Heat Conduction Using Green's Function

Cole, Kevin D.; Beck, James V.; Haji‐Sheikh, A.; Litkouhi, Bahman

doi:10.1201/9780429258367

Cited by 366 publications

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“…As a natural choice we use a spherical geometry and, motivated by the observation that astrocytic Ca 2+ signals do not change significantly in Ca 2+ free medium [ 24 ], no-flux boundary condition that reflects the properties of the plasma membrane for our in silico cell. With these specifications we could develop the analytic solution by coupled Green's functions [ 52 , 53 ] describing Ca 2+ and buffer dynamics where open channels correspond to source terms. For mathematical details of the implemented Green's cell algorithm (GCA) see [ 27 ] where we have shown how microscopic channel fluctuations are transmitted onto the level of the cell by diffusion thereby matching the experimental findings on cell variability and buffer influence.…”

Section: Resultsmentioning

confidence: 99%

Multiscale Modeling Indicates That Temperature Dependent [Ca²⁺]_iSpiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity

et al. 2015

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Changes in the cytosolic Ca2+ concentration ([Ca2+]i) are the most predominant active signaling mechanism in astrocytes that can modulate neuronal activity and is assumed to influence neuronal plasticity. Although Ca2+ signaling in astrocytes has been intensively studied in the past, our understanding of the signaling mechanism and its impact on tissue level is still incomplete. Here we revisit our previously published data on the strong temperature dependence of Ca2+ signals in both cultured primary astrocytes and astrocytes in acute brain slices of mice. We apply multiscale modeling to test the hypothesis that the temperature dependent [Ca2+]i spiking is mainly caused by the increased activity of the sarcoendoplasmic reticulum ATPases (SERCAs) that remove Ca2+ from the cytosol into the endoplasmic reticulum. Quantitative comparison of experimental data with multiscale simulations supports the SERCA activity hypothesis. Further analysis of multiscale modeling and traditional rate equations indicates that the experimental observations are a spatial phenomenon where increasing pump strength leads to a decoupling of Ca2+ release sites and subsequently to vanishing [Ca2+]i spikes.

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Section: Resultsmentioning

confidence: 99%

Multiscale Modeling Indicates That Temperature Dependent [Ca²⁺]_iSpiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity

et al. 2015

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“…As α → 0, both dispersion coefficient and velocity become constant. Then, equation 33becomes the solution of Yeh (1981), Haberman (1987) and Basha and El-Habel (1993) in case of instantaneous point source, and equation 34becomes that of Yeh (1981), Beck et al (1992) and Yeh and Yeh (2007) for continuous point source. (ii) Linear dispersion coefficient:…”

Section: Spatially Dependent Velocity and Constant Coefficient Of Decmentioning

confidence: 99%

Analytical solution of advection–diffusion equation in heterogeneous infinite medium using Green’s function method

Sanskrityayn

Kumar

2016

J Earth Syst Sci

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Some analytical solutions of one-dimensional advection-diffusion equation (ADE) with variable dispersion coefficient and velocity are obtained using Green's function method (GFM). The variability attributes to the heterogeneity of hydro-geological media like river bed or aquifer in more general ways than that in the previous works. Dispersion coefficient is considered temporally dependent, while velocity is considered spatially and temporally dependent. The spatial dependence is considered to be linear and temporal dependence is considered to be of linear, exponential and asymptotic. The spatio-temporal dependence of velocity is considered in three ways. Results of previous works are also derived validating the results of the present work. To use GFM, a moving coordinate transformation is developed through which this ADE is reduced into a form, whose analytical solution is already known. Analytical solutions are obtained for the pollutant's mass dispersion from an instantaneous point source as well as from a continuous point source in a heterogeneous medium. The effect of such dependence on the mass transport is explained through the illustrations of the analytical solutions.

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“…They are frequently combined to simulate heat conduction in the time domain or in a transform space defined by the Laplace transform, in half-spaces, infinite plates, rectangular 2D spaces, wedges, and rectangular 3D spaces [1][2][3]. Solutions have also been proposed to deal with multilayer systems, and they include the matrix method [1], the thermal quadrupole method [3], the thin layer method [4], and methods based on the definition of potentials [5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Green’s Functions for Heat Conduction for Unbounded and Bounded Rectangular Spaces: Time and Frequency Domain Solutions

Simões

Tadeu

Simões

2016

Journal of Applied Mathematics

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This paper presents a set of fully analytical solutions, together with explicit expressions, in the time and frequency domain for the heat conduction response of homogeneous unbounded and of bounded rectangular spaces (three-, two-, and one-dimensional spaces) subjected to point, line, and plane heat diffusion sources. Particular attention is given to the case of spatially sinusoidal, harmonic line sources. In the literature this problem is often referred to as the two-and-a-half-dimensionalfundamental solutionor 2.5D Green’s functions. These equations are very useful for formulating three-dimensional thermodynamic problems by means of integral transforms methods and/or boundary elements. The image source technique is used to build up different geometries such as half-spaces, corners, rectangular pipes, and parallelepiped boxes. The final expressions are verified here by applying the equations to problems for which the solution is known analytically in the time domain.

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Heat Conduction Using Green's Function

Cited by 366 publications

References 0 publications

Multiscale Modeling Indicates That Temperature Dependent [Ca²⁺]_iSpiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity

Multiscale Modeling Indicates That Temperature Dependent [Ca²⁺]_iSpiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity

Analytical solution of advection–diffusion equation in heterogeneous infinite medium using Green’s function method

Green’s Functions for Heat Conduction for Unbounded and Bounded Rectangular Spaces: Time and Frequency Domain Solutions

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Heat Conduction Using Green's Function

Cited by 366 publications

References 0 publications

Multiscale Modeling Indicates That Temperature Dependent [Ca2+]iSpiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity

Multiscale Modeling Indicates That Temperature Dependent [Ca2+]iSpiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity

Analytical solution of advection–diffusion equation in heterogeneous infinite medium using Green’s function method

Green’s Functions for Heat Conduction for Unbounded and Bounded Rectangular Spaces: Time and Frequency Domain Solutions

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Multiscale Modeling Indicates That Temperature Dependent [Ca²⁺]_iSpiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity

Multiscale Modeling Indicates That Temperature Dependent [Ca²⁺]_iSpiking in Astrocytes Is Quantitatively Consistent with Modulated SERCA Activity