mathematics ,Discrete Fourier series ,Mathematical analysis ,Conjugate Fourier series ,Fourier sine and cosine series ,symbols ,Fourier series ,Mathematics ,Parseval's theorem - Abstract
V. B. Stanojevic suggested in her recent paper that it would be of interest to prove a corresponding L1-convergence theorem for Fourier series with complex O-regularly varying quasimonotonc coefficients. The present paper will discuss this question and establish L1-convergence and. furthermore. L1-approximation theorems for complex-valued integrable functions.
- Published
- 1996
18. On the absolutely continuous subspaces of Floquet operators
- Author
-
Min-Jei Huang
- Subjects
Floquet theory ,Algebra ,Operator (computer programming) ,General Mathematics ,Mathematical analysis ,Absolute continuity ,Linear subspace ,Subspace topology ,Mathematics - Abstract
The purpose of this paper is to describe various subspaces that are closely related to the absolutely continuous subspace of a Floquet operator. This paper generalises and extends several known results.
- Published
- 1994
19. On the analysis and control of hyperbolic systems associated with vibrating networks
- Author
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Günter Leugering, J. E. Lagnese, and E. J. P. G. Schmidt
- Subjects
Timoshenko beam theory ,Controllability ,General linear model ,Control theory ,General Mathematics ,Mathematical analysis ,Vibration control ,Control (linguistics) ,Closed loop ,Hyperbolic partial differential equation ,Hyperbolic systems ,Mathematics - Abstract
In this paper a general linear model for vibrating networks of one-dimensional elements is derived. This is applied to various situations including nonplanar networks of beams modelled by a three-dimensional variant on the Timoshenko beam, described for the first time in this paper. The existence and regularity of solutions is established for all the networks under consideration. The methods of first-order hyperbolic systems are used to obtain estimates from which exact controllability follows for networks containing no closed loops.
- Published
- 1994
20. Hyperasymptotics and the Stokes' phenomenon
- Author
-
A. B. Olde Daalhuis
- Subjects
symbols.namesake ,Integral representation ,General Mathematics ,Phenomenon ,Confluence ,Mathematical analysis ,symbols ,Hypergeometric function ,Series expansion ,Bessel function ,Mathematics - Abstract
SynopsisHyperasymptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric U-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.
- Published
- 1993
21. On the commutativity of certain quasi-differential expressions II
- Author
-
H. Frentzen, D. Race, and Anton Zettl
- Subjects
Pure mathematics ,Computer Science::Information Retrieval ,General Mathematics ,Mathematical analysis ,Commutative property ,Differential (mathematics) ,Mathematics - Abstract
SynopsisWe consider the question: when do two ordinary, linear, quasi-differential expressions commute? For classical differential expressions, answers to this question are well known. The set of all expressions which commute with a given such expression form a commutative ring. For quasi-differential expressions less is known and such an algebraiastructure can no longer be exploited. Using the theory of very general quasi-differential expressions with matrix-valued coefficients, we prove some general results concerning commutativity of such expressions. We show how, when specialised to scalar expressions, these results include a proof of the conjecture that if a 2nth-order scalar, J-symmetric (or real symmetric) quasi-differential expression commutes with a second order expression having the same properties, then the former must be an nth-order polynomial in the latter. This result was conjectured in a paper by Race and Zettl, to which this paper is a sequel.
