Search results for "differentiaalilaskenta"
showing 6 items of 6 documents
Approximating hidden chaotic attractors via parameter switching.
2018
In this paper, the problem of approximating hidden chaotic attractors of a general class of nonlinear systems is investigated. The parameter switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated attractor approximates the attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic attractor. The hidden chaotic attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration. In Refs. 1–3, it is proved that the attractors of a chaotic system, considered as the unique numerical …
Differential structure associated to axiomatic Sobolev spaces
2020
The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure. peerReviewed
New degrees of freedom for differential forms on cubical meshes
2022
We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Significantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices.
Hyperreaaliluvut
2017
Hyperreaaliluvut ovat reaalilukujen joukon laajennus, jossa on olemassa äärettömän pieniä ja suuria lukuja. Hyperreaalilukuja käytetään differentiaali- ja integraalilaskennassa. Metodi on suosittu erityisesti fyysikoiden keskuudessa. Analyysin osa-aluetta, jossa hyödynnetään hyperreaalilukuja, kutsutaan epästandardiksi analyysiksi. Epästandardissa analyysissä käytetään analyysille epästandardeja työkaluja, josta nimi juontuu. Hyperreaalilukujen edut verrattaessa reaalilukuihin tulevat esille epästandardissa analyysissä. Varsinkin fysiikassa hyödynnetään yhtälöiden differentiaalimuotoja ja integroinnissa lähtökohtana pidetään infinitesimaalin valintaa. Tutkielmassa hyperreaaliluvut määritell…
Integraalista ja joukon mitan käsitteestä
2012
Systematic implementation of higher order Whitney forms in methods based on discrete exterior calculus
2022
AbstractWe present a systematic way to implement higher order Whitney forms in numerical methods based on discrete exterior calculus. Given a simplicial mesh, we first refine the mesh into smaller simplices which can be used to define higher order Whitney forms. Cochains on this refined mesh can then be interpolated using higher order Whitney forms. Hence, when the refined mesh is used with methods based on discrete exterior calculus, the solution can be expressed as a higher order Whitney form. We present algorithms for the three required steps: refining the mesh, solving the coefficients of the interpolant, and evaluating the interpolant at a given point. With our algorithms, the order of…