LecturesDiffus.pdf

"!
#%$'&)(*&+ ",.-/$
0.13241"57698;:=<?>A@CBD8FEG<96
HIKJMLNJOPJ7QMRNSKTNJMLVUDTWJXOYUZTN[JMOP\^]4UZLNJ7_a`DbdcXI4UDefRNJ7Thgi`DjlkmUDbfnlopLqRNJsrRtvuwyxqzibK`DR{IfJ7TA|DJ7T}\
SKLWJMjGSKO~TNJMjGJMTNJMbKQ7JPL€‚`[
UDbƒo„L…RNJsrR+tvuwy†q‡/IKPQˆI)‰L…SfLNJ7_ŠPb)‹UZRNIŠŒDŽF‘
’“GR{IfJUDeKeKOP‰JM_
” UZR{IKJ ” U=R{‰QMLmLNJM•SKJ7bKQMJDx—–
˜JDopTNJ[D`‰bK[+R{` bKJ7JM_)U™‡/If`O‰Jš]KSKbKQˆI)`DjXR{J7T ” PbK`OP`[D\†‚LW` ” J›`Djm‡/If‰QˆIœˆopO‰OVL}R{‰Qˆn
IKJMTNJDxŸžqPTNL}R/LN` ” J^bK`DR{UZR{P`b+UDbK_+j3UDQMRNLjGTN` ” Q;UDOPQ7SKOPSKL7 “¢¡F– ŸUDb+`SfR£‡XUDTN_™e9`D‰b
R{PbK[SKbK¤RbK`T ” UZO9|JMQMRN`T/_KJM¥4bfJ7_¦U=RJ;UDQˆI™el`‰b
R/`bR{IfJ
n
]l`SKbK_KUDTW\
`Dj~U `T ‘_K ” J7bfLN‰`Db4UDOCLN`OP‰_
‡/IK‰QˆI§PL¨SKLNSKUDO‰O¤\¦UDLWLNS ” J7_
∂Ω
2
3
Ω
RN`š]lJ©UDb™`e9JMb+Q7`bKbfJ7QMRNJ7_¦LWJMR/‡/PRNI™UDb+`TNPJ7b
RˆUD]fO‰J^]l`SKbK_4UZTW\x
“yŽ–
‚R{IKJm[DT{UD_KPJ7b
RA`Dj4R{IKJ~LNQ7UDOYUDTCjGSKbKQsR{‰`Db
`DjfR{IKJmLWe4UZRNYUDO
|ZUDTN‰UD]KOPJ7L7xAªx„[fxP†Pb
5f
f
RNIKJ¨Q;UZLNJ«`Zj‚R{IKTWJ7J«LWe4UDQMJ¨_K ” J7bfLN‰`DbKL7†
‡~J¨`]fR{UD‰b"R{IKJ¬|J7QsR{`T~|DUZO‰SKJM_›jGSKbKQsR{‰`Db
D
E
∂f ∂f
5f = ∂f
,
,
= hfx , fy , fz i
“3’
–
Š∂x
R{IKJ§∂y_K¤|∂z
J7TW[J7bfQ7J­`ZjU)|JMQMR{`DT™jGSKbfQMR{P`b
‡/IKJMTNJ*J;UDQˆIQ7``TN_KPb4UZRNJ
5
F
_KJMe9· JMF
bK_KL`b®R{IKJ LNe4U=R{YUZO/|ZUDTWYUD]fO‰J7LMxª/xp[Kx¤†Pj
†mR{IKJMb
F = hF, Gi
5·F =
x
∂F
∂G
+
=F +G
∂x jGSfbKQM∂y
“G–¯z°
RN‰`bPL¬x±s²›³´³=yµ·¶PbU…_K` ” UDPb›¤j‚‰L~I4UDLmQM`b
R{‰bSK`DSKLme4UDT}R{‰UDO¸_KJ7TWP|ZUZR{¤|JML
Pb UZO‰O?¤R{Lm|ZUDTN‰UD]KOPJ7LmPb+RNIKJ^_K` ” UDPbSKbK_KJMTiQM`bKLNP_KJ7TNUZR{P`b?x
“y¹–º©UDSKLWLi»iP|JMTN[JMbKQ7J%HIKJM`TNJ ” Ÿ¼ibf_KJ7T/LNSfPRˆUZ]KO‰J^L ” ``DR{IfbKJ7LWL¬QM`bK_K¤R{P`bKL
Z
Z
5 · Fdx =
F · ndx.
