Search results for "Differentia"
showing 10 items of 8428 documents
On a numerical solution of the Maxwell equations by discrete exterior calculus
2014
p-Laplacen operaattorin ominaisarvo-ongelmasta
2016
Tämän tutkielman tarkoitus on tutustua epälineaarisiin ominaisarvo-ongelmiin p-Laplacen operaattorin ominaisarvo-ongelman kautta. p-Laplacen operaattori on Laplacen operaattorin eräs yleistys ja tarkastelun kohteena oleva ominaisarvo-ongelma on Dirichletin ominaisarvo-ongelman yleistys. Tutkielmassa kerrataan ensin tarvittavia taustatietoja Sobolevin avaruuksista ja funktionaalianalyysistä, ja keskitytään sitten itse ongelmaan. Päätulokset koskevat ensimmäistä ominaisarvoa, ja ne ovat ensimmäisen ominaisarvon olemassaolo, ensimmäisen ominaisarvon karakterisointi Rayleighin osamäärän avulla, ensimmäisen ominaisfunktion yksinkertaisuus, ja se, että ensimmäinen ominaisfunktio on ainoa ominaisf…
Guaranteed error bounds and local indicators for adaptive solvers using stabilised space-time IgA approximations to parabolic problems
2019
The paper is concerned with space–time IgA approximations to parabolic initial–boundary value problems. We deduce guaranteed and fully computable error bounds adapted to special features of such type of approximations and investigate their efficiency. The derivation of error estimates is based on the analysis of the corresponding integral identity and exploits purely functional arguments in the maximal parabolic regularity setting. The estimates are valid for any approximation from the admissible (energy) class and do not contain mesh-dependent constants. They provide computable and fully guaranteed error bounds for the norms arising in stabilised space–time approximations. Furthermore, a p…
On the second-order regularity of solutions to the parabolic p-Laplace equation
2022
AbstractIn this paper, we study the second-order Sobolev regularity of solutions to the parabolic p-Laplace equation. For any p-parabolic function u, we show that $$D(\left| Du\right| ^{\frac{p-2+s}{2}}Du)$$ D ( D u p - 2 + s 2 D u ) exists as a function and belongs to $$L^{2}_{\text {loc}}$$ L loc 2 with $$s>-1$$ s > - 1 and $$1<p<\infty $$ 1 < p < ∞ . The range of s is sharp.
Notes on the p-Laplace equation
2017
2. p.
Approximation of heat equation and backward SDEs using random walk : convergence rates
2018
This thesis addresses questions related to approximation arising from the fields of stochastic analysis and partial differential equations. Theoretical results regarding convergence rates are obtained by using discretization schemes where the limiting process, the Brownian motion, is approximated by a simple discrete-time random walk. The rate of convergence is derived for a finite-difference approximation of the solution of a terminal value problem for the backward heat equation. This weak approximation result is proved for a terminal function which has bounded variation on compact sets. The sharpness of the according rate is achieved by applying some new results related to the first exit time …
Static instability analysis of an elastic band travelling in the gravitational field
2011
Static instability analysis is performed for an axially moving elastic band, which is travelling at a constant velocity in a uniform gravitational field between two supports. The buckling of the band is investigated with the help of admitting small transverse deflections. The model of a thin elastic beam (panel) subjected to bending, centrifugal forces and nonhomogeneous tension (including a gravitational term) is used. Buckling analysis and estimation of the critical velocities of elastic instability are based on variational principles and variational inequalities. As a result, explicit formulas for upper and lower limits for critical velocities are found. It is shown analytically that a c…
Functional Type Error Control for Stabilised Space-Time IgA Approximations to Parabolic Problems
2018
The paper is concerned with reliable space-time IgA schemes for parabolic initial-boundary value problems. We deduce a posteriori error estimates and investigate their applicability to space-time IgA approximations. Since the derivation is based on purely functional arguments, the estimates do not contain mesh dependent constants and are valid for any approximation from the admissible (energy) class. In particular, they imply estimates for discrete norms associated with stabilised space-time IgA approximations. Finally, we illustrate the reliability and efficiency of presented error estimates for the approximate solutions recovered with IgA techniques on a model example. peerReviewed
Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
2018
Let $\mathscr{L}$ be a smooth second-order real differential operator in divergence form on a manifold of dimension $n$. Under a bracket-generating condition, we show that the ranges of validity of spectral multiplier estimates of Mihlin--H\"ormander type and wave propagator estimates of Miyachi--Peral type for $\mathscr{L}$ cannot be wider than the corresponding ranges for the Laplace operator on $\mathbb{R}^n$. The result applies to all sub-Laplacians on Carnot groups and more general sub-Riemannian manifolds, without restrictions on the step. The proof hinges on a Fourier integral representation for the wave propagator associated with $\mathscr{L}$ and nondegeneracy properties of the sub…
Partial Data Problems and Unique Continuation in Scalar and Vector Field Tomography
2022
AbstractWe prove that if P(D) is some constant coefficient partial differential operator and f is a scalar field such that P(D)f vanishes in a given open set, then the integrals of f over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.