Long-Time Solutions to Heat-Conduction Transients with Time-Dependent Inputs

1980 ◽  
Vol 102 (1) ◽  
pp. 115-120 ◽  
Author(s):  
H. T. Ceylan ◽  
G. E. Myers
Keyword(s):  
Heat Conduction ◽  
Lower Cost ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Time Dependent ◽  
Time Interval ◽  
Piecewise Linear Functions ◽  
One Dimensional ◽  
Long Time

An economical method for obtaining long-time solutions to one, two, or three-dimensional heat-conduction transients with time-dependent forcing functions is presented. The conduction problem is spatially discretized by finite differences or by finite elements to obtain a system of first-order ordinary differential equations. The time-dependent input functions are each approximated by continuous, piecewise-linear functions each having the same uniform time interval. A set of response coefficients is generated by which a long-time solution can be carried out with a considerably lower cost than for conventional methods. A one-dimensional illustrative example is included.

Multiquadric Collocation Method for Time-Dependent Heat Conduction Problems With Temperature-Dependent Thermal Properties

2006 ◽  
Vol 129 (2) ◽  
pp. 109-113 ◽  
Author(s):  
Somchart Chantasiriwan
Keyword(s):  
Thermal Properties ◽  
Heat Conduction ◽  
Collocation Method ◽  
Heat Conduction Problem ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Time Dependent ◽  
Piecewise Linear Functions ◽  
Temperature Dependent ◽  
Kirchhoff Transformation

Abstract The multiquadric collocation method is a meshless method that uses multiquadrics as its basis function. Problems of nonlinear time-dependent heat conduction in materials having temperature-dependent thermal properties are solved by using this method and the Kirchhoff transformation. Variable transformation is simplified by assuming that thermal properties are piecewise linear functions of temperature. The resulting nonlinear equation is solved by an iterative scheme. The multiquadric collocation method is tested by a heat conduction problem for which the exact solution is known. Results indicate satisfactory performance of the method.

CONTINUOUS PIECEWISE LINEAR FUNCTIONS

2005 ◽  
Vol 10 (1) ◽  
pp. 77-99 ◽  
Author(s):  
CHARALAMBOS D. ALIPRANTIS ◽  
DAVID HARRIS ◽  
RABEE TOURKY
Keyword(s):  
Piecewise Linear ◽  
Continuous Functions ◽  
Linear Functions ◽  
Dimensional Euclidean Space ◽  
Piecewise Linear Functions ◽  
Space Of Continuous Functions ◽  
One Dimensional ◽  
Mathematical Background ◽  
Linear Regressions ◽  
Special Case

The paper studies the function space of continuous piecewise linear functions in the space of continuous functions on them-dimensional Euclidean space. It also studies the special case of one dimensional continuous piecewise linear functions. The study is based on the theory of Riesz spaces that has many applications in economics. The work also provides the mathematical background to its sister paper Aliprantis, Harris, and Tourky (2006), in which we estimate multivariate continuous piecewise linear regressions by means of Riesz estimators, that is, by estimators of the the Boolean formwhereX=(X1,X2, …,Xm) is some random vector, {Ej}j∈Jis a finite family of finite sets.

Modeling of three-dimensional surfaces using high-level canonical piecewise-linear functions in cylindrical coordinates

2018 ◽  
Vol 37 (4) ◽  
pp. 5500-5513
Author(s):  
Victor M. Tlapa-Carrera ◽  
Victor M. Jimenez-Fernandez ◽  
Hector Vazquez-Leal ◽  
Uriel A. Filobello-Nino ◽  
Jesus Garcia-Guzman ◽  
...  
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Cylindrical Coordinates ◽  
High Level

Apparent Shape of the Mach Book

Perception ◽  
1996 ◽  
Vol 25 (3) ◽  
pp. 313-319 ◽  
Author(s):  
Richard A Clement
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Two Dimensional ◽  
Piecewise Linear Functions ◽  
Common Edge ◽  
Edge Orientation ◽  
Line Drawings ◽  
Alternative Hypotheses

The Mach book is a two-dimensional figure which looks three-dimensional. Despite the impression of depth in the figure, the apparent shape has not been determined. It has been suggested that the book appears as part of a ‘cubic corner’, ‘as flat as possible’, or with each half rotated about its long diagonal. Alternative hypotheses as to the three-dimensional orientation of the book were tested by means of a probe-line technique. It was found that, although no hypothesis matched the results of all of the subjects, the probe-line settings of individual subjects were approximately linear or piecewise linear functions of the angles in the picture. The technique was also applied to asymmetric versions of the figure and it was found that the subjects modified their settings in accord with the constraint that the two halves of the figure must join in depth along their common edge. The findings are in agreement with models of the interpretation of line drawings in which local estimates of edge orientation in depth are formed, and subsequently checked for consistency.

