832:
is used to rewrite the anisotropic tensor diffusion equation, in standard discretization schemes, because direct discretization of the diffusion equation with only first order spatial central differences leads to checkerboard artifacts. The rewritten diffusion equation used in image filtering:
803:
The diffusion equation is continuous in both space and time. One may discretize space, time, or both space and time, which arise in application. Discretizing time alone just corresponds to taking time slices of the continuous system, and no new phenomena arise. In discretizing space alone, the
493:
1041:
603:
198:
1117:
781:
698:
836:
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640:, which states that a change in density in any part of the system is due to inflow and outflow of material into and out of that part of the system. Effectively, no material is created or destroyed:
515:
1127:
711:
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77:
1376:
488:{\displaystyle {\frac {\partial \phi (\mathbf {r} ,t)}{\partial t}}=\sum _{i=1}^{3}\sum _{j=1}^{3}{\frac {\partial }{\partial x_{i}}}\left}
704:
is the flux of the diffusing material. The diffusion equation can be obtained easily from this when combined with the phenomenological
1036:{\displaystyle {\frac {\partial \phi (\mathbf {r} ,t)}{\partial t}}=\nabla \cdot \left\nabla \phi (\mathbf {r} ,t)+{\rm {tr}}{\Big }}
1371:
24:
708:, which states that the flux of the diffusing material in any part of the system is proportional to the local density gradient:
1381:
1337:
1287:
56:
272:. If the diffusion coefficient depends on the density then the equation is nonlinear, otherwise it is linear.
1351:
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32:
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598:{\displaystyle {\frac {\partial \phi (\mathbf {r} ,t)}{\partial t}}=D\nabla ^{2}\phi (\mathbf {r} ,t),}
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1242:
1075:. The spatial derivatives can then be approximated by two first order and a second order central
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1044:
266:
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1118:
Radiative transfer equation and diffusion theory for photon transport in biological tissue
193:{\displaystyle {\frac {\partial \phi (\mathbf {r} ,t)}{\partial t}}=\nabla \cdot {\big },}
8:
1092:
805:
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1148:"Advanced Analytic Self-Similar Solutions of Regular and Irregular Diffusion Equations"
48:
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44:
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16:
Equation that describes density changes of a material that is diffusing in a medium
776:{\displaystyle \mathbf {j} =-D(\phi ,\mathbf {r} )\,\nabla \phi (\mathbf {r} ,t).}
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813:
693:{\displaystyle {\frac {\partial \phi }{\partial t}}+\nabla \cdot \mathbf {j} =0,}
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27:. In physics, it describes the macroscopic behavior of many micro-particles in
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31:, resulting from the random movements and collisions of the particles (see
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with a varying kernel (stencil) of size 3 Ă 3 in 2D and 3 Ă 3 Ă 3 in 3D.
1080:
817:
40:
1347:
A tutorial on the theory behind and solution of the
Diffusion Equation.
625:
52:
1052:
287:, and the equation is written (for three dimensional diffusion) as:
1164:
276:
219:
1352:
Classical and nanoscale diffusion (with figures and animations)
1079:. The resulting diffusion algorithm can be written as an image
1048:
275:
The equation above applies when the diffusion coefficient is
1335:
Diffusion
Calculator for Impurities & Dopants in Silicon
636:
The diffusion equation can be trivially derived from the
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The diffusion equation has numerous analytic solutions.
269:
505:
is constant, then the equation reduces to the following
1294:
The
Diffusion Handbook: Applied Solutions for Engineers
1128:
Numerical solution of the convectionâdiffusion equation
816:. In discretizing both time and space, one obtains the
839:
823:
714:
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518:
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when bulk velocity is zero. It is equivalent to the
1243:"Heroes and Highlights in the History of Diffusion"
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192:
55:. The diffusion equation is a special case of the
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1303:. Long Island, NY, USA: Dover Publication Inc
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1067:) are symmetric matrices constructed from the
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43:, and applied in many other fields, such as
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1257:Carslaw, H. S. and Jaeger, J. C. (1959).