Ω
∂Ω
½¬J7TNJŸœˆo ” SKLNPbK[/R{IKJ¾Q7`Db|DJ7b
R{P`b¯RNI4UZR?¿
TWJ7eKTWJ7LNJMb
R{LARNIKJŸUDeKeKTW`eKTWYUZRNJV_KPÁ9J7T¢
dx
À
JMbRNYUDO.JMO‰J ” J7b
R7x/ž¸`DTiJÂrKU ” eKOPJD†¸Pj
‰Li’=‘_K ” J7bKLW‰`bKUDOÆ
TNJ7efTNJ7LWJ7b
R{L«_KPÁ9J7T¢
JMbRNYUDOÄ|`OPS ” J/UZbK_d‰bdQ;UDOPQ7SKOPSKLhbK`ZF
RˆUZRN‰`b‡~J¬‡~`SKOP_dI4U;|dx
J/SKLNJM_šRNIKTNJMJ¬Pb
R{J7[DT{UDO
LW‰[bKLMx
œ£bKQM‰_KJMbR{UDO‰O¤\†fº%»iHPLmT{U=R{IKJMT ” `TNJi`TN_KPb4UDT}\"RNI4UDb\D`S ” P[I
R~RNIK‰bKn¸xŸcX`bfLN‰_fJ7TX‡/I4U=R
PR ” J;UZbKL‰b+`bKJ©_K ” JMbKLNP`b?x
ÅÂÆ ÇÂÆfÈ4ɃʂËsÌÍZÎ
ÏÄÐKÑYɃʧÏ9Ê9ÒÓfσÔDÏlʂÈKÌÔZÏZÕ"Ë
Ö
0‚1‰×?1د<96CÙ7ڏ:´ÛKÜf8´E3<l6ÝÜ46C>ށÜ4ßGÜ46CBÚáàŸÜ
âšÙ
œ ” UD[PbKJ¨R{I4U=R¨UdeKI
\LN‰Q7UDO.•S4UDb
R{¤R£\+‰L/_K‰L}R{TW‰]KSfRNJ7_™`´|J7TiUšLWe4UZRNYUDO?TWJ7[P`b
xXHIKPL
TNJM[‰`DbáQ7`SKOP_§]9J¡M£’_K ” J7bfLN‰`Db4UDOƃUDbK_*U[J7bfJ7TNPQae9`D‰b
R^‰L«TNJMjGJMTNJMbKQ7JM_­]
\¦R{IfRJaO‰JMRWR{JMT
xãHISKL7†
Q7`SKOP_TNJsjGJ7TNJMbKQ7J*UOP`ÄQ7UZR{P`b
`bR{IfJ¦TNJ7UDO¨O‰PbKJD†/Uel`PbR
PbU
x
(x, y)
R£‡~`Z£_K ” JMbKLNxP`b4UDO9TNJM[‰`b+`T
Pb URNIKxTNJMJs£_f ” J7bKLW‰`b4UZOƒTNJM[‰`Db?x
(x,|ZUDy,TN‰z)
ä¬JsrR;†LWSKeKel`LNJRNI4UZR"R{IKJ+
UD]KOPJ
PL"RN`­TNJMeKTNJMLNJMbRUeKI
\LNPQ;UDO¬•S4UDb
R{¤R£\OP‰nZJ
N_KPLWR{TW‰]KSÄR{J7_™`´|JMTiLWe4UDQ7J…UDbK_
el`eKSKO‰UZR{P`b?† ” UDLWL7†¸J7bKJMTN[D\D†¸JMR{QZxP†4RNI4UZR¬‰L/
” U;\|ZUDT}\‡/PRNI
R{ ” JZxåHIfJ7bPRPLbKJMQ7J7LWL{UDT}\R{` ” J;UDLWSKTNJ¦RNIKJÝ戳=çƒæˆèsç4µyéWêDµ3ëó=çì“G`T§íèsçf±së·µ3îf–"`Dj©R{IKPL
LNSf]KLWR{UDbKQ7JZx+ž4`T^JsrfU ” eKO‰JZ†.‰b§R{I4UZR%Q7UDLNJ"`DjU+LNPbK[OPJaLNe4UDQMJ›_K ” JMbKLNP`b“GRNIKJdLN ” eKOPJ7LWR
Q;UZLNJ´–i‡~J‡~`SKO‰_áIKU´|DJR{I4U=R
UDbK_á¤jVR{IKJSfbKPRNL¨`Dj¾R{IKJ…eKI
\ÄLW‰Q7UDOh•S4UDb
R{¤R£\
N
=
N
(x,
t)
UDTWJ
†fRNIKJ7b™R{IKJ©SKbfPR{L`Zj
UDTWJ%[DP|JMb+]\
Q
N
Q
[N ] = ,
L
‡/IKJMTNJ
PLO‰JMbK[DRNI?x­k~Hm˜ï†
UZbK_
‡~`SKOP_®]lJQ7UDO‰OPJ7_ðÂñçlí
êD²›èsç4µ‘êDòŸñç4ë·µG±sx­óÄ` ” J
L
Q
L
`DRNIKJ7TjGSKbK_KU ” JMb
RˆUDO^SKbfPR{L+UZTNJ ” UZLNL
UDbf_RN J
xô˜Já‡/‰OPO^UDLNLWS J§RNI4UZR+R{IfJ
M`Tq_KJ7L}R{TN`´” \DJ7_"TSKbKO‰JMLNLq‡~J/UDTWJm[P|DJ7” bdUDb…JsrÄeKO‰PQ7¤R
PLhbKJ7¤R{IKJMThQ7TWJ;UZRNJ7_a
eKI
\LNPQ;UDOK•S4UZbRNPR£\
N
” J7QˆI4UDbf‰L ” jG`T_K`PbK[aLN`K†4OP‰nZJ©UdLW`SKTWQ7J%`T_KJ7Q7U;\jGSKbKQMRN‰`bƒx