Boundary crossing probability for Brownian motion and general boundaries

1997 ◽  
Vol 34 (1) ◽  
pp. 54-65 ◽  
Author(s):  
Liqun Wang ◽  
Klaus Pötzelberger
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Boundary Crossing ◽  
Time Interval ◽  
Linear Boundary ◽  
Piecewise Linear Functions ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Crossing Probabilities

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

scholarly journals A posteriori error estimates for semilinear optimal control problems

2021 ◽  
Author(s):  
Alejandro Allendes ◽  
Francisco Fuica ◽  
Enrique Otarola ◽  
Daniel Quero
Keyword(s):  
Optimal Control ◽  
Piecewise Linear ◽  
Control Variable ◽  
Three Dimensional ◽  
Posteriori Error ◽  
Linear Functions ◽  
Adjoint Equations ◽  
Error Estimator ◽  
Piecewise Linear Functions ◽  
A Posteriori Error

In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we devise and analyze a reliable and efficient a posteriori error estimator for a semilinear optimal control problem; control constraints are also considered. We consider a fully discrete scheme that discretizes the state and adjoint equations with piecewise linear functions and the control variable with piecewise constant functions. The devised error estimator can be decomposed as the sum of three contributions which are associated to the discretization of the state and adjoint equations and the control variable. We extend our results to a scheme that approximates the control variable with piecewise linear functions and also to a scheme that approximates the solution to a nondifferentiable optimal control problem. We illustrate the theory with two and three-dimensional numerical examples.

Boundary crossing probability for Brownian motion and general boundaries

1997 ◽  
Vol 34 (01) ◽  
pp. 54-65 ◽  
Author(s):  
Liqun Wang ◽  
Klaus Pötzelberger
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Linear Functions ◽  
Boundary Crossing ◽  
Time Interval ◽  
Linear Boundary ◽  
Piecewise Linear Functions ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Crossing Probabilities

An explicit formula for the probability that a Brownian motion crosses a piecewise linear boundary in a finite time interval is derived. This formula is used to obtain approximations to the crossing probabilities for general boundaries which are the uniform limits of piecewise linear functions. The rules for assessing the accuracies of the approximations are given. The calculations of the crossing probabilities are easily carried out through Monte Carlo methods. Some numerical examples are provided.

scholarly journals DEMOGRAPHIC PHASE SPACE OF THE TVER REGION FROM 1989 TO 2020 AND TVER REGION POPULATION

2020 ◽  
pp. 170-180
Author(s):  
Алексей Никифорович Кудинов ◽  
Сергей Александрович Михеев ◽  
Владимир Николаевич Рыжиков ◽  
Виктор Павлович Цветков ◽  
Илья Викторович Цветков
Keyword(s):  
Time Series ◽  
Phase Space ◽  
Piecewise Linear ◽  
Rate Of Change ◽  
Linear Functions ◽  
Time Interval ◽  
Specific Time ◽  
Piecewise Linear Functions ◽  
Demographic Dynamics ◽  
Tver Region

Авторами предложен метод исследования особенностей демографической динамики на основе демографического фазового пространства. Целью работы является анализ динамики народонаселения Тверской области и оценка возможности ее стабилизации в будущем с использованием демографического фазового пространства. Научная новизна работы состоит в применении нового метода исследования особенностей демографической динамики на основе демографического фазового пространства на интересующем промежутке времени. В статье построены кусочно-линейные функции, которые непрерывно аппроксимируют временные ряды численности народонаселения и скорости изменения численности народонаселения Тверской области с 1989 по 2020 гг. Сконструировано демографическое фазовое пространство Тверского региона с 1989 по 2020 гг. Получены аналитические выражения, описывающие динамику тренда численности народонаселения Тверской области. Найдена асимптотическая стабилизация численности народонаселения Тверского региона на уровне 1.0998 млн человек к 2060 году. The article proposes a method for studying the features of demographic dynamics based on the demographic phase space. The aim of the work is to analyze the dynamics of the population of the Tver region and to assess the possibility of its stabilization in the future using the demographic phase space. The scientific novelty of the work consists in the application of a new method for studying the characteristics of demographic dynamics based on the demographic phase space at the specific time interval. The article constructs piecewise linear functions that continuously approximate the time series of the population and the rate of change in the population of the Tver region from 1989 to 2020. The authors present a demographic phase space of the Tver region from 1989 to 2020t. An analytical expression describes the dynamics of the trend in the population of the Tver region. The investigation highlights an asymptotic stabilization of the population of the Tver region at the level of 1.0998 million people by 2060.

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