1,720,971 research outputs found
Volterra integral equations with highly oscillatory kernels: A new numerical method with applications
No full textThe aim of this paper is to present a Nyström-type method for the numerical approximation of the solution of Volterra integral equations of the second kind having highly oscillatory kernels. The method is based on a mixed quadrature scheme which combines the classical product rule with a dilation quadrature formula. The convergence and the stability of the method are investigated and the accuracy of the presented approach is assessed by some numerical tests. The proposed procedure is also applied to the computation of initial scattering data related to the initial value problem associated to the Korteweg-de Vries equation
A Matrix Schrodinger Approach to Focusing Nonlinear Schrodinger Equations with Nonvanishing Boundary Conditions
Full text linkWe relate the scattering theory of the focusing AKNS system with equally sized nonvanishing boundary conditions to that of the matrix Schrodinger equation. This (shifted) Miura transformation converts the focusing matrix nonlinear Schrodinger (NLS) equation into a new nonlocal integrable equation. We apply the matrix triplet method of solving the Marchenko integral equations by separation of variables to derive the multisoliton solutions of this nonlocal equation, thus proposing a method to solve the reflectionless matrix NLS equation
Reflectionless Solutions for Square Matrix NLS with Vanishing Boundary Conditions
Full text linkIn this article we derive the reflectionless solutions of the 2 + 2 matrix NLS equation with vanishing boundary conditions and four different symmetries by using the matrix triplet method of representing the Marchenko integral kernel in separated form. Apart from using the Marchenko method, these solutions are also verified by direct substitution in the 2 + 2 NLS equation
A numerical method to compute the scattering solution for the KdV equation
No full textThe authors propose a numerical method to approximate the solution of specific bivariate Volterra integral equations which arise in the numerical solution of the initial value problem for the Korteweg-de Vries equation. A preliminary study of the domain of the unknown function is carried out and a general algorithm based on the Gauss-Legendre and Newton-Cotes quadrature formulae is developed. Numerical tests are also given in order to show the efficiency of the method
Boundedness in a nonlinear attraction-repulsion Keller–Segel system with production and consumption
Full text linkThis paper is focused on the zero-flux attraction-repulsion chemotaxis model {ut=∇⋅((u+1)mjavax.xml.bind.JAXBElement@6e9df6a4−1∇u−χu(u+1)mjavax.xml.bind.JAXBElement@7c934d62−1∇v in Ω×(0,Tmax),+ξu(u+1)mjavax.xml.bind.JAXBElement@6630c70d−1∇w)vt=Δv−f(u)v in Ω×(0,Tmax),0=Δw−δw+g(u) in Ω×(0,Tmax), defined in Ω, which is a bounded and smooth domain of Rn, for n≥2, with χ,ξ,δ>0, m1,m2,m3∈R, and f(u) and g(u) reasonably regular functions generalizing the prototypes f(u)=Kuα and g(u)=γul, with K,γ>0 and appropriate α,l>0. Moreover Tmax is finite or infinite and (0,Tmax) stands for the maximal temporal interval where solutions to the related initial problem exist. Our main interest is to identify constellations of the impacts m1,m2 and m3 of the diffusion and drift terms, as well as of the growth l of the production g for the chemorepellent (i.e., w) and the rate α of the consumption f for the chemoattractant (i.e., v), which ensure boundedness of cell densities (i.e., u). Precisely, for any fixed [Formula presented] and l≥1, we prove that whenever [Formula presented] any sufficiently smooth initial data u(x,0)=u0(x)≥0 and v(x,0)=v0(x)≥0 produce a unique classical solution (u,v,w) to problem (◇) such that its life span Tmax=∞ and, moreover, u,v and w are uniformly bounded in Ω×(0,∞)
The inverse scattering transform for the focusing nonlinear Schrödinger equation with asymmetric boundary conditions
No full textThe inverse scattering transform (IST) as a tool to solve the initial-value problem for the focusing nonlinear Schr ̈odinger (NLS) equation with non-zero boundary values as x →∓∞ is presented in the fully asymmetric case for both asymptotic amplitudes and phases, i.e., with and .
The direct problem is shown to be well-defined for NLS solutions q(x, t) such that with respect to x for all t ≥ 0, and the corresponding analyticity properties of eigenfunctions and scattering data are established. The inverse scattering problem is formulated both via (left and right) Marchenko integral equations, and as a Riemann-Hilbert problem on a single sheet of the scattering variables λ_{l/r} =\sqrt{k^2 + A^2_{l/r} , where k is the usual complex scattering parameter in the IST. The time evolution of the scattering coefficients is then derived, showing that, unlike the case of solutions with equal amplitudes as x →±∞, here both reflection and transmission coefficients have a nontrivial (although explicit) time dependence. The results presented in this paper will be instrumental for the investigation of the longtime asymptotic behavior of fairly general NLS solutions with nontrivial boundary conditions via the nonlinear steepest descent method on the Riemann-Hilbert problem, or via matched asymptotic expansions on the Marchenko integral equations
The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter
Full text linkWe present the complete version including proofs of the results announced in [van der Mee C., Pivovarchik V.: A Sturm-Liouville spectral problem with boundary conditions depending on the spectral parameter. Funct. Anal. Appl. 36 (2002), 315–317 [Funkts. Anal. Prilozh. 36 (2002), 74–77 (Russian)]]. Namely, for the problem of small transversal vibrations of a damped string of nonuniform stiffness with one end fixed we give the description of the spectrum and solve the inverse problem: find the conditions which should be satisfied by a sequence of complex numbers to be the spectrum of a damped string
Numerical methods for Cauchy singular integral equations in spaces of weighted continuous functions
No full textSome convergent and stable numerical procedures for Cauchy singular
integral equations are given. The proposed approach consists of solving
the regularized equation and is based on the weighted polynomial interpolation.
The convergence estimates are sharp and the obtained linear systems
are well conditioned
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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