ä¬`F‡õLWSKeKel`LNJ R{I4UZR
‰LšU­LWSK]KTNJM[‰`DbÝ`Dj
‡/PRNIÝ]l`SKbK_4UZTW\
x°œ£b݇~`TW_KL7†/U
Ω
R
∂Ω
Q7`DbKLNJMTW|ZUZR{P`b+OYU;‡RˆUDnZJ7L/R{IKJ^jG`T ” H ” J^TNUZR{J©`DjAQˆI4UDbK[DJ%`ZjARN`DRˆUDO
‰bKLW‰_KJ
J7•S4UDOPL
Q
Ω
Ÿz ” `SfbR¬`Dj
`´‡/PbK[d`SfR/`Dj
UDQMTN`LWL
e9JMT/SKbKPRRN ” J©÷
z ” `SfbRX`Dj Q]9JMö ‰bK[¯LN`SKTWQ7JM_™`DTXΩLNPbKnDJM_*“3QM∂Ω
TNJ;U=R{J7_`DTX_KJMLWR{TW`´\J7_+PLX`bKJ«‰b
R{JMTNeKTWJMRˆUF
R{P`b¸–~Pb
“3`ZjqQMQ
`SKTNLWJD†¸LN`DSKTNQMJ7_™‰L/e9`DLNPRNP|DJ^UDbK_™LNPbKnDJM_‰LbfJ7[
UZRNP|DJDx—–
zQsR{S4Ω
UDOPOP\D†AR{IfJ›b4U ” J›PL…U™LNOP‰[I
R ” PLNbK` ” J7TMxHATNUD_K¤R{‰`Db4UDO‰O¤\†?RNIKJ›O‰U´‡ôœ^IKU´|DJ%øWSKL}R
_KJMLNQ7TW‰]lJ7_Š‰L…UŠùˆêDò¤êDçƒæˆèšò¤êDúV†h‡/IK‰OPJU戳=çf±´èséÂû=êDµyëó=çŠò¤êDúô‰L…U¦]4UDO‰UDbKQ7J›OYU;‡ô‰b­‡/If‰QˆI
R{IfJ7TNJ©PLbK`šLW`SKTWQ7J^R{JMT ” x
˜J¯bfJ7J7_`bKJ ” `DTNJ^R{JMT ” †4b4U ” J7OP\U¿ SÄr R{JMT ”
R{I4UZR¬TWJ7eKTWJ7LNJMb
R{L«RNIKJ…U ” `SfbR
`Dj
RNI4UZR `´‡/LŸUDQ7TW`LNLŸU^SKbK¤RVLNeKUZR{‰UDOKJ7OPJ ” ö J7b
R¾À `Dj
elΦJ7TVSKbfPRVR{ ” JZxhóSKQˆIU¨jGSKbKQsR{P`b
Φ]lJa|J7T}ö \§Q7` eKOP‰Q;U=R{J7_§UZbK__fJ7elJ7bK_bK`DR©`bKO¤\á∂Ω
Q;UZb
`b§LNeKUZR{‰UDOhQ7``TW_K‰b4U=R{J7L©UZbK_*RN ” JD†
”
]KSfRJM|JMb
UDbK_PRNL_KJMTNP|ZUZRNP|DJ7L7xåäi`ZR{‰QMJ RNI4UZR
‰LU­|DJ7QsR{`T•S4UDb
RNPR£\†i]9JMQ;UDSfLNJ
N
Φ
`´‡/L¨I4U;|DJaU›_KPTNJ7QsR{P`bUZLi‡~J7O‰OAUDL«U ” UZ[bKPRNSK_KJZxižA‰b4UDOPOP\D†f‡XJ¯‡mUDb
R¨RN`UDOPO‰`´‡°jG`DTiR{IfJ
öel`LNLW‰]KPO‰PR£\ R{I4U=R¯RNIKJ
” UZR{JMTN‰UDOV‰L©]lJ7‰bf[LN`DSKTNQMJ7_)`T%LNPbKnDJM_)UDQ7QM`TN_f‰bK[™R{`¦U¿¢LW`SKTNQMJ À
“3TNUZR{J;–ijGSKbKQsR{‰`Db
‡/IK`LWJšSfbKPRNL©UDTNJ
xaHIK‰L^jGSKbKQsR{P`b§Q7`SKOP_§_KJ7elJ7bK_`báR{ ” JZ†
el`LN¤R{‰`Db¯UZbK_JM|DJ7fb
xA˜JDopO‰OLNSfeKeKTNJMLN[N
Lh¤R{]LA/T
UDTN[DS ” JMb
RqOP‰LWR7xCäi`´‡)‡XJZo„TWJX[`D‰bK[/R{`iJsrÄeKTNJMLNL
PR/PbLW\ ” ]l`O‰LMxh˜­JZN
o„TWJ¯TWJ;UD_Ä\jG`TmR{IfJ%Pb
R{J7[DT{UDOƒjG`DT ” SKO‰UZR{P`b?†
‡/IKPQˆI L}RˆUDT}R{L¬UDL
Z
Z
Z
d
N (x, t)dx = −
Φ · ndx + f dx
dt Ω
∂Ω
Ω
UDbK_‰LdR{TNUDbKLWjG`DT ” J7_Ý]
\ݺ©»«HüR{`*R{IKJ LN`Z‘Q;UDOPO‰J7_®‰b
RNJ7[TNUDOmjG`T ” SKOYUZRN‰`bŠ`Dj¨Q7`bKLWJ7T}|DUF
R{P`b¸ýZ]KUDOYUDbfQ7J
Z
Z
d
N (x, t)dx =
(− 5 ·Φ + f ) dx.
