A class of generalized trapezoidal formulas for the numerical integration of

Al-Zanaidi, M. A.; Evans, David

doi:10.1080/00207169608804530

Cited by 27 publications

(11 citation statements)

References 5 publications

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“…It then follows that cx$J -zX$$ where we have set Now, writing the scheme (4.7) in scalar form, and applying it to the Fourier mode (4.13), in view of (4.14) we obtain where the symbol of the scheme sx is given by Noting that aX>O, and following the arguments given for Theorem 2.1 in Chawla et al [2], it can be shown that lsxl < 1. In a similar manner, for the symbol s, of the scheme (4.12), Is,,[ 5 1.…”

Section: Stability Of the Lod-gtf(a) Schemementioning

confidence: 95%

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Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

Al-Zanaidi

Evans

2001

International Journal of Computer Mathematics

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We describe locally one-dimensional (LOD) time integration schemes for parabolic differential equations in two space dimensions, based on the generalized trapezoidal formulas (GTF(a)). We describe the schemes for the diffusion equation with Dirichlet and Neumann boundary conditions, for nonlinear reaction-diffusion equations, and for the convection-diffusion equation in two space dimensions. The obtained schemes are second order in time and unconditionally stable for all a E [0, 11. Numerical experiments are given to illustrate the obtained schemes and to compare their performance with the better known LOD CrankNicolson scheme. While the LOD Crank-Nicolson scheme can give unwanted oscillations in the computed solution, our present LOD-GTF(a) schemes provide both stable and accurate approximations for the true solution.

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Section: Stability Of the Lod-gtf(a) Schemementioning

confidence: 95%

“…Recently, Chawla et al [3,4] have described time integration schemes for parabolic differential equations and for the convection-diffusion equation, in one space dimension, based on the generalized trapezoidal formulas (GTF(a)) given in Chawla et al [2].…”

Section: Introductionmentioning

confidence: 99%

Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

Al-Zanaidi

Evans

2001

International Journal of Computer Mathematics

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“…We define q 2,1,0 as a linear combination of the Bernoulli function at each point x j +1 , i.e., (6) with m = 2. By using the properties of Bernoulli function in Section 2, we can verify that the coefficients (18) satisfy the conditions (16). In addition, we can write Eq.…”

Section: Case Ii: M =mentioning

confidence: 97%

“…Chawla et al [15] derived fifth order extended one-step methods, which are A-and/or L-stable. Chawla et al [16] obtained a one-parameter family of double-stride L-stable methods of fourth order by the coupling of three linear multistep methods. De Meyer et al [17] derived a technique based on mono-implicit Runge-Kutta methods.…”

Section: Introductionmentioning

confidence: 99%

Optimal extended one-step schemes of exponential type for stiff initial-value problems

Salama¹,

Bakr²

2004

International Journal of Computer Mathematics

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“…Cross-sectional area (CSA) of the CAT was computed for each of the images selected. In addition, volume comparisons were made based on a calculation of CAT volume using the slice thickness and trapezoidal integration [25] . CSA was used as the primary metric, but it is a derived measurement and does not adequately determine if contours agree on a point-by-point basis.…”

Section: Validation Sample and Statistical Analysismentioning

confidence: 99%

Automated segmentation of cardiac adipose tissue in Dixon magnetic resonance images

Klingensmith

Elliott

Fernández-del-Valle

et al. 2017

JBGC

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Objective: Increasing evidence suggests a strong link between excess cardiac adipose tissue (CAT) and the risk of a cardiovascular event. Multi-echo Dixon magnetic resonance imaging (MRI), providing fat-only and water-only images, is a useful tool for quantification but requires the segmentation of CAT from a large number of images. The intent of this study was to evaluate an automated technique for CAT segmentation from Dixon MRI by comparing the contours identified by the automated algorithm to those manually traced by an observer.Methods: An automated segmentation algorithm, based on optimal thresholds and custom morphological processing, was applied to the registered fat-only and water-only images to identify CAT in the volume scans. CAT contours in 446 images, from 10 MRI scans, were selected for validation analysis. Cross-sectional area (CSA) and volume were computed and compared using Bland-Altman analysis. In addition, Hausdorff distance and Dice Similarity Coefficient (DSC) were used for assessment.Results: Linear regression analysis yielded correlation of R 2 = 0.381 for CSA and R 2 = 0.879 for volume. When compared to the observer, the computer algorithm under-estimated CSA by 27.5 ± 40.0% and volume by 26.4 ± 10.4%. The average bidirectional Hausdorff distance was 26.2 ± 16.0 mm while the average unidirectional Hausdorff distances were 24.5 ± 15.7 mm and 12.4 ± 11.7 mm. The average DSC was 0.561 ± 0.100. The time required for manual tracing was 15.84 ± 3.73 min and the time required for the computer algorithm was 2.81 ± 0.12 min. Conclusions:This study provided a technique, faster and less tedious than manual tracing (p < 0.00001), for quantification of CAT in Dixon MRI data, demonstrating feasibility of this approach for cardiac risk stratification.

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A class of generalized trapezoidal formulas for the numerical integration of

Cited by 27 publications

References 5 publications

Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

Lod generalized trapezoidal formula schemes for parabolic differential equations in two space dimensions

Optimal extended one-step schemes of exponential type for stiff initial-value problems

Automated segmentation of cardiac adipose tissue in Dixon magnetic resonance images

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