> 584 jbjbj=G=G 4$_-_-jpp@@@@@TTTT$xT sssbdddddd,#L-@sssssk@@kkks@@bksbkk
ϫjN0! k!
k@
Dsssksss! sssssssssp:
Introduction to fractional partial differential equations: modeling, computation and analysis
Hong Wang
Department of Mathematics, University of South Carolina, and IBM Visiting Fellow, Division of Applied Mathematics, Brown University
Fractional partial differential equations (FPDEs) are emerging as a powerful tool for modeling challenging phenomena including anomalous transport, and long range time memory or spatial interactions. Compared to integer-order PDEs, the fractional order of the derivatives in FPDEs may be a function of space and time or even a distribution, opening up great opportunities for modeling and simulation of multiphysics phenomena, e.g. seamless transition from wave propagation to diusion, or from local to nonlocal dynamics. Nevertheless, FPDEs present new mathematical and numerical difficulties that were not encountered in the context of integer-order PDEs and so require rigorous modeling, numerical and mathematical analysis. In this sequences of lectures, we will give an overview of FPDEs in terms of modeling, computation and mathematical analysis, especially the open problems in these fields. No a priori knowledge of fractional calculus is assumed.
Part I: From normal to anomalous diffusion
Date: April 12, 2018 (noon 1 pm, 170 Hope St, Room 118)
We begin by reviewing the historical development of classical Fickian diffusion PDE of Einstein and Pearson and, especially, the assumptions under which the Fickian PDE was derived. Then we discuss when these assumptions hold and when they may fail. Then we go over why FPDEs may provide a competitive modeling tool in these scenarios. We will also discuss the new difficulties that occur in the FPDE modeling and the related modeling issues.
Part II: Computational issues of FPDEs
Date: April 19, 2018 (noon 1 pm, 170 Hope St, Room 118)
Because of the nonlocal nature of fractional differential operators, numerical methods for FPDEs generate dense stiffness matrices with complicated structures or/and long tails in time that require numerical solutions at all the previous time steps (for time-fractional PDE). Consequently, O(N^2) memory, where N is the number of spatial unknowns (plus O(MN) memory where M is the number of the time steps for time-fractional PDE), is required. In addition, a direct solver has an O(N^3) computational complexity (per time step for time-dependent FPDEs) plus (O(M^2N) for time-fractional PDEs). A Krylov subspace iterative solver has an O(N^2) computational complexity per matrix-vector multiplication and a large number of iterations is often needed, and may diverge even for very simple problems due to the impact of round-off errors. This is in sharp contrast to the numerical methods for integer-order PDEs, which yield sparse stiffness matrices that have an O(N) memory requirement and O(N) or O(N log N) computational complexity to invert (per time step for time-dependent problems).
The significantly increased computational complexity and memory requirement of FPDE models make their realistic applications computationally very expensive, which calls for the development of fast and accurate numerical methods with efficient storage for FPDEs. We will go over some recent developments in this area.
Part III: Mathematical issues of FPDEs
Date: April 26, 2018 (noon 1 pm, 170 Hope St, Room 118)
FPDEs present new mathematical difficulties that are not encountered in the context of integer-order PDEs. For example, the solutions to linear elliptic FPDEs with smooth data (imposed in smooth domains for multimensional problems) have low regularity (not even in Sobolev space H^1) and boundary layer. The Galerkin formulation of linear variable-coefficient elliptic FPDEs may lose its coercivity even though its constant-coefficient analogue has proved coercivity. The inhomogeneous Dirichlet boundary-value problem of a constant-coefficient Riemann-Liouville elliptic FPDE is not well posed even if its homogeneous analogue is. Apparently, these issues, in turn, cause new mathematical and hence increased numerical and computational difficulties. We will analyze the reasons of these issues and report some recent progress in this direction.