dt Ω
Ω
äi`´‡ã‡XJ ” UDnDJdUQMTNSKQMYUDOqUDLNLWS ” eÄR{‰`Db§UD]9`DSfR
†‚bKU ” JMOP\™R{I4U=R^PR^PL^L ” `Ä`ZR{I¦Pb§UDO‰OAPj
PRNL¬|ZUDTWYUD]fO‰J7L¬UDbf_ ‡~JaUDLNLWS ” J¯R{IKUZR
‰LiUš]l`SKNbK_KJM_¦TNJM[‰`Db?xHIKJ7b¦‡~J ” U;\RˆUDnZJ…R{IfJ
_KJMTNP|ZUZRNP|DJ^LN‰[Db™PbKLNP_KJ©UDbK_™`]fR{UD‰b Ω
Z Ω
∂N (x, t)
− (− 5 ·Φ + f ) dx.
∂t
ÅÂÆ ÖÂÆ4ËÂÑ·þmÿÔDÌ+ÈKσËMÌÄË
ÑPʂËMÐDÏ9Ê?ÐDÏ9Ê?ÌÄɂËiË7É9ÍlÈKÌfË
k SfRXR{IKPLX‰L~R{TNSfJ«jG`TmUZO‰OlLNSK]KTWJ7[P`bKL
X
`Dj xVziLNLWS ” PbK[…RNIKJ¨‰b
RNJ7[TNUDbK_‰L~Q7`b
RN‰bSK`SKLM†
Ω
R{IfJ%`DbKOP\š‡mU;\jG`TmRNIK‰LmRN`"]lJ^R{TWSKJ©‰LmR{IKUZR R
Nt (x, t) + 5 · Φ = f,
‡/IKPQˆI PLmR{IKJ©LN`=£Q;UZO‰O‰JM_*íDë ¨èsé{èsç4µyë3êDòDð;³=é²ü`ZjhUdQ7`bfLNJ7T}|ZUZR{P`b¸ýZ]4UZOYUDbKQMJ¨OYU;‡©x
HIKJabfJsrR^•SKJ7L}R{P`bR{`›]9JaUZbKLW‡~J7TWJ7_§‰L¬R{IKJ%jG`T ” `DjVRNIKJ SÄr R{JMT ” x CJ7TNIKUDeKL¨R{IfJ
ö
” `LWR/j3U ” `SKL/`DjAR{IKJMLNJ…PL¬žq‰QˆnZJDopL«OYU;‡©†4R{IKUZRiUDLNLWJ7TWRNLiRNI4UZR¬R{IKJ%•S4UDb
R{¤R£\R{J7bf_KLiRN` `´‡
jGTN` ” If‰[IKJMT¯QM`bKQMJ7b
R{TNUZR{P`b­R{`O‰`´‡XJMTUZRUT{U=R{J›efTN`el`TWRN‰`bKUDOhR{`RNIKJ›QM`bKQMJ7b
R{TNUZöR{P`b
[TNUD_K‰JMb
R;xVœ£b™LW\ ” ]l`O‰L
Φ = −D 5 N.
œ£bR{IKJ ” `L}R[J7bfJ7T{UZO?Q;UDLWJ
‰L/U…R{JMbKLN`DT•SKUDb
R{PR£\D†KR{I4U=R‰LM†K‰b`SfTmLNJsRNR{PbK[K†4U ” UZRNTN¤r9x
‹ `TWJ7`´|JMT7†l‰b UDO‰OƒQ;UDLWJ7L7† D‰L¬UDLNLWS ” J7_ RN`š]lJ¯U"L}\ ”"” JMR{TW‰Q^el`LN¤R{¤|J©_KJM¥KbKPRNJ ” UZRNTN¤r9†
LN`RNI4UZR^]
\LN` ” J`T}R{IK`[DD
`b4UDOCQˆI4UDbK[DJ`DjŸQ7``TN_f‰b4UZRNJ7L¬‡XJaQ7UDbá|ÄPJM‡
UDL^Uš_KYUD[D`b4UDO
D
Z
U
{
R
W
T
¤

r
/
‡
¤

{
R
+
I
l
e
`
N
L
¤

{
R
P

D
|
©
J
K
_
Y

D
U
D
[
`
4
b
D
U
?
O
M
J
b
{
R
N
T
P

7
J
M
L
X
x
‘
œ
q
j
N
R
K
I
M
J
™
\
Z
U
N
T
%
J
D
U
‰
O
ƒ
O
7
J

•
4
S
D
U
y
O
K
†
X
‡
¯
J
;
Q
D
U
+
b
RˆUDnZJ
R{`"]lJ
”
D
Uae9`DLNPRNP|DJ¨LNQ7UDOYUDTM†Ä‡/IKPQˆI+Q7`SKOP_+]9J©U…jGSKbfQMR{P`b+`DjCe9`DLNPRN‰`bUDbK_+`DR{IfJ7Tm|ZUDTN‰UD]KOPJ7L7†JM|DJ7b
PRNLNJMOPj¢x
N œ£b+R{IKPLQ;UDLWJ%RNIKJ©_KPÁ9J7TWJ7b
R{‰UDOƒjG`T
” `DjAQ7`DbKLNJMTW|ZUZR{P`b+]9JMQ7` ” J7L
Nt (x, t) = 5 · (D 5 N ) + f.