- Published
- 1993
22. Solutions in Lebesgue spaces of the Navier-Stokes equations with dynamic boundary conditions
- Author
-
Niko Sauer and Marié Grobbelaar-Van Dalsen
- Subjects
General Mathematics ,Mathematical analysis ,Rigid body ,Lebesgue integration ,Lebesgue–Stieltjes integration ,Physics::Fluid Dynamics ,symbols.namesake ,Compressibility ,symbols ,Boundary value problem ,Lp space ,Navier–Stokes equations ,Rotation (mathematics) ,Mathematics - Abstract
SynopsisThis paper, although self-contained, is a continuation of the work done in [8], where the motion of a viscous, incompressible fluid is considered in conjunction with the rotation of a rigid body which is immersed in the fluid. The resulting mathematical model is a Navier-Stokes problem with dynamic boundary conditions. In [8] a uniqueL2,3solution is constructed under certain regularity assumptions on the initial states. In this paper we consider the Navier-Stokes problem with dynamic boundary conditions in the Lebesgue spacesLr,3(3
- Published
- 1993
23. Existence of solution for elliptic equations with supercritical Trudinger–Moser growth
- Author
-
Luiz F. O. Faria and Marcelo Montenegro
- Subjects
010101 applied mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Mathematics::Analysis of PDEs ,0101 mathematics ,01 natural sciences ,Supercritical fluid ,Mathematics - Abstract
This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution.
- Published
- 2021
24. Attractors of partial differential evolution equations in an unbounded domain
- Author
-
M. I. Vishik and A. V. Babin
- Subjects
Stochastic partial differential equation ,Sobolev space ,Pure mathematics ,Nonlinear system ,Partial differential equation ,Distributed parameter system ,General Mathematics ,Mathematical analysis ,Attractor ,Domain (mathematical analysis) ,Mathematics ,Numerical partial differential equations - Abstract
SynopsisThere is a large number of papers in which attractors of parabolic reaction-diffusion equations in bounded domains are investigated. In this paper, these equations are considered in the whole unbounded space, and a theory of attractors of such equations is built. While investigating these equations, specific difficulties arise connected with the noncompactness of operators, with the continuity of their spectra, etc. Therefore some new conditions on nonlinear terms arise. In this paper weighted spaces are widely applied. An important feature of this problem is worth mentioning: namely, properties of semigroups corresponding to equations with solutions in spaces of growing and of decreasing functions essentially differ.
- Published
- 1990
25. Continuity of solutions for the Δϕ-Laplacian operator
- Author
-
Juan F. Spedaletti, Ariel Martin Salort, and Natalí Ailín Cantizano
- Subjects
010101 applied mathematics ,Class (set theory) ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Point (geometry) ,0101 mathematics ,Type (model theory) ,01 natural sciences ,Laplace operator ,Domain (mathematical analysis) ,Mathematics - Abstract
In this paper we give sufficient conditions to obtain continuity results of solutions for the so called ϕ-Laplacian Δϕ with respect to domain perturbations. We point out that this kind of results can be extended to a more general class of operators including, for instance, nonlocal nonstandard growth type operators.
- Published
- 2020
26. Semi-classical solutions for Kirchhoff type problem with a critical frequency
- Author
-
Xu Zhang and Qilin Xie
- Subjects
010101 applied mathematics ,Critical frequency ,Kirchhoff type ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Vanish at infinity ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In the present paper, we consider the following Kirchhoff type problem $$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.
- Published
- 2020
27. Existence results for the Kudryashov–Sinelshchikov–Olver equation
- Author
-
Lorenzo di Ruvo and Giuseppe Maria Coclite
- Subjects
Cauchy problem ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Existence ,01 natural sciences ,Kudryashov-Sinelshchikov-Olver equation ,010305 fluids & plasmas ,Physics::Fluid Dynamics ,Viscosity ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,0103 physical sciences ,Heat transfer ,Initial value problem ,0101 mathematics ,Mathematics - Abstract
The Kudryashov–Sinelshchikov–Olver equation describes pressure waves in liquids with gas bubbles taking into account heat transfer and viscosity. In this paper, we prove the existence of solutions of the Cauchy problem associated with this equation.
