scholarly journals Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions

2021 ◽  
Vol 4 (3) ◽  
pp. 1-17
Author(s):  
Carmen Cortázar ◽  
◽  
Fernando Quirós ◽  
Noemí Wolanski ◽  
◽  
...  
Keyword(s):  
Heat Equation ◽  
Growth Rates ◽  
Time Scale ◽  
Time Derivative ◽  
Space Time ◽  
Heat Equations ◽  
Time Behavior ◽  
Forcing Term ◽  
Large N ◽  
With Memory

<abstract><p>We study the decay/growth rates in all $ L^p $ norms of solutions to an inhomogeneous nonlocal heat equation in $ \mathbb{R}^N $ involving a Caputo $ \alpha $-time derivative and a power $ \beta $ of the Laplacian when the dimension is large, $ N &gt; 4\beta $. Rates depend strongly on the space-time scale and on the time behavior of the spatial $ L^1 $ norm of the forcing term.</p></abstract>

scholarly journals Fully discrete finite element data assimilation method for the heat equation

2018 ◽  
Vol 52 (5) ◽  
pp. 2065-2082 ◽  
Author(s):  
Erik Burman ◽  
Jonathan Ish-Horowicz ◽  
Lauri Oksanen
Keyword(s):  
Finite Element ◽  
Initial Data ◽  
Heat Equation ◽  
Time Derivative ◽  
Classical Problem ◽  
Classical Case ◽  
Space Time ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Element Discretization

We consider a finite element discretization for the reconstruction of the final state of the heat equation, when the initial data is unknown, but additional data is given in a sub domain in the space time. For the discretization in space we consider standard continuous affine finite element approximation, and the time derivative is discretized using a backward differentiation. We regularize the discrete system by adding a penalty on the H2-semi-norm of the initial data, scaled with the mesh-parameter. The analysis of the method uses techniques developed in E. Burman and L. Oksanen [Numer. Math. 139 (2018) 505–528], combining discrete stability of the numerical method with sharp Carleman estimates for the physical problem, to derive optimal error estimates for the approximate solution. For the natural space time energy norm, away from t = 0, the convergence is the same as for the classical problem with known initial data, but contrary to the classical case, we do not obtain faster convergence for the L2-norm at the final time.

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale ◽  
Serife Bayramoglu
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Numerical Experiments ◽  
Time Derivative ◽  
Heat Equations ◽  
First Order ◽  
Fractional Heat Equation

AbstractIn this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2-α + h 2), conditionally. Numerical experiments are carried out to support the theoretical claims.

scholarly journals A heat equation with memory: large-time behavior

2021 ◽  
pp. 109174
Author(s):  
Carmen Cortázar ◽  
Fernando Quirós ◽  
Noemí Wolanski
Keyword(s):  
Heat Equation ◽  
Large Time ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Heat Equation With Memory ◽  
With Memory

scholarly journals Exploring the Characteristics and Influencing Factors of Leisure Walking Based on the Demand of Behavior

2021 ◽  
Vol 13 (8) ◽  
pp. 4105
Author(s):  
Yupei Jiang ◽  
Honghu Sun
Keyword(s):  
Health Benefits ◽  
Health Benefit ◽  
Public Health Research ◽  
Space Time ◽  
Time Behavior ◽  
Walking Behavior ◽  
Face To Face ◽  
Residential Neighborhood ◽  
The Impact

Leisure walking has been an important topic in space-time behavior and public health research. However, prior studies pay little attention to the integration and the characterization of diverse and multilevel demands of leisure walking. This study constructs a theoretical framework of leisure walking behavior demands from three different dimensions and levels of activity participation, space-time opportunity, and health benefit. On this basis, through a face-to-face survey in Nanjing, China (N = 1168, 2017–2018 data), this study quantitatively analyzes the characteristics of leisure walking demands, as well as the impact of the built environment and individual factors on it. The results show that residents have a high demand for participation and health benefits of leisure walking. The residential neighborhood provides more space opportunities for leisure walking, but there is a certain constraint on the choice of walking time. Residential neighborhood with medium or large parks is more likely to satisfy residents’ demands for engaging in leisure walking and obtaining high health benefits, while neighborhood with a high density of walking paths tends to limit the satisfaction of demands for space opportunity and health benefit. For residents aged 36 and above, married, or retired, their diverse demands for leisure walking are more likely to be fulfilled, while those with high education, medium-high individual income, general and above health status, or children (<18 years) are less likely to be fulfilled. These finding that can have important implications for the healthy neighborhood by fully considering diverse and multilevel demands of leisure walking behavior.

A parallel space–time boundary element method for the heat equation

2019 ◽  
Vol 78 (9) ◽  
pp. 2852-2866 ◽  
Author(s):  
Stefan Dohr ◽  
Jan Zapletal ◽  
Günther Of ◽  
Michal Merta ◽  
Michal Kravčenko
Keyword(s):  
Boundary Element Method ◽  
Heat Equation ◽  
Boundary Element ◽  
Space Time ◽  
Time Boundary ◽  
Element Method

scholarly journals Regularity and time behavior of the solutions to weak monotone parabolic equations

2021 ◽  
Author(s):  
Maria Michaela Porzio
Keyword(s):  
Heat Equation ◽  
Parabolic Equations ◽  
Parabolic Problems ◽  
Summable Function ◽  
Time Behavior ◽  
Decay Estimates ◽  
Laplacian Equation ◽  
Evolution Problems ◽  
Degenerate Equations ◽  
Uniqueness Results

AbstractIn this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

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