HIKPLAPL.RNIKJ ” `L}RA[DJ7bKJMT{UDO
jG`T ” `DjKR{IfJ~_K¤ÁƒSfLN‰`Db¯JM•SKUZR{P`b?xA€.JsR;opLqLWe9JMQ7YUZO‰MJhR{`¬R{IKJŸQ;UDLWJ
‡/IKJMTNJ
‰L©UQM`bKLWR{UDb
R;†CU›|JMTW\áQM` ”"” `b§UZbK_áSKLNJsjGSKOVQ;UZLNJDx…HIKJd_KPÁ9SKLNP`b¦JM•SKUZR{P`b
D
bK`´‡°R{UDnDJML¬RNIKJ^jG`T ”
Nt (x, t) = D 4 N + f,
‡/IKJMTNJ
‰L"R{IKJ¦€CUZeKOYUDQMYUDb`elJ7T{U=R{`TMxãóÄ` ” JMR{ ” JML›`DRNIKJ7T›Q7``TN_KPb4UZRNJLW\L}R{J ” L+UDTWJ
eKTWJMjGJ7TNUD4
]KOPJ«R{`dQ;UZTWR{JMLN‰UDb?xVœ£b+QM\O‰PbK_KTW‰Q;UZOƒUDbf_LNeKIKJMTNPQ;UDO?QM`Ä`DTN_KPb4UZR{JML7†fjG`DTmJsrfU ” eKO‰JZ†Ä‡~J
I4U;|J©R{IfJ©jG`DT ” SKO‰UDL
∂N
1 ∂2N
1 ∂
r
+
4N =
r ∂r
∂r
r 2 ∂θ 2
∂2N
∂
1 ∂
1
∂N
1
2 ∂N
4N = 2
r
+ 2
sin φ
+ 2 2
r ∂r
∂r
r sin φ ∂φ
∂φ
r sin φ ∂θ 2
œ£bšRNIKJ«Q7UDLNJi`Dj?LWeKIKJMTN‰Q7UDOlQ7``TN_KPb4UZRNJ7LV‡/PRNIšjGSKbKQsR{P`bKL¾R{I4UZRXUDTWJiPbK_KJ7elJ7bf_KJ7b
Rm`Zj.UDbK[OPJD†
R{IfJ¯€.UDefOYUDQMYUDbLW ” eKO‰¤¥4J7LŸR{`
∂2N
2 ∂N
4N =
+
2
∂r
r ∂r
UDbK_™‰bQs\O‰‰bf_KTNPQ;UDO?QM`Ä`DTN_KPb4UZR{JMLXR{`
4N =
∂2N
1 ∂N
.
+
2
∂r
r ∂r
0‚1
?1?EqßGڧدÜ4ÙMÚÄÙ%5M6CÙM8´Ü4698;Ü46Cڏ<l@Cك<9@.:=BÚÙ
ñé{è ˜­J^L}RˆUDT}R/‡/PR{IRNIK‰LeKSfTNJ©œ
∂N
∂2N
= D 2 , −∞ < x < ∞, t > 0
∂t
∂x
N (x, 0) = f (x), −∞ < x < ∞.
ÅÂÆ ÖÂÆ4ËÂÑ·þmÿÔDÌ+ÈKσËMÌÄË
ÑPʂËMÐDÏ9Ê?ÐDÏ9Ê?ÌÄɂËiË7É9ÍlÈKÌfË
½ JMTNJ
i
‰L%UDLNLWS ” J7_§R{`™]9Ja‡~J7O‰OV]lJ7IKU´|DJ7_)UZbK_§_KJ7Q7U´\PbK[TNUDeK‰_fOP\†‚LW`™RNI4UZR%ž4`SKTW‰JMT
f
(x)
R{TNUDbKL}jG`T ” LQ;UDb™]lJ¯UZeKeKO‰PJ7_?xh»«`šLW`"UDbK_™`]fRˆUZ‰b
b
∂N
b,
= −Dξ 2 N
∂t
LN`dR{IKUZR/`bKJ©`]fR{UD‰bfL
−x
2 /(4Dt)
\
e
b = fbe
N
= fb √
4πDt
UDbK_+R{ISKLRNIKJ©Q7`b
|`DO‰SfRN‰`bR{IfJ7`TWJ ” \‰JMO‰_KLmRNI4UZR
−Dξ 2 t
Z ∞
2
2
1
e−x /(4Dt)
√
=
e−(x−y) /(4Dt) f (y)dy.