- Published
- 2020
28. Distributions as initial values in a triangular hyperbolic system of conservation laws
- Author
-
A. Paiva and C. O. R. Sarrico
- Subjects
Shock wave ,Cauchy problem ,Conservation law ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Extension (predicate logic) ,Space (mathematics) ,01 natural sciences ,Hyperbolic systems ,Product (mathematics) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Solution concept ,Mathematics - Abstract
The present paper concerns the systemut + [ϕ(u)]x = 0,vt + [ψ(u)v]x = 0 having distributions as initial conditions. Under certain conditions, and supposingϕ,ψ: ℝ → ℝ functions, we explicitly solve this Cauchy problem within a convenient space of distributionsu,v. For this purpose, a consistent extension of the classical solution concept defined in the setting of a distributional product (not constructed by approximation processes) is used. Shock waves,δ-shock waves, and also waves defined by distributions that are not measures are presented explicitly as examples. This study is carried out without assuming classical results about conservation laws. For reader's convenience, a brief survey of the distributional product is also included.
- Published
- 2019
29. A priori bounds and existence of non-real eigenvalues for singular indefinite Sturm–Liouville problems with limit-circle type endpoints
- Author
-
Jiangang Qi and Fu Sun
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,Sturm–Liouville theory ,Mathematics::Spectral Theory ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,A priori and a posteriori ,Limit (mathematics) ,Boundary value problem ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The present paper deals with non-real eigenvalues of singular indefinite Sturm–Liouville problems with limit-circle type endpoints. A priori bounds and the existence of non-real eigenvalues of the problem associated with a special separated boundary condition are obtained.
- Published
- 2019
30. Wave propagation for a class of non-local dispersal non-cooperative systems
- Author
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Jia-Bing Wang, Fei-Ying Yang, and Wan-Tong Li
- Subjects
Class (set theory) ,Wave propagation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Bilinear interpolation ,Non local ,01 natural sciences ,010101 applied mathematics ,Traveling wave ,Biological dispersal ,0101 mathematics ,Epidemic model ,Mathematics ,Incidence (geometry) - Abstract
This paper is concerned with the travelling waves for a class of non-local dispersal non-cooperative system, which can model the prey-predator and disease-transmission mechanism. By the Schauder's fixed-point theorem, we first establish the existence of travelling waves connecting the semi-trivial equilibrium to non-trivial leftover concentrations, whose bounds are deduced from a precise analysis. Further, we characterize the minimal wave speed of travelling waves and obtain the non-existence of travelling waves with slow speed. Finally, we apply the general results to an epidemic model with bilinear incidence for its propagation dynamics.
- Published
- 2019
31. Ground state solutions of Hamiltonian elliptic systems in dimension two
- Author
-
Jianjun Zhang, João Marcos do Ó, and Djairo G. de Figueiredo
- Subjects
Elliptic systems ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Of the form ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Bounded function ,symbols ,0101 mathematics ,Nehari manifold ,Hamiltonian (quantum mechanics) ,Ground state ,Schrödinger's cat ,Mathematics - Abstract
The aim of this paper is to study Hamiltonian elliptic system of the form 0.1$$\left\{ {\matrix{ {-\Delta u = g(v)} & {{\rm in}\;\Omega,} \cr {-\Delta v = f(u)} & {{\rm in}\;\Omega,} \cr {u = 0,v = 0} & {{\rm on}\;\partial \Omega,} \cr } } \right.$$ where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane 0.2$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).
- Published
- 2019
32. Non-collapsing in homogeneity greater than one via a two-point method for a special case
- Author
-
Heiko Kröner
- Subjects
General Mathematics ,Homogeneity (statistics) ,010102 general mathematics ,Mathematical analysis ,Curvature ,01 natural sciences ,Maximum principle ,Two point method ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Special case ,Normal ,Mathematics - Abstract
We study the mechanism of proving non-collapsing in the context of extrinsic curvature flows via the maximum principle in combination with a suitable two-point function in homogeneity greater than one. Our paper serves as the first step in this direction and we consider the case of a curve which isC2-close to a circle initially and which flows by a power greater than one of the curvature along its normal vector.
