125,175 research outputs found
Runge-Kutta Fehlberg Method for Solving Linear and Nonlinear Fuzzy Fredholm Integro-Differential Equations
No full textRunge-Kutta Fehlberg Method for Solving Linear and Nonlinear Fuzzy Fredholm Integro-Differential Equation
Mixed collocation methods for y” = f(x , y)
No full textThe second-order initial value problem y" = f(x,y), y(x(_0)) = y(_0), y'(x(_0)) = z(_0) which does not contain the first derivative explicitly and where the solution is oscillatory has been of great interest for many years. Our aim is to construct numerical methods which are tuned to act efficiently on strongly oscillating functions. The frequencies involved determine the oscillatory character of the function and as the frequencies approach zero, the classical methods are obtained. The exponential- fitting tool has become increasingly popular as it is specially tailored for oscillating functions. Many classes of methods have been used with exponential-fitting and this will be discussed in more detail in the thesis. Collocation methods are considered for which the basis functions are combinations of polynomial and trigonometric terms. The resulting methods can be regarded as Runge-Kutta-Nyström methods with steplength dependent coefficients. We show how order conditions may be obtained, investigate the stability and other properties of particular methods and present some numerical results
Collocation methods for a class of second order initial value problems with oscillatory solutions
No full textWe derive and analyse two families of multistep collocation methods for periodic initial-value problems of the form y" = f(x, y); y((^x)o) = yo, y(^1)(xo) = zo involving ordinary differential equations of second order in which the first derivative does not appear explicitly. A survey of recent results and proposed numerical methods is given in chapter 2. Chapter 3 is devoted to the analysis of a family of implicit Chebyshev methods proposed by Panovsky k Richardson. We show that for each non-negative integer r, there are two methods of order 2r from this family which possess non-vanishing intervals of periodicity. The equivalence of these methods with one-step collocation methods is also established, and these methods are shown to be neither P-stable nor symplectic. In chapters 4 and 5, two families of multistep collocation methods are derived, and their order and stability properties are investigated. A detailed analysis of the two-step symmetric methods from each class is also given. The multistep Runge-Kutta-Nystrom methods of chapter 4 are found to be difficult to analyse, and the specific examples considered are found to perform poorly in the areas of both accuracy and stability. By contrast, the two-step symmetric hybrid methods of chapter 5 are shown to have excellent stability properties, in particular we show that all two-step 27V-point methods of this type possess non-vanishing intervals of periodicity, and we give conditions under which these methods are almost P-stable. P-stable and efficient methods from this family are obtained and demonstrated in numerical experiments. A simple, cheap and effective error estimator for these methods is also given
Efficient numerical methods for Volterra integral equations of Hammerstein type
No full textVolterra integral equations (VIEs) are the mathematical model of many evolutionary problems with memory arising from biology, chemistry, physics, engineering. It is known that the numerical treatment of VIEs has an high computational cost, due mainly to the computation of the ``lag term'' or ``tail term'' which contains the past history of the phenomenon. Since it depends on time, the ``lag term'' has to be computed for each time step and its cost increases when time passes. Among the Volterra equations, the Hammerstein type ones, are particularly interesting for the applications.
The aim of this thesis is the construction of numerical methods for VIEs of Hammerstein type which produce accurate solution at a low computational cost and ``catch'' the qualitative behaviour of the exact solution.
The study developed has been concerned at first with the construction and analysis of efficient methods for the numerical treatment of VIEs of Hammerstein type where the Laplace transform of the kernel rather than the convolution kernel itself is a priori known. This is not an anomalous or restricting situation, as a matter of fact these kind of problems arise in chemical absorption kinetics in the determination of non reflecting boundary conditions, and in general in situations when Laplace transform tecnique are used to reduce systems of ordinary or partial differential equations in VIEs.
It is known that a classical numerical method for computing the numerical solution of such equations over Nt time steps requires O(N2t) operations and O(Nt) memory space.
In this thesis we construct two classes of fast numerical methods based on collocation and Runge-Kutta formulas respectively. These methods have a computational cost of O(NtlogNt) operations, O(logNt) memory requirement and they have an high order of accuracy. In both cases the knowledge of the Laplace transform of the kernel and the convolution nature of the kernel itself are exploited in order to obtain a fast computation of the lag term. This is possible by using an opportune inverse Laplace transform approximation formula for computing the kernel evaluations.
The fast numerical methods constructed in this thesis tend to the corresponding classical methods when the inverse Laplace transfrom approximation formula is exact. The convergence analysis of the fast collocation and Runge-Kutta methods shows that their order of convergence coincides with the order of the corresponding classical methods.
We also analyse the stability properties of the fast Runge-Kutta methods with respect to test equations.
We prove that the stability regions depend on the approxiamation of the inverse Laplace transform and that the stability properties of the classical Runge-Kutta methods are obtained when the error of the inverse Laplace transform approximation formula tends to zero.