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1173:
1163:
785:If drift must be taken into account, the
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1377:Parabolic partial differential equations
1278:Mathews, Jon; Walker, Robert L. (1970).
789:provides an appropriate generalization.
279:; in the case of anisotropic diffusion,
1182:
25:parabolic partial differential equation
1359:
1320:Gillespie, D.T.; Seitaridou, E (2013)
222:of the diffusing material at location
1308:Transport by Advection and Diffusion.
1282:(2nd ed.), New York: W. A. Benjamin,
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615:
71:The equation is usually written as:
35:). In mathematics, it is related to
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1280:Mathematical methods of physics
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1051:, and superscript "T" denotes
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1195:Annalen der Physik und Chemie
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812:, rather than the continuous
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57:convectionâdiffusion equation
1273:The Mathematics of Diffusion
1259:Conduction of Heat in Solids
507:linear differential equation
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7:
1382:Functions of space and time
1268:Berlin/Heidelberg: Springer
1086:
622:particle diffusion equation
10:
1400:
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624:was originally derived by
608:which is identical to the
63:under some circumstances.
1324:. Oxford University Press
1322:Simple Brownian Diffusion
1275:. Oxford: Clarendon Press
1216:10.1002/andp.18551700105
1108:Fick's laws of diffusion
810:discrete Gaussian kernel
799:Discrete Gaussian kernel
285:positive definite matrix
33:Fick's laws of diffusion
1261:Oxford: Clarendon Press
1113:MaxwellâStefan equation
1043:where "tr" denotes the
1306:Bennett, T.D: (2013)
1247:Diffusion fundamentals
1103:FokkerâPlanck equation
1037:
787:FokkerâPlanck equation
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265:represents the vector
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1310:John Wiley & Sons
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267:differential operator
249:diffusion coefficient
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1264:Jacobs. M.H. (1935)
1189:Fick, Adolf (1855).
1175:10.3390/math10183281
1123:Streamline diffusion
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1315:Adventure Diffusion
1301:Diffusion Phenomena
1266:Diffusion Processes
1207:1855AnP...170...59F
1093:Continuity equation
638:continuity equation
1340:2009-05-02 at the
1271:Crank, J. (1956).
1077:finite differences
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49:information theory
21:diffusion equation
1191:"Ueber Diffusion"
1073:structure tensors
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1069:eigenvectors
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830:product rule
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41:random walks
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1152:Mathematics
1081:convolution
818:random walk
1361:Categories
1165:2204.04895
1134:References
797:See also:
632:Derivation
626:Adolf Fick
53:biophysics
39:, such as
1367:Diffusion
1225:0003-3804
1053:transpose
1000:ϕ
991:∇
987:∇
966:ϕ
920:ϕ
917:∇
898:ϕ
884:⋅
881:∇
869:∂
847:ϕ
844:∂
751:ϕ
748:∇
733:ϕ
724:−
674:⋅
671:∇
659:∂
654:ϕ
651:∂
628:in 1855.
573:ϕ
564:∇
548:∂
526:ϕ
523:∂
465:∂
443:ϕ
440:∂
423:ϕ
386:∂
382:∂
360:∑
339:∑
326:∂
304:ϕ
301:∂
277:isotropic
228:and time
161:ϕ
158:∇
141:ϕ
125:⋅
122:∇
110:∂
88:ϕ
85:∂
67:Statement
1338:Archived
1317:Springer
1087:See also
1253:: 1â32.
1203:Bibcode
220:density
218:is the
1286:
1223:
1049:tensor
700:where
261:; and
203:where
155:
51:, and
1160:arXiv
1045:trace
23:is a
1284:ISBN
1221:ISSN
828:The
620:The
232:and
19:The
1211:doi
1199:170
1170:doi
501:If
270:del
1363::
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