Numer. Methods Partial Differential Eq.
 1999
DOI: 10.1002/(sici)1098-2426(199911)15:6<697::aid-num6>3.0.co;2-#
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A finite difference scheme for solving the heat transport equation at the microscale

Weizhong Dai
1
,
Raja Nassar
2

Abstract: Heat transport at the microscale is of vital importance in microtechnology applications. In this study, we develop a finite difference scheme of the Crank-Nicholson type by introducing an intermediate function for the heat transport equation at the microscale. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results show that the solution is accurate.

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Cited by 75 publications

(34 citation statements)
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“…2. From this figure, we can see that the numerical solutions are more accurate than those obtained by the scheme in [8].…”
Section: Numerical Examples
mentioning
confidence: 81%
Exaggerated anticipatory anxiety is common in social anxiety disorder (SAD). Neuroimaging studies have revealed altered neural activity in response to social stimuli in SAD, but fewer studies have examined neural activity during anticipation of feared social stimuli in SAD. The current study examined the time course and magnitude of activity in threat processing brain regions during speech anticipation in socially anxious individuals and healthy controls (HC). Method Participants (SAD n = 58; HC n = 16) underwent functional magnetic resonance imaging (fMRI) during which they completed a 90s control anticipation task and 90s speech anticipation task.
“…2. From this figure, we can see that the numerical solutions are more accurate than those obtained by the scheme in [8].…”
Section: Numerical Examples
mentioning
confidence: 81%
Exaggerated anticipatory anxiety is common in social anxiety disorder (SAD). Neuroimaging studies have revealed altered neural activity in response to social stimuli in SAD, but fewer studies have examined neural activity during anticipation of feared social stimuli in SAD. The current study examined the time course and magnitude of activity in threat processing brain regions during speech anticipation in socially anxious individuals and healthy controls (HC). Method Participants (SAD n = 58; HC n = 16) underwent functional magnetic resonance imaging (fMRI) during which they completed a 90s control anticipation task and 90s speech anticipation task.
“…The direct, adjoint and sensitivity problems present in the CGM described in section 5.1 are solved using the Crank-Nicolson finite-difference method (FDM) [14] in one-dimension (d = 1) with a uniform mesh size L/M and time step T/N. The two-point first-order backward finite difference formula is used to approximate the time-derivative u t (t, x) in (57).…”
Section: Numerical Results and Discussion
mentioning
confidence: 99%
Exaggerated anticipatory anxiety is common in social anxiety disorder (SAD). Neuroimaging studies have revealed altered neural activity in response to social stimuli in SAD, but fewer studies have examined neural activity during anticipation of feared social stimuli in SAD. The current study examined the time course and magnitude of activity in threat processing brain regions during speech anticipation in socially anxious individuals and healthy controls (HC). Method Participants (SAD n = 58; HC n = 16) underwent functional magnetic resonance imaging (fMRI) during which they completed a 90s control anticipation task and 90s speech anticipation task.
“……”
Section: Numerical Results and Discussion
mentioning
confidence: 99%
“……”
Section: Example
mentioning
confidence: 94%
Exaggerated anticipatory anxiety is common in social anxiety disorder (SAD). Neuroimaging studies have revealed altered neural activity in response to social stimuli in SAD, but fewer studies have examined neural activity during anticipation of feared social stimuli in SAD. The current study examined the time course and magnitude of activity in threat processing brain regions during speech anticipation in socially anxious individuals and healthy controls (HC). Method Participants (SAD n = 58; HC n = 16) underwent functional magnetic resonance imaging (fMRI) during which they completed a 90s control anticipation task and 90s speech anticipation task.
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