1,080 research outputs found
Volterra-type convolution of classical polynomials
We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence {Pk(x)}k?0 with degPk(x)=k. Based on this framework, series representations for the convolutions of classical orthogonal polynomials, including Jacobi and Laguerre families, are derived, along with some relevant results pertaining to these new formulas
Spectral approximation of convolution operator
We develop a unified framework for constructing matrix approximations for the convolution operator of Volterra type defined by functions that are approximated using classical orthogonal polynomials on [?1, 1]. The numerically stable algorithms we propose exploit recurrence relations and symmetric properties satisfied by the entries of these convolution matrices. Laguerrebased convolution matrices that approximate Volterra convolution operator defined by functions on [0, ?] are also discussed for the sake of completeness
Existence and uniqueness of nontrivial collocation solutions of implicitly linear homogeneous Volterra integral equations
We analyze collocation methods for nonlinear homogeneous Volterra-Hammerstein
integral equations with non-Lipschitz nonlinearity. We present different kinds
of existence and uniqueness of nontrivial collocation solutions and we give
conditions for such existence and uniqueness in some cases. Finally we
illustrate these methods with an example of a collocation problem, and we give
some examples of collocation problems that do not fit in the cases studied
previously.Comment: 18 pages, 4 figure
Analysis of bilinear oscillators under harmonic loading using nonlinear output frequency response functions
In this paper, the new concept of Nonlinear Output Frequency Response Functions (NOFRFs) is extended to the harmonic input case, an input-independent relationship is found between the NOFRFs and the Generalized Frequency Response Functions (GFRFs). This relationship can greatly simplify the application of the NOFRFs. Then, beginning with the demonstration that a bilinear oscillator can be approximated using a polynomial type nonlinear oscillator, the NOFRFs are used to analyze the energy transfer phenomenon of bilinear oscillators in the frequency domain. The analysis provides insight into how new frequency generation can occur using bilinear oscillators and how the sub-resonances occur for the bilinear oscillators, and reveals that it is the resonant frequencies of the NOFRFs that dominate the occurrence of this well-known nonlinear behaviour. The results are of significance for the design and fault diagnosis of mechanical systems and structures which can be described by a bilinear oscillator model
Stability and convergence analysis of a class of continuous piecewise polynomial approximations for time fractional differential equations
We propose and study a class of numerical schemes to approximate time
fractional differential equations. The methods are based on the approximation
of the Caputo fractional derivative by continuous piecewise polynomials, which
is strongly related to the backward differentiation formulae for the
integer-order case. We investigate their theoretical properties, such as the
local truncation error and global error analyses with respect to a sufficiently
smooth solution, and the numerical stability in terms of the stability region
and -stability by refining the technique proposed in
\cite{LubichC:1986b}. Numerical experiments are given to verify the theoretical
investigations.Comment: 34 pages, 3 figure
Basics of Qualitative Theory of Linear Fractional Difference Equations
Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.This doctoral thesis concerns with the fractional calculus on discrete settings, namely in the frame of the so-called (q,h)-calculus and its special case h-calculus. First, foundations of the theory of linear fractional difference equations in (q,h)-calculus are established. In particular, basic properties, such as existence, uniqueness and structure of solutions, are discussed and a discrete analogue of the Mittag-Leffler function is introduced via eigenfunctions of a fractional difference operator. Further, qualitative analysis of a scalar and vector test fractional difference equation is performed in the frame of h-calculus. The results of stability and asymptotic analysis enable us to specify the connection to other mathematical disciplines, such as continuous fractional calculus, Volterra difference equations and numerical analysis. Finally, a possible generalization of the fractional calculus to more general settings is outlined.
p-adic Difference-Difference Lotka-Volterra Equation and Ultra-Discrete Limit
In this article, we have studied the difference-difference Lotka-Volterra
equations in p-adic number space and its p-adic valuation version. We pointed
out that the structure of the space given by taking the ultra-discrete limit is
the same as that of the -adic valuation space.Comment: AMS-Tex Use. Title change
Sparse spectral methods for integral equations and equilibrium measures
In this thesis, we introduce new numerical approaches to two important types of integral equation problems using sparse spectral methods. First, linear as well as nonlinear Volterra integral and integro-differential equations and second, power-law integral equations on d-dimensional balls involved in the solution of equilibrium measure problems.
These methods are based on ultraspherical spectral methods and share key properties and advantages as a result of their joint starting point: By working in appropriately weighted orthogonal Jacobi polynomial bases, we obtain recursively generated banded operators allowing us to obtain high precision solutions at low computational cost.
This thesis consists of three chapters in which the background of the above-mentioned problems and methods are respectively introduced in the context of their mathematical theory and applications, the necessary results to construct the operators and obtain solutions are proved and the method's applicability and efficiency are showcased by comparing them with current state-of-the-art approaches and analytic results where available. The first chapter gives a general scope introduction to sparse spectral methods using Jacobi polynomials in one and higher dimensions. The second chapter concerns the numerical solution of Volterra integral equations. The introduced method achieves exponential convergence and works for general kernels, a major advantage over comparable methods which are limited to convolution kernels.
The third chapter introduces an approximately banded method to solve power law kernel equilibrium measures in arbitrary dimensional balls. This choice of domain is suggested by the radial symmetry of the problem and analytic results on the supports of the resulting measures. For our method, we obtain the crucial property of computational cost independent of the dimension of the domain, a major contrast to particle simulations which are the current standard approach to these problems and scale extremely poorly with both the dimension and the number of particles.Open Acces
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