- Published
- 2019
33. Twists and shear maps in nonlinear elasticity: explicit solutions and vanishing Jacobians
- Author
-
Sandra Kabisch and Jonathan J. Bevan
- Subjects
Global energy ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Stationary point ,010101 applied mathematics ,49N60, 74G40 ,symbols.namesake ,Mathematics - Analysis of PDEs ,Shear (geology) ,Jacobian matrix and determinant ,FOS: Mathematics ,symbols ,Uniqueness ,0101 mathematics ,Twist ,Nonlinear elasticity ,Analysis of PDEs (math.AP) ,Mathematics ,Energy functional - Abstract
In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine in particular the relationship between the positivity of the Jacobian $\det \nabla u$ and the uniqueness and regularity of energy minimizers $u$ that are either twist maps or shear maps. We exhibit \emph{explicit} twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer $u_{\sigma}: \Omega\to \mathbb{R}^2$ in a model, two-dimensional case. The shear map minimizer has the properties that (i) $\det \nabla u_{\sigma}$ is strictly positive on one part of the domain $\Omega$, (ii) $\det \nabla u_{\sigma} = 0$ necessarily holds on the rest of $\Omega$, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma}$ is not continuous on the whole domain., Comment: 2 figures
- Published
- 2019
34. On the classification of standing wave solutions to a coupled Schrödinger system
- Author
-
Zhi You Chen, Yu Jen Huang, and Yong Li Tang
- Subjects
Dirichlet problem ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,010101 applied mathematics ,Standing wave ,symbols.namesake ,Nonlinear system ,symbols ,Uniqueness ,Ball (mathematics) ,0101 mathematics ,Nonlinear Schrödinger equation ,Stationary state ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we consider a nonlinear elliptic system which is an extension of the single equation derived by investigating the stationary states of the nonlinear Schrödinger equation. We establish the existence and uniqueness of solutions to the Dirichlet problem on the ball and entire space as the parameters within certain regions. In addition, a complete structure of different types of solutions for the radial case is also provided.
- Published
- 2019
35. Logarithmic upper bounds for weak solutions to a class of parabolic equations
- Author
-
Xiangsheng Xu
- Subjects
Class (set theory) ,Mathematics - Analysis of PDEs ,Logarithm ,General Mathematics ,Weak solution ,Mathematical analysis ,FOS: Mathematics ,Boundary value problem ,Parabolic partial differential equation ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
It is well known that a weak solution φ to the initial boundary value problem for the uniformly parabolic equation $\partial _t\varphi - {\rm div}(A\nabla \varphi ) +\omega \varphi = f $ in $\Omega _T\equiv \Omega \times (0,T)$ satisfies the uniform estimate $$\Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi\Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{q,\Omega_T}, \ \ \ c=c(N,\lambda, q, \Omega_T), $$ provided that $q \gt 1+{N}/{2}$, where Ω is a bounded domain in ${\open R}^N$ with Lipschitz boundary, T > 0, $\partial _p\Omega _T$ is the parabolic boundary of $\Omega _T$, $\omega \in L^1(\Omega _T)$ with $\omega \ges 0$, and λ is the smallest eigenvalue of the coefficient matrix A. This estimate is sharp in the sense that it generally fails if $q=1+{N}/{2}$. In this paper, we show that the linear growth of the upper bound in $\Vert f \Vert_{q,\Omega _T}$ can be improved. To be precise, we establish $$ \Vert \varphi \Vert_{\infty,\Omega_T}\les \Vert \varphi_0 \Vert_{\infty,\partial_p\Omega_T}+c \Vert f \Vert_{1+{N}/{2},\Omega_T} \left(\ln(\Vert f \Vert_{q,\Omega_T}+1)+1\right). $$
- Published
- 2019
36. Periodic solutions for a second-order differential equation with indefinite weak singularity
- Author
-
Manuel Zamora and José Godoy
- Subjects
010101 applied mathematics ,Singularity ,Differential equation ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Order (group theory) ,Function (mathematics) ,0101 mathematics ,01 natural sciences ,Mathematics ,Term (time) - Abstract
As a consequence of the main result of this paper efficient conditions guaranteeing the existence of a T −periodic solution to the second-order differential equation $${u}^{\prime \prime} = \displaystyle{{h(t)} \over {u^\lambda }}$$are established. Here, h ∈ L(ℝ/Tℤ) is a piecewise-constant sign-changing function and the non-linear term presents a weak singularity at 0 (i.e. λ ∈ (0, 1)).