N (x, t) = f (x) ? √
4πDt
4πDt
−∞
³=ë·ç4µ"!?³=ñÄé{æˆèˆ± cX`DbKLNP_KJ7T%U+e9`Pb
R^LN`SfTNQ7Jd‡/PRNI§‰b
R{JMbKLN¤R£\
†.R{I4UZR%PL7†.UDb*‰bK¤R{‰UDO
M
_KPLWR{TW‰]KSÄR{‰`Db
f (x) = M δ(x)
‡/IKPQˆI [¤|JML¬LW`O‰SÄR{‰`Db
N (x, t) = √
HCT{UDbKLWe9`DTWR¬HAJMT ” L
ϣb
R{TW`Ä_fSKQ7J¯UR{T{UZbKLNel`TWR/RNJ7T ”
M
x2
e− 4Dt .
4πDt
‰b
R{`RNIKJ©eKTN`]fO‰J ” †KLW`dRNI4UZR/R{IfJ SÄrR{JMT ”
ö
PLbK`´‡
Φ = −D 5 N + v0 ,
‡/IKJMTNJ
‰L/UQ7`DbKLWR{UDb
RiLWe9JMJ7_ UZRm‡/IKPQˆIR{IKJ ” UZR{JMTN‰UDO9PLm]lJ7‰bf[R{T{UZbKLNel`TWRNJ7_+LNeKUZR{‰UDO‰O¤\x
“3œ£báIf‰[vIK0JMT¨_K ” JMbKLNP`bKLiRNIK‰L«LNelJ7J7_á‡~`SKOP_*]lJaU›|DJ7O‰`Q7¤R£\ |DJ7QsR{`TMxv–§HIKJaQ7`bfLNJ7T}|ZUZR{P`b
LW\L}R{J ” bf`F‡ ]9JMQ7` ” J7L
∂N
∂2N
= D 2 − v0 N, −∞ < x < ∞, t > 0
∂t
∂x
N (x, 0) = f (x), −∞ < x < ∞.
z°QˆIKUDbK[J¨`Zj‚|DUZTNYUZ]KO‰JML
OPJ;UD_KLŸR{`…RNIKJ¨JM•SKUZR{P`b
UDbK_IfJ7bKQMJ^]\
z
=
x
−
v
t
N
=
DN
0
t
zz
‡/I4UZR/‡~J©I4U;|J¯UDO‰TWJ;UD_f\›_K`bKJZ†¸U"LW`O‰SÄR{‰`Db
Z
2
1
e−(x−v0 t−y) /(4Dt) f (y)dy.
4πDt
−∞
# jqQM`SKTWLNJD†4RNIK‰L[¤|JML¬SfLiUabKPQ7J^jG`T ” SKOYU¯‰bR{IKJ©Q7UDLNJ©R{I4U=R
†lbKU ” JMOP\D†
f (x) = M δ(x)
N (x, t) =
∞
√
(x−v0 t)2
M
e− 4Dt .
4πDt
$mê
íDë3êDò%©ë /ñ
±sëó=çúVë·µ·¶'&)(+*4³=çƒèsç4µyë3êDò-,mé{³=úVµ·¶ œ£b+R{IKPLQ;UDLWJ%RNIKJ.¾»¨ª ]9JMQ7` ” J7L
2
∂ N
2 ∂N
∂N
=D
+
+ aN,
∂t
∂r 2
r ∂r
‡/IKJMTNJ
PL%U [TW`F‡RNIT{U=R{J“3`T%U _fJ7Q;U;\T{UZRNJDx—– HA` [JMR…U™jGJMJ7O¾jG`T^R{IKJš[TN`´‡R{I*R{JMT ” †
 ” UD[PbKaJ›RNI4UZRdR{IKJMTNJ™‰LdbK`*_K‰LWe9JMTNLW‰`b?†Ÿ]KSfRšLN ” efOP\JÂrÄe9`bfJ7b
R{‰UDO[TN`´‡RNI UZRšUá¥frÄJ7_
el`‰b
R;xh˜IKUZR‡XJ©R{IfJ7b™I4U;|J%‰L/UDb # »¨ª
x~HIKPL/‰LSKLWJMjGSKO‚Pb™el`eKSfOYUZRN‰`b
dN/dt
=
aN
Ä
`
K
_
M
J
‰
O
¬
L
D
`
¾
j
D
U
f
b

D
U
‚
O
[
N
T
´
`
‡
N
R
I
{
R
4
I
=
U
i
R
N
R
W
T
+
\
{
R
`
Z
U
7
Q
7
Q
D
`
K
S
b
«
R
G
j
`
¬
T
N
T
M
J
eKTN`_KSKQsR{‰`DbáUDbK_¦_f‰LNelJ7TWLNP`bPb
”
”
N (x, t) = √
ÅsÆ ˆÆKþɃÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DɃͧËMÐ19Ò32
4
U LNPbK[OPJ ” `Ä_KJMOÃxAœ‘RQ;UDb+]lJ¨LNIf`F‡/b“GLNJ7J"tvuwY–ŸR{IKUZR/U…LW`O‰SfRN‰`bšR{`R{If‰LmeKTW`]KOPJ ”

‰bfLWRˆUZbR{UDbKP`SKLel`‰b
RLW`SKTNQMJ…U=R
UDbK_
‰L/[¤|J7b+]
\
r=0
t = 0 M δ(r)
r2
M
exp at −
N (x, t) =
.
