Continuous symmetries of three-dimensional diffusion equations

1985 ◽  
Vol 24 (3) ◽  
pp. 237-253 ◽  
Author(s):  
W. -H. Steeb
Keyword(s):  
Three Dimensional ◽  
Diffusion Equations ◽  
Dimensional Diffusion ◽  
Continuous Symmetries

Accurate numerical schemes based on the lattice Boltzmann methods for multi-dimensional diffusion equations

2014 ◽  
Vol 25 (04) ◽  
pp. 1350104
Author(s):  
Shinsuke Suga
Keyword(s):  
Fourth Order ◽  
Lattice Boltzmann ◽  
Three Dimensional ◽  
Relaxation Parameter ◽  
Distribution Functions ◽  
Diffusion Equations ◽  
Numerical Schemes ◽  
Velocity Models ◽  
Dimensional Diffusion ◽  
Error Terms

We propose accurate explicit numerical schemes based on the lattice Boltzmann (LB) method for multi-dimensional diffusion equations. In LB schemes, the velocity models D2Q9 and D2Q13 are used for two-dimensional equations and D3Q19 and D3Q25 for three-dimensional equations. We introduce free parameters that characterize the weight of the equilibrium distribution functions to reduce numerical errors. Consistency analysis through the fourth-order Chapman–Ensgok expansion of the distribution functions gives an approximate diffusion equation with error terms up to fourth-order. The relaxation parameter and weight parameters are determined so that second-order error terms are eliminated in the approximate equation. Stability analysis shows that we can find a relaxation parameter so that each of the presented schemes is stable for given diffusion coefficients and discretizing parameters. Numerical experiments for the isotropic and anisotropic benchmark problems show that the presented schemes derived from the velocity models D2Q13 and D3Q25 are useful for numerical simulations of practical problems governed by two- and three-dimensional diffusion equations, respectively. In particular, schemes in which the value of the relaxation parameter is set to be 1 demonstrate a fourth-order accuracy under the stability condition.

scholarly journals Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1195 ◽  
Author(s):  
Imtiaz Ahmad ◽  
Hijaz Ahmad ◽  
Phatiphat Thounthong ◽  
Yu-Ming Chu ◽  
Clemente Cesarano
Keyword(s):  
Meshless Method ◽  
Three Dimensional ◽  
Fractional Part ◽  
Diffusion Equations ◽  
Numerical Treatment ◽  
Physical Processes ◽  
Dimensional Diffusion ◽  
Symmetry Properties ◽  
Fractional Pde ◽  
Fractional Differential

Fractional differential equations depict nature sufficiently in light of the symmetry properties which describe biological and physical processes. This article is concerned with the numerical treatment of three-term time fractional-order multi-dimensional diffusion equations by using an efficient local meshless method. The space derivative of the models is discretized by the proposed meshless procedure based on the multiquadric radial basis function though the time-fractional part is discretized by Liouville–Caputo fractional derivative. The numerical results are obtained for one-, two- and three-dimensional cases on rectangular and non-rectangular computational domains which verify the validity, efficiency and accuracy of the method.

scholarly journals A noninvasive fluorescence imaging-based platform measures 3D anisotropic extracellular diffusion

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Peng Chen ◽  
Xun Chen ◽  
R. Glenn Hepfer ◽  
Brooke J. Damon ◽  
Changcheng Shi ◽  
...  
Keyword(s):  
Anisotropic Diffusion ◽  
Biological Systems ◽  
Three Dimensional ◽  
Molecular Transport ◽  
Light Sheet ◽  
Fundamental Limitations ◽  
Disease Mechanisms ◽  
Dimensional Diffusion ◽  
Compositional Changes ◽  
Diffusion Tensors

AbstractDiffusion is a major molecular transport mechanism in biological systems. Quantifying direction-dependent (i.e., anisotropic) diffusion is vitally important to depicting how the three-dimensional (3D) tissue structure and composition affect the biochemical environment, and thus define tissue functions. However, a tool for noninvasively measuring the 3D anisotropic extracellular diffusion of biorelevant molecules is not yet available. Here, we present light-sheet imaging-based Fourier transform fluorescence recovery after photobleaching (LiFT-FRAP), which noninvasively determines 3D diffusion tensors of various biomolecules with diffusivities up to 51 µm2 s−1, reaching the physiological diffusivity range in most biological systems. Using cornea as an example, LiFT-FRAP reveals fundamental limitations of current invasive two-dimensional diffusion measurements, which have drawn controversial conclusions on extracellular diffusion in healthy and clinically treated tissues. Moreover, LiFT-FRAP demonstrates that tissue structural or compositional changes caused by diseases or scaffold fabrication yield direction-dependent diffusion changes. These results demonstrate LiFT-FRAP as a powerful platform technology for studying disease mechanisms, advancing clinical outcomes, and improving tissue engineering.

Tidal three-dimensional diffusion in a model of the Lagoon of Venice and reliability conditions for its numerical integration

1987 ◽  
Vol 37 (1-2) ◽  
pp. 81-101 ◽  
Author(s):  
Camillo Dejak ◽  
Ileana Mazzei Lalatta ◽  
Marina Molin ◽  
Giovanni Pecenik
Keyword(s):  
Numerical Integration ◽  
Three Dimensional ◽  
Lagoon Of Venice ◽  
Dimensional Diffusion

scholarly journals Fractional Diffusion Equations for Lattice and Continuum: Grünwald-Letnikov Differences and Derivatives Approach

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Derivatives ◽  
Three Dimensional ◽  
Lattice Models ◽  
Fractional Diffusion ◽  
Lattice Diffusion ◽  
Diffusion Equations ◽  
Difference Operators ◽  
Fractional Diffusion Equations ◽  
Nonlocal Media ◽  
Derivatives Of

Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grünwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to another. In continuum limit, the suggested lattice diffusion equations with noninteger order differences give the diffusion equations with the Grünwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grünwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.

Multifractal spectrum of off-lattice three-dimensional diffusion-limited aggregation

1992 ◽  
Vol 46 (6) ◽  
pp. R3016-R3019 ◽  
Author(s):  
Stefan Schwarzer ◽  
Marek Wolf ◽  
Shlomo Havlin ◽  
Paul Meakin ◽  
H. Eugene Stanley
Keyword(s):  
Three Dimensional ◽  
Multifractal Spectrum ◽  
Diffusion Limited Aggregation ◽  
Diffusion Limited ◽  
Dimensional Diffusion
Sign in / Sign up

Export Citation Format

Share Document