- Published
- 2019
37. On the boundary conditions in estimating ∇ω by div ω and curl ω
- Author
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Dhanya Rajendran, Olivier Kneuss, and Gyula Csató
- Subjects
Curl (mathematics) ,General Mathematics ,Mathematical analysis ,Boundary value problem ,Mathematics - Abstract
In this paper, we study under what boundary conditions the inequality$${\rm \Vert }\nabla \omega {\rm \Vert }_{L^2(\Omega )}^2 \les C({\rm \Vert }{\rm curl}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }{\rm div}\omega {\rm \Vert }_{L^2(\Omega )}^2 + {\rm \Vert }\omega {\rm \Vert }_{L^2(\Omega )}^2 )$$holds true. It is known that such an estimate holds if either the tangential or normal component ofωvanishes on the boundary ∂Ω. We show that the vanishing tangential component condition is a special case of a more general one. In two dimensions, we give an interpolation result between these two classical boundary conditions.
- Published
- 2018
38. A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects
- Author
-
Wen Yang and Aleks Jevnikar
- Subjects
Inequality ,Turbulence ,General Mathematics ,media_common.quotation_subject ,Mathematical analysis ,Mathematics::Analysis of PDEs ,01 natural sciences ,010305 fluids & plasmas ,Vortex ,Mathematics - Analysis of PDEs ,Argument ,Mean field equation ,Phenomenon ,0103 physical sciences ,FOS: Mathematics ,35J61, 35J20, 35R01, 35B44 ,010306 general physics ,Analysis of PDEs (math.AP) ,Variable (mathematics) ,Mathematics ,Probability measure ,media_common - Abstract
We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper, we describe the blow-up picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. In the second part, we discuss the Moser–Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser–Trudinger inequalities.
- Published
- 2018
39. On the well-posedness and asymptotic behaviour of the generalized Korteweg–de Vries–Burgers equation
- Author
-
Fernando A. Gallego and Ademir F. Pazoto
- Subjects
General Mathematics ,Mathematical analysis ,01 natural sciences ,010305 fluids & plasmas ,Burgers' equation ,Term (time) ,Multiplier (Fourier analysis) ,Compact space ,0103 physical sciences ,Exponent ,Exponential decay ,010306 general physics ,Real line ,Mathematics ,Interpolation theory - Abstract
In this paper we are concerned with the well-posedness and the exponential stabilization of the generalized Korteweg–de Vries–Burgers equation, posed on the whole real line, under the effect of a damping term. Both problems are investigated when the exponent p in the nonlinear term ranges over the interval [1, 5). We first prove the global well-posedness in Hs(ℝ) for 0 ≤ s ≤ 3 and 1 ≤ p < 2, and in H3(ℝ) when p ≥ 2. For 2 ≤ p < 5, we prove the existence of global solutions in the L2-setting. Then, by using multiplier techniques and interpolation theory, the exponential stabilization is obtained with an indefinite damping term and 1 ≤ p < 2. Under the effect of a localized damping term the result is obtained when 2 ≤ p < 5. Combining multiplier techniques and compactness arguments, we show that the problem of exponential decay is reduced to proving the unique continuation property of weak solutions. Here, the unique continuation is obtained via the usual Carleman estimate.