4πDt
4Dt
‡/PRNI+UDb
0‚165.1'‚E78Aß3ÚáدÜ4Ù7ÚÙ9aد<96l8FEy6ƒ@C<l@CÙ:ƒ<l@C:=BÚÙ
HIKJMLNJaLW`SKTNQMJ7L ” UDnZJ¯R{IfJ7‰TiUDeKelJ;UZT{UDbKQMJ7Li‰b]l`SKbf_4UDTW\™QM`bK_K¤R{‰`DbKLi`DTiQM`b
R{‰bSK`DSKL
‰b
RNJ7TNP`TmLW`SKTNQMJ7LMxŸ˜JDopO‰O?L}R{PQˆn›R{`aR{IfJ©jG`DT ” J7T7x
$/èˆæsµ3ë·ò ë·çƒèNêD;
é %©ë /ñ
±sëó=ç°úVë·µ·¶=
ê <~³=çf±sµÃêDç4µ?>¨³=ñçlí
êDésî@<~³=çlíDë·µ3ëó=ç ½iJ7TWJ*U®LW ” eKO‰J
Ä
`
K
_
M
J
?
O
K
e
W
T
`
K
]
‰
O
J
P

™
b
D
`
K
b
©
J
N
L
4
e
D
U
M
Q
©
J
K
_

”
”
” J7bKLW‰`bPL
∂N
∂t
N (x, 0)
N (0, t)
HIKJ©LN`DO‰SfRN‰`bRN`"RNIK‰LeKTW`]KOPJ ”
∂2N
, 0 < x < ∞, t > 0
∂x2
= f (x), 0 < x < ∞
= N∗ .
PL`]fR{UD‰bKJM_+]\™€.UDefOYUDQMJ«R{TNUDbKLWjG`DT ” LiUZbK_+‰L
= D
N = N∗ erfc √
‡/IKJMTNJ
x
4Dt
2
erfc z = 1 − erf z = 1 − √
π
0‚1A?1CB
Z
z
2
e−ξ dξ
0
<‚>CÚßED%:F<GFqßGÚHÙJI‘<¸:;98F@A>LK
˜J¬‡/PO‰OjG`ÄQMSKL¾`bšTNUD_K‰UDOK_K¤ÁƒSKLW‰`bƒxC˜JDopTNJ¬bK`DR¾[`D‰bK[^R{`©QM`bKLNP_KJ7TŸU«¿¢el`‰b
RVLW`SKTWQ7J
UDbK_)‡~J ” U;\­‡XUDb
RdR{`§`bKO¤\­QM`bKLNP_KJ7T"U¦¥4bK¤R{J_K` ” UDPb?xž4SKT}R{IKJMT ” `TNJZ†V‡XJ‡XUDb
R"RN` À
UDOPO‰`´‡jG`TLN` ” J^Q7` ” eKOPJsrTNJ;UZQMR{P`bR{JMT ” L7x~óÄ`dIKJ7TWJ©‰LmR{IfJ%[DJ7bKJMT{UDO‚L}RˆUZRNJ ” J7b
R; B <‚>.Ú
ß MÜ4>CE3Ü4N
ß D¯:FG
< FAß3H
Ú O
2
∂N
∂ N
1 ∂N
+ f (r, t, N ), 0 < r0 < r < R ≤ ∞, t > 0
= D
+
∂t
∂r 2
r ∂r
N (r, 0) = f (r),
R{`D[JMRNIKJ7T›‡/¤R{I LWSKPR{UD]KOPJ ]l`SKbK_KUDTW\Q7`bf_KPRN‰`bKL›UZR
UDbf_
xï½i`´‡XJs|JMT7†
r
=
r
r
=
R
0
eKTW`]KO‰J ” L^‡/¤R{I
`T
eKTNJMLNJ7b
RaLWe9JMQ7‰UDO¾eKTN`]fO‰J ” L©PjŸ‡XJš‰b
R{JMbK_­RN` QM`bĐ
LWRNTNSKQsR«bS ” J7TW‰Q7UDrO.0UD=
eKeK0TW`;rf ” RU=R{=
‰`DbK∞
LmR{`dR{IKJMLNJ¯LN`OPSfR{P`bKL7x~äi`ZR{J©R{I4UZRikmUZbKnL¬jG`Q7SKLWJ7Li`b
eKTW`]KO‰J ” L%‡/IKJ7TWJ›R{IfJ7TNJ+UZTNJ›JÂrÄeKO‰PQ7PR…UDb4UZOP\R{PQ;UDO¾LN`DO‰SfRN‰`bKLM†hJM|DJ7bŠPj¢†CjG`TUZO‰OŸeKT{UDQsR{PQ;UDO
eKSKTWe9`DLNJ7LM†ARNIKJ7LWJšLW`O‰SfRN‰`bfL©UDTNJaR{IfJ ” LNJ7O¤|JML%|DJ7TW\*_K6 P"Q7SKO¤R^R{`™Q;UDOPQ7SKO‰UZR{JZx›ó`™‡~JšIKU´|DJ
R{`"¥4bKJMLNLiRNIKJ%LNelJ7Q7‰UDO.]9`DSKbK_4UDT}\+Q7`bf_KPRN‰`bKLM†¸UDbK_™R{IKJMTNJUZTNJ ” UDb
\‡XU;\ÄL¬R{I4U=Ri‡~J…Q7UDb
_K`aR{If‰L7x