1:FQ^_`ajklvG \ ` p
IF
V
ǻǰǊwowwghvCJ aJ h:CJ aJ h-h-CJ aJ h>CJ aJ hvhvCJ aJ hv6CJ aJ hH6CJ aJ hTB6CJ aJ hTB6CJ aJ o(h Oht6CJ aJ h OhK6CJ aJ h OhKCJ aJ h Ohthi*h>h~y1h~y1h~y1hvh$'`al
56yjgdgd*mgd|GSgd:Dcgd7=gdgdKgdKV
h
t
u
,>G%'Rensʼ}u}u}u}u}u}u}mu}u}u}euhCJ aJ hHCJ aJ h:DcCJ aJ h
CJ aJ h5QCJ aJ hHhH5>*CJ aJ h|GS5>*CJ aJ hH5>*CJ aJ hv5>*CJ aJ hHhk5>*CJ aJ hHh5>*CJ aJ h)CJ aJ h+CJ aJ hvCJ aJ h>CJ aJ hvCJ aJ $456r^_l:Y_'(Kvwxy*}ƻhQCJ aJ hAWCJ aJ h5Qh*mCJ aJ h*mCJ aJ hHCJ aJ hm5iCJ aJ h5QCJ aJ h5Qh5QCJ aJ h:DcCJ aJ hHh|GS5>*CJ aJ h|GS5>*CJ aJ h:Dc5>*CJ aJ hHh:Dc5>*CJ aJ 2LO6IJK%'NWWijhBsCJ aJ hHCJ aJ hCJ aJ hhCJ aJ hHh|GS5>*CJ aJ h|GS5>*CJ aJ h5>*CJ aJ hHh5>*CJ aJ h*mCJ aJ hQhQCJ aJ hm5iCJ aJ &21h:pK/ =!"#$%666666666vvvvvvvvv666666>6666666666666666666666666666666666666666666666666hH6666666666666666666666666666666666666666666666666666666666666666662 0@P`p2( 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p 0@P`p8XV~ OJPJQJ_HmH nH sH tH N`NNormaldCJ_HaJmH nHsH tH^@^t Heading 1$<@&"5CJ KH OJPJQJ\^JaJ DA D
Default Paragraph FontRiR
0Table Normal4
l4a(k (
0No ListVoVtHeading 1 Char"5CJ KH OJPJQJ\^JaJ PK!pO[Content_Types].xmlj0Eжr(]yl#!MB;.n̨̽\A1&ҫ
QWKvUbOX#&1`RT9<l#$>r `С-;c=1g~'}xPiB$IO1Êk9IcLHY<;*v7'aE\h>=^,*8q;^*4?Wq{nԉogAߤ>8f2*<")QHxK
|]Zz)ӁMSm@\&>!7;wP3[EBU`1OC5(F\;ܭqpߡ 69&MDO,ooVM M_ո۹U>7eo >ѨN6}
bvzۜ6?ߜŷiLvm]2SFnHD]rISXO]0 ldC^3شd$s#2.h565!v.chNt9W
dumԙgLStf+]C9P^%AW̯f$Ҽa1Q{B{mqDl
u" f9%k@f?g$p0%ovkrt ֖ ? &6jج="MN=^gUn.SƙjмCR=qb4Y" )yvckcj+#;wb>VD
Xa?p
S4[NS28;Y[,T1|n;+/ʕj\\,E:!
t4.T̡e1
}; [z^pl@ok0e
g@GGHPXNT,مde|*YdT\Y䀰+(T7$ow2缂#G֛ʥ?qNK-/M,WgxFV/FQⷶO&ecx\QLW@H!+{[|{!KAi
`cm2iU|Y+ި [[vxrNE3pmR
=Y04,!&0+WC܃@oOS2'Sٮ05$ɤ]pm3FtGɄ-!y"ӉV
.
`עv,O.%вKasSƭvMz`3{9+e@eՔLy7W_XtlPK!
ѐ'theme/theme/_rels/themeManager.xml.relsM
0wooӺ&݈Э5
6?$Q
,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-!pO[Content_Types].xmlPK-!֧6-_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!!Z!theme/theme/theme1.xmlPK-!
ѐ'(
theme/theme/_rels/themeManager.xml.relsPK]#
j$V
j
jL#@0(
B
S ?U\
$+OW
DMl7C{XZ
l::::::::6H^`OJQJo(8^8`OJQJo(^`OJQJo(o p^ `OJQJo(@^`OJQJo(x^x`OJQJo(H^H`OJQJo(o^`OJQJo(^`OJQJo(KFAW_4f
Z')M+MqhO5 ,L"-^//11~y1$5:8:7=
?TBMD O
P:P5Q|GSC[_?5_:Dchm5i8oGo>bqOsBsrv?I|*m?p&wb#>Hv$%r'Ok
!ctv i*KF;}y
Qjl@ { (jX@X
X@Unknown GTimes New Roman5Symbol3Arial7Calibri;(SimSun[SO7Cambria? Courier New;WingdingsACambria Math"qhc[dgs
s
!20bbKHP $PhO2!xxhwangAnna Lischke
Oh+'0x
(4@
LX`hp'hwangNormal.dotmAnna Lischke3Microsoft Macintosh Word@F#@8"@.!@.fs
՜.+,0hp
' University of South CarolinabTitle
!"#%&'()*+-./012367:Root Entry FЫ9@1Table1 WordDocument 4$SummaryInformation($DocumentSummaryInformation8,MsoDataStoreـΫ0ΫLRRKRNA==2ΫΫItem
PropertiesUCompObj`
F Microsoft Word 97-2004 DocumentNB6WWord.Document.8