scholarly journals A Lipschitz Stability Estimate for the Inverse Source Problem and the Numerical Scheme

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Xianzheng Jia
Keyword(s):  
Heat Equation ◽  
Numerical Algorithm ◽  
Numerical Scheme ◽  
Source Term ◽  
Stability Estimate ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Inverse Source Problem ◽  
Lipschitz Stability ◽  
Piecewise Constant

We consider the inverse source problem for heat equation, where the source term has the formf(t)ϕ(x). We give a numerical algorithm to compute unknown source termf(t). Also, we give a stability estimate in the case thatf(t)is a piecewise constant function.

scholarly journals Current-phase relation in layered superconducting structures of SIS’IS type

2021 ◽  
Vol 24 (2) ◽  
pp. 23701
Author(s):  
A. M. Shutovskyi ◽  
V. E. Sakhnyuk
Keyword(s):  
Phase Difference ◽  
Current Density ◽  
Phase Relation ◽  
Dielectric Layer ◽  
Intermediate Layer ◽  
Order Parameter ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Current Phase ◽  
Piecewise Constant

The dependence of the current density on the phase difference is investigated considering the layered superconducting structures of a SIS’IS type. To simplify the calculations, the quasiclassical equations for the Green’s functions in a t-representation are derived. An order parameter is considered as a piecewise constant function. To consider the general case, no restrictions on the dielectric layer transparency and the thickness of the intermediate layer are imposed. It was found that a new analytical expression for the current-phase relation can be used with the aim to obtain a number of previously known results arising in particular cases.

scholarly journals Determination of source term for fractional heat equation on the sphere

2020 ◽  
Vol 24 (Suppl. 1) ◽  
pp. 361-370
Author(s):  
Nguyen Phuong ◽  
Tran Binh ◽  
Nguyen Luc ◽  
Nguyen Can
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Exact Solutions ◽  
Source Term ◽  
Fractional Diffusion ◽  
Inverse Source Problem ◽  
Fractional Heat Equation ◽  
Ill Posed ◽  
Time Fractional Diffusion Equation

In this work, we study a truncation method to solve a time fractional diffusion equation on the sphere of an inverse source problem which is ill-posed in the sense of Hadamard. Through some priori assumption, we present the error estimates between the regularized and exact solutions.

scholarly journals Continuous CM-regularity of semihomogeneous vector bundles

2020 ◽  
Vol 20 (3) ◽  
pp. 401-412
Author(s):  
Alex Küronya ◽  
Yusuf Mustopa
Keyword(s):  
Vector Bundle ◽  
Abelian Variety ◽  
Upper Bound ◽  
Vector Bundles ◽  
Projective Variety ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Piecewise Constant ◽  
Generic Vanishing

AbstractWe ask when the CM (Castelnuovo–Mumford) regularity of a vector bundle on a projective variety X is numerical, and address the case when X is an abelian variety. We show that the continuous CM-regularity of a semihomogeneous vector bundle on an abelian variety X is a piecewise-constant function of Chern data, and we also use generic vanishing theory to obtain a sharp upper bound for the continuous CM-regularity of any vector bundle on X. From these results we conclude that the continuous CM-regularity of many semihomogeneous bundles — including many Verlinde bundles when X is a Jacobian — is both numerical and extremal.

scholarly journals PERIODIC ORBITS OF SINGLE NEURON MODELS WITH INTERNAL DECAY RATE 0 < Β ≤ 1

2013 ◽  
Vol 18 (3) ◽  
pp. 325-345 ◽  
Author(s):  
Aija Anisimova ◽  
Maruta Avotina ◽  
Inese Bula
Keyword(s):  
Dynamical System ◽  
Decay Rate ◽  
Single Neuron ◽  
Discrete Dynamical System ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Sigmoid Function ◽  
Signal Function ◽  
Neuron Models ◽  
Piecewise Constant