The numerical experiments on some significant problems taken from the ``Test Set'' collection project confirm the expected accuracy, computational cost and the stability properties of the constructed methods.
The second part of the thesis concerns with the numerical treatment of problems of SIS epidemic diffusion with periodic immigration flow. The mathematical model of such problems is represented by an Hammerstein type VIE with convolution kernel.
We consider problems caracterized by the relapse of the epidemic which implies that the VIE has an asymptotically periodic solution.
It is clear that an efficient numerical method has to reproduce the asymptotically periodic solution whenever applied to equations that show this behaviour.
For this reason we analyse the discrete Volterra equation (DVE) corresponding to such VIEs and we prove a theorem which establishes the existence and the uniqueness of the asymptotically periodic solution of the DVE.
Moreover we consider SIS epidemic models with periodic immigration flow and constant contact rate. Also in this case we prove, for the DVE corresponding to the problem, the existence and the uniqueness of the asymptotically periodic solution when the DVE satisfies some significant hypothesis depending only on its kernel and forcing term.
In order to analyse if the existing numerical methods satisfy these conditions, that is if they are AP-stable, we consider the class of θ-methods and we prove that they are AP-stable if the integration step satisfies an inequality depending only on some parameters that are characteristic of the problem
Modified Order Theory For Partitioned Runge-Kutta And Runge-Kutta-Nyström Methods
No full textA parametrization technique for deriving general symplectic and/or selfadjoint (symmetric) Runge-Kutta methods of standard type was discussed in a recent paper by the authors. Here these results are extended to the derivation of order conditions for partitioned and Nystrom-Runge-Kutta methods. These methods are of particular interest for the symplectic time-integration of Hamiltonian systems in separable form. Key words. partitioned Runge-Kutta methods, Runge-Kutta-Nystrom methods, symplectic, symmetric, self-adjoint, symplectic integration AMS(MOS) subject classifications. primary 65L05; secondary 70H05 1 Introduction The equations of motion for a conservative classical mechanical system with n degrees of freedom may be written as a set of 2n first order differential equations in Hamiltonian form: dq dt = r p H(q;p; t) ; dp dt = \Gammar q H(q;p; t) : (1.1) The configuration coordinates and momenta q; p 2 R n together define the instantaneous state of the system and may be inte..
Rahel Electoralis Brandenburgica Oder Unvormuteter/ jedoch seliger Todesfall/ der weiland ... Frawen Eleonorae, gebornen und Vermäleten Marggräffin/ auch Churfürstin zu Brande[n]burgk ... : Welche den 22. Martii ... eines Fürstlichen Frewleins genesen/ den 31. desselben ... zu Cöln an der Sprew/ selig eingeschlaffen/ und den 16. April: im Newen Stifft doselbsten/ zu ihrer Ruhe/ beygesetzet worden / In Fünff Predigten gehandelt/ Durch M. Johannem Fleck. Churf. Brand. Hoffprediger ...
No full textConstruction of explicit and generalized Runge-Kutta formulas of arbitrary order with rational parameters
No full textsummary:In the article containing the algorithm of explicit generalized Runge-Kutta formulas of arbitrary order with rational parameters two problems occuring in the solution of ordinary differential equaitions are investigated, namely the determination of rational coefficients and the derivation of the adaptive Runge-Kutta method. By introducing suitable substitutions into the nonlinear system of condition equations one obtains a system of linear equations, which has rational roots. The introduction of suitable symbols enables the authors to generalize the Runge-Kutta formulas. The starting point for the construction of adaptive R. K. method was the consistent -stage R. K. formula. Finally, the S-stability of the ARK method is investigated
Numerical Solution of Fuzzy Differential Equations of 2nd-Order by Runge-Kutta Method
No full text. In this paper, solving fuzzy ordinary differential equations
of the n th order by Runge-Kutta method have been done, and the convergence
of the proposed method is proved. This method is illustrated
by some numerical examples
A Multi-Language Comparison of Influences on Author Verification using Character N-Grams
No full textWe create a new multi-language corpus for author verification based on Wikipedia talkpages, and evaluate the influence that differences in topic and time have on character n-gram author profiles. Topic alignment between two texts is found to increase author verification precision, and an authors writing style is found to change over time, but not more significantly after 3 years than after 1 year.Information ArchitectureWISElectrical Engineering, Mathematics and Computer Scienc
Practical guidance on the application of R-K integration method in finite element analysis of creep damage problem
No full textA practical user guidance of Runge-Kutta (R-K) integration method with the context of non-linear time dependent finite element analysis (FEA) was proposed in this paper. Following the literature review of different integration method within the finite element analysis framework, detailed numerical experiments were conducted to find out the right balance between computing accuracy and efficiency. It contributes to knowledge to the numerical analysis software development in general and specific to computational creep damage mechanics
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