- Published
- 2018
40. Sharp decay estimates in a bioconvection model with quadratic degradation in bounded domains
- Author
-
Xinru Cao and Michael Winkler
- Subjects
Cauchy problem ,Solenoidal vector field ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Quadratic equation ,Bounded function ,Neumann boundary condition ,Fluid dynamics ,0101 mathematics ,Mathematics ,Degradation (telecommunications) - Abstract
The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given byunder Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.
- Published
- 2018
41. On the finite-element approximation of ∞-harmonic functions
- Author
-
Tristan Pryer
- Subjects
Discretization ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Mathematical analysis ,Infinity ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Harmonic function ,Limit (mathematics) ,0101 mathematics ,Galerkin method ,Laplace operator ,media_common ,Mathematics - Abstract
In this paper we show that conforming Galerkin approximations for p-harmonic functions tend to ∞-harmonic functions in the limit p → ∞ and h → 0, where h denotes the Galerkin discretization parameter.
- Published
- 2018
42. Local Hölder estimates for non-uniformly variable exponent elliptic equations in divergence form
- Author
-
Fengping Yao
- Subjects
010101 applied mathematics ,Variable exponent ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,0101 mathematics ,Divergence (statistics) ,01 natural sciences ,Mathematics - Abstract
In this paper we obtain the local Hölder regularity of the gradients of weak solutions for a class of non-uniformly nonlinear variable exponent elliptic equations in divergence formincluding the following special modelunder some proper assumptions on Ai and the Hölder continuous functions f, pi(x) for i = 1, 2.
- Published
- 2017
43. Some sharp results about the global existence and blowup of solutions to a class of pseudo-parabolic equations
- Author
-
Fuyi Li, Yuhua Li, and Xiaoli Zhu
- Subjects
Class (set theory) ,Semigroup ,Computer Science::Information Retrieval ,General Mathematics ,Weak solution ,010102 general mathematics ,Mathematical analysis ,Space (mathematics) ,01 natural sciences ,Parabolic partial differential equation ,010101 applied mathematics ,0101 mathematics ,Exponential decay ,Nehari manifold ,Energy functional ,Mathematics - Abstract
In this paper we are interested in a sharp result about the global existence and blowup of solutions to a class of pseudo-parabolic equations. First, we represent a unique local weak solution in a new integral form that does not depend on any semigroup. Second, with the help of the Nehari manifold related to the stationary equation, we separate the whole space into two components S+ and S– via a new method, then a sufficient and necessary condition under which the weak solution blows up is established, that is, a weak solution blows up at a finite time if and only if the initial data belongs to S–. Furthermore, we study the decay behaviour of both the solution and the energy functional, and the decay ratios are given specifically.
- Published
- 2017
44. Morse index and symmetry breaking for an elliptic equation with negative exponent in expanding annuli
- Author
-
Linfeng Mei, Zhitao Zhang, and Zongming Guo
- Subjects
Elliptic curve ,Index (economics) ,Negative exponent ,law ,General Mathematics ,Mathematical analysis ,Symmetry breaking ,Morse code ,Mathematics ,law.invention - Abstract
Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.
- Published
- 2017
45. Asymptotic behaviour of the lifespan of solutions for a semilinear heat equation in hyperbolic space
- Author
-
Jingxue Yin and Zhiyong Wang
- Subjects
010101 applied mathematics ,General Mathematics ,Hyperbolic space ,010102 general mathematics ,Mathematical analysis ,Geodetic datum ,Heat equation ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with the asymptotic behaviour of the lifespan of solutions for a semilinear heat equation with initial datum λφ(x) in hyperbolic space. The growth rates for both λ → 0 and λ → ∞ are determined.
