scholarly journals Adaptive isogeometric methods with hierarchical splines: Optimality and convergence rates

2017 ◽  
Vol 27 (14) ◽  
pp. 2781-2802 ◽  
Author(s):  
Annalisa Buffa ◽  
Carlotta Giannelli
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Convergence Rates ◽  
Main Tool ◽  
Adaptive Methods ◽  
Interpolation Operator ◽  
B Splines ◽  
Optimal Convergence Rates ◽  
Error Indicators ◽  
Suitable Approximation

We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and continue the study of its numerical properties. We prove that our AIGM is optimal in the sense that delivers optimal convergence rates as soon as the solution of the underlying partial differential equation belongs to a suitable approximation class. The main tool we use is the theory of adaptive methods, together with a local upper bound for the residual error indicators based on suitable properties of a well selected quasi-interpolation operator on hierarchical spline spaces.

Monotone methods and attractivity results for Volterra integro-partial differential equations

1981 ◽  
Vol 89 (1-2) ◽  
pp. 135-142 ◽  
Author(s):  
Andrea Schiaffino ◽  
Alberto Tesei
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Past History ◽  
Main Tool ◽  
Parabolic Type ◽  
Supremum Norm ◽  
Partial Differential ◽  
Diffusion Effects ◽  
Monotone Methods ◽  
Globally Attractive

SynopsisA Volterra integro-partial differential equation of parabolic type, which describes the time evolution of a population in a bounded habitat, subject both to past history and space diffusion effects, is investigated; general homogeneous boundary conditions are admissible. Under suitable conditions, the unique nontrivial nonnegative equilibrium is shown to be globally attractive in the supremum norm. Monotone methods are the main tool of the proof.

scholarly journals A New Proof of Central Limit Theorem for i.i.d. Random Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Zhaojun Zong ◽  
Feng Hu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Main Tool ◽  
Fundamental Result ◽  
Solution Theory ◽  
Independent Identically Distributed

Central limit theorem (CLT) has long and widely been known as a fundamental result in probability theory. In this note, we give a new proof of CLT for independent identically distributed (i.i.d.) random variables. Our main tool is the viscosity solution theory of partial differential equation (PDE).

scholarly journals LIBERATION, FREE MUTUAL INFORMATION AND ORBITAL FREE ENTROPY

2018 ◽  
Vol 239 ◽  
pp. 205-231
Author(s):  
TAREK HAMDI
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Main Tool ◽  
Stochastic Calculus ◽  
Boundary Values ◽  
Spectral Measures ◽  
Orthogonal Projections ◽  
Free Entropy ◽  
Orbital Free ◽  
Subordinate Function

In this paper, we perform a detailed spectral study of the liberation process associated with two symmetries of arbitrary ranks: $(R,S)\mapsto (R,U_{t}SU_{t}^{\ast })_{t\geqslant 0}$, where $(U_{t})_{t\geqslant 0}$ is a free unitary Brownian motion freely independent from $\{R,S\}$. Our main tool is free stochastic calculus which allows to derive a partial differential equation (PDE) for the Herglotz transform of the unitary process defined by $Y_{t}:=RU_{t}SU_{t}^{\ast }$. It turns out that this is exactly the PDE governing the flow of an analytic function transform of the spectral measure of the operator $X_{t}:=PU_{t}QU_{t}^{\ast }P$ where $P,Q$ are the orthogonal projections associated to $R,S$. Next, we relate the two spectral measures of $RU_{t}SU_{t}^{\ast }$ and of $PU_{t}QU_{t}^{\ast }P$ via their moment sequences and use this relationship to develop a theory of subordination for the boundary values of the Herglotz transform. In particular, we explicitly compute the subordinate function and extend its inverse continuously to the unit circle. As an application, we prove the identity $i^{\ast }(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q))=-\unicode[STIX]{x1D712}_{\text{orb}}(P,Q)$.

Qxygen transfer in gravity flow sewers

2000 ◽  
Vol 42 (3-4) ◽  
pp. 417-422 ◽  
Author(s):  
T.Y. Pai ◽  
C.F. Ouyang ◽  
Y.C. Liao ◽  
H.G. Leu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Dissolved Oxygen ◽  
Flow Velocity ◽  
Molecular Diffusion ◽  
Eddy Diffusion ◽  
Partial Differential ◽  
Diffusion Term ◽  
Sewer Pipes ◽  
Gravity Sewer

Oxygen diffused to water in gravity sewer pipes was studied in a 21 m long, 0.15 m diameter model sewer. At first, the sodium sulfide was added into the clean water to deoxygenate, then the pump was started to recirculate the water and the deoxygenated water was reaerated. The dissolved oxygen microelectrode was installed to measure the dissolved oxygen concentrations varied with flow velocity, time and depth. The dissolved oxygen concentration profiles were constructed and observed. The partial differential equation diffusion model that considered Fick's law including the molecular diffusion term and eddy diffusion term were derived. The analytic solution of the partial differential equation was used to determine the diffusivities by the method of nonlinear regression. The diffusivity values for the oxygen transfer was found to be a function of molecular diffusion, eddy diffusion and flow velocity.

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