In this paper we consider a discrete dynamical system x n+1=βx n – g(x n ), n=0,1,..., arising as a discrete-time network of a single neuron, where 0 &lt; β ≤ 1 is an internal decay rate, g is a signal function. A great deal of work has been done when the signal function is a sigmoid function. However, a signal function of McCulloch-Pitts nonlinearity described with a piecewise constant function is also useful in the modelling of neural networks. We investigate a more complicated step signal function (function that is similar to the sigmoid function) and we will prove some results about the periodicity of solutions of the considered difference equation. These results show the complexity of neurons behaviour.

scholarly journals The Kirchhoff Transformation and the Fick’s Second Law with Concentration-dependent Diffusion Coefficient

2021 ◽  
Vol 16 ◽  
pp. 59-67
Author(s):  
R. M. S. Gama ◽  
R. Pazetto S. Gama
Keyword(s):  
Diffusion Coefficient ◽  
Mathematical Structure ◽  
Constant Function ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Piecewise Constant Function ◽  
Second Law ◽  
Piecewise Constant ◽  
Numerical Examples ◽  
Kirchhoff Transformation

In this work it is considered the Fick’s second law in a context in which the diffusion coefficient depends on the concentration. It is employed the Kirchhoff transformation in order to simplify the mathematical structure of the Fick’s second law, giving rise to a more convenient description. In order to provide a general protocol, the diffusion coefficient will be assumed a piecewise constant function of the concentration. Exact formulas are presented for both the Kirchhoff transformation and its inverse, in such a way that there is no limit of accuracy. Some numerical examples are presented with the aid of a semi-implicit procedure associated with a finite difference approximation.

scholarly journals A Production Inventory Model for deteriorating items with backlog-dependent demand

2019 ◽  
Author(s):  
Chayanika Rout ◽  
Debjani Chakraborty ◽  
Prof Adrijit Goswami
Keyword(s):  
Inventory Model ◽  
Constant Function ◽  
Deteriorating Items ◽  
Piecewise Constant Function ◽  
High Quality ◽  
Piecewise Constant ◽  
Production Inventory ◽  
Demand Rate ◽  
Theoretical Results ◽  
Dependent Demand

This paper investigates a production inventory model under classical EPQ framework with the assumption that the customer demand during the stock out period is affected by the accumulated back-orders. The backlog rate is not fixed; instead, the demand rate during stock-out is assumed to decrease proportionally to the existing backlog which is thereby approximated by a piecewise constant function. Deteriorating items are taken into consideration in this proposed work. For better illustration of the theoretical results and to highlight managerial insights, numerical examples arepresentedwhicharethencomparedtotheresultsobtainedbyconsideringanexact (non-approximated) backlogging rate (from literature). The comparisons indicate high quality results for the approximated model.

scholarly journals Stability estimate for a strongly coupled parabolic system

2012 ◽  
Vol 43 (1) ◽  
pp. 137-144 ◽  
Author(s):  
Kun-Chu Chen
Keyword(s):  
Parabolic System ◽  
Stability Estimate ◽  
Carleman Estimates ◽  
Inverse Source Problem ◽  
Lipschitz Stability ◽  
Strongly Coupled

We consider an inverse source problem for a 2×2 strongly coupled parabolic system. The Lipschitz stability is proved and the proof is based on the Carleman estimates with two large parameters.

scholarly journals Geodesic X-ray tomography for piecewise constant functions on nontrapping manifolds

2018 ◽  
Vol 168 (1) ◽  
pp. 29-41 ◽  
Author(s):  
JOONAS ILMAVIRTA ◽  
JERE LEHTONEN ◽  
MIKKO SALO
Keyword(s):  
Higher Dimensions ◽  
Uniqueness Result ◽  
Constant Function ◽  
Piecewise Constant Function ◽  
Two Dimensional ◽  
Local Uniqueness ◽  
Piecewise Constant ◽  
Convex Boundary ◽  
X Ray ◽  
Strictly Convex

AbstractWe show that on a two-dimensional compact nontrapping manifold with strictly convex boundary, a piecewise constant function is determined by its integrals over geodesics. In higher dimensions, we obtain a similar result if the manifold satisfies a foliation condition. These theorems are based on iterating a local uniqueness result. Our proofs are elementary.

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