- Published
- 2016
46. A convergence result for the ergodic problem for Hamilton–Jacobi equations with Neumann-type boundary conditions
- Author
-
Hitoshi Ishii, Ebraheem O. Alzahrani, Arshad M. M. Younas, and Eman S. Al-Aidarous
- Subjects
Asymptotic analysis ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Function (mathematics) ,01 natural sciences ,Hamilton–Jacobi equation ,010101 applied mathematics ,Kolmogorov equations (Markov jump process) ,Ergodic theory ,Applied mathematics ,Convergence problem ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by uλ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.
- Published
- 2016
47. Dynamics in the fundamental solution of a non-convex conservation law
- Author
-
Young-Ran Lee and Yong-Jung Kim
- Subjects
Conservation law ,Astrophysics::High Energy Astrophysical Phenomena ,General Mathematics ,010102 general mathematics ,Scalar (mathematics) ,Mathematical analysis ,Regular polygon ,01 natural sciences ,Convexity ,010101 applied mathematics ,Classical mechanics ,Inflection point ,Jump ,Fundamental solution ,0101 mathematics ,Mathematics - Abstract
There is a huge jump in the theory of conservation laws if the convexity assumption is dropped. In this paper we study a scalar conservation law without the convexity assumption by monitoring the dynamics in the fundamental solution. Three extra shock types are introduced other than the usual genuine shock, which are left, right and double sided contacts. There are three kinds of phenomena of these shocks, which are called branching, merging and transforming. All of these shocks and phenomena can be observed if the flux function has two inflection points. A comprehensive picture of a global dynamics of a nonconvex flux is discussed in terms of characteristic maps and dynamical convex-concave envelopes.
- Published
- 2015
48. A note on linear ordinary quasi-differential equations
- Author
-
W. N. Everitt
- Subjects
Differential equation ,General Mathematics ,Mathematical analysis ,First-order partial differential equation ,Linear equation ,Mathematics - Abstract
SynopsisThe theory of differential equations is largely concerned with properties of solutions of individual, or classes of, equations. This paper is given over to the converse problem - that of seeking properties of functions which require them to be, in some respect, solutions of a differential equation, and to determining all possible such differential equations.From this point of view this paper discusses only linear ordinary quasi-differential equations of the second order. However, the methods can be extended to quasi-differential equations of general order.
- Published
- 1985
49. Polynomial interpolation at points of a geometric mesh on a triangle
- Author
-
S. L. Lee and George M. Phillips
- Subjects
Discrete mathematics ,Inverse quadratic interpolation ,General Mathematics ,Mathematical analysis ,Bilinear interpolation ,Linear interpolation ,Birkhoff interpolation ,Polynomial interpolation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Bicubic interpolation ,Spline interpolation ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics ,Interpolation - Abstract
SynopsisIn an earlier paper [8], I. J. Schoenberg discussed polynomial interpolation in one dimension at the points of a geometric progression, which was originally proposed by James Stirling. In the present paper, these ideas are generalised to two-dimensional polynomial interpolation at the points of a geometric mesh on a triangle. A Lagrange form is obtained for this interpolating polynomial and an algorithm is derived for evaluating it efficiently.
- Published
- 1988
50. Pointwise convergence of eigenfunction expansions, associated with a pair of ordinary differential expressions
- Author
-
Hsv Desnoo, Ea Coddington, and Aad Dijksma
- Subjects
Pointwise convergence ,General Mathematics ,Uniform convergence ,Mathematical analysis ,Eigenfunction ,Modes of convergence ,Differential (mathematics) ,Compact convergence ,Mathematics - Abstract
SynopsisFor the differential equation Lf = λMf on an open interval of ℝ, a theory in terms of relations in a Hilbert space associated with M was developed in a paper by Coddington and de Snoo, and eigenfunction expansions were derived in a paper by Dijksma and de Snoo. In the case of a regular problem on a compact interval, pointwise convergence of the expansions was shown in another paper by Coddington and de Snoo. Here, we show pointwise convergence in the general singular case.
- Published
- 1984
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