QSRUT)VWGXHY[Zfxv¹xP]
¡ \.˜­J¬Q;UDb› ” e9`LWJ¬»«‰TW‰QˆIKOPJMRŸ]9`DSKbK_4UDT}\"Q7`bK_fPR{P`bKLM†‡/If‰QˆI ” J;UDbfL
R{IKUZR«‡~JLNelJ7QMPj·\+R{IfJ¯|ZUDO‰SfJ…`Dj
UZR«J;UDQˆIá]l`SKbK_KUDTW\DxiHIKJMLNJ|ZUDOPSKJ7LiQ;UDb¦]lJ…QM`bKL}RˆUDb
R
N
`T/JM|DJ7b¦|ZUDT}\‡/PR{I xmœ‘jhRNIKJ7LWJUDTWJ©`TN_KPb4UDT}\jGSKbKQsR{‰`DbKL7†4R{IfJ7LNJ¯eKTN`D]KO‰J ” L¬UDTWJ¯I4UZbK_KO‰JM_
j3UDPTNOP\šJ;UZLN‰O¤\]
\R{IfJ t # QsRˆU;|J¯LNQ7TW‰ef"
R ^`_badcdegf]cghEi7j§ž4`TJÂrKU ” eKOPJD†KLWSKeKel`LNJ%‡XJ%I4UD_øWSKL}R
‡/TN¤RNRNJ7b¯RNIKJŸQ7`_KJXUZbK_¯‡XUDb
R{JM_…R{`/R{JMLWRh¤R;xC€‚JMR7o„LAeKSKO‰ORNIKJŸLWRˆUZbK_4UDTW_…eKTW`[T{U ”š” J7TMo„L‚R{TW‰Qˆn
UDbK_¦RNTW\PR¨`DSfR^`b§UeKTW`]KOPJ ” ‡/PR{I§UnbK`´‡/b§LN`DO‰SfRN‰`b?x%ó`LWRˆUZTWR¨‡/¤R{I¦R{IfJLN`OPSfR{P`b?x
ÅsÆ ˆÆKþɃÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DɃͧËMÐ19Ò32
k
ófU;\
UDbf_
x¨ziLNLNS ” J_fPÁ9SKLNP`b QM`ÄJlP"Q7‰JMb
R
x
2 /2−t †
t)
=
eRNrIKJ©
1|ZUZ≤
rJML/
≤UD2bK_™[JMRt‡/≥¤R{I™
0 UdO‰PRWR{OPJi
D=1
äi`´‡°NQ7` (r,
K
e
f
S
N
R
©
J
K
_
7
J
W
T
P

N
R
P

|
~
‡
`
N
T
›
n
N
R
4
I
Z
U
R
”
LN`dR{IfJ©RNJ7L}RiefTN`]KOPJ ”
‰L
Nt −
1
(rNr )r = − 3 + r 2 N,
r
1
Nt = − (rNr )r − 3 + r 2 N, 1 < r < 2, t > 0,
r
1/2−t
N (1, t) = e
,
N (2, t) = e2−t , t > 0,
2
N (r, 0) = er /2 .
äi`´‡ ” UDLNLNUD[J›RNIKJ›¥4OPJ:^`_adcdemf]c]hEi7j®LW`¦R{I4UZR…PR…L{UZRN‰LW¥KJ7LR{IfJ7LNJQM`bK_K¤R{P`bKL7x)“ÃcXIKJMQˆn
R{IfJOYUZRNJ7L}R«|JMTNLW‰`b¦Pb ” ;
\ VSK]KOP‰Q…_f‰TNJMQMRN`TW\Dxv–HIKJMb¦R£\e9JPb
R{J7TNUDQMRNP|DJ7OP\™PbáUDb¦`QMR{U´|DJ
LNJMLNLW‰`b­“G‡/IKPQˆIk~Hm˜ œXTWJ7QM`TN_KJM_SKLW‰bK[aRNIKJ©_KYUDT}\›jGJ7UZR{SKTWJ´–Â n _oempqsrut]v=wyx]_zcG{b|HoEr
cdegfG{ed}z~{cG{g_b€3}mq9oEi7j
n _oempqsrut]v=wog€1q‚x n }ƒdo3{ n a‚o n og€{]x„|dc n… }mqzj†{gxˆ‡-‰ŠcN‹Œo3zŽHqgm|N‰Šc`mtd‘mt1’bo
n _oempqsrut]v=w”“dq9o=og€1q•{–a{o3{ed}d{z—de9o3{ n a3x n ƒHo n ^og€1q@˜em™
n _oempqsrut]vašŽ=tH›œucma n fdqxŽž}{–a3xb|1e1_9q)‰ Ÿ¡‹¢t‹a£3Ÿg¥¤]œucma n fdqx‚Ž@cga n fdqx¦‰¢tsra¦œ
n _oempqsrŒ§Hv=w@a1qmgožog€`q‚{–a{bo3{ed}š_ n adfG{bo3{ n a
n _oempqsrŒ§Hv=™›Ž=qmm|N‰Šcga n fdqxi¢¨zcma n fdqxg‘mt`¥œ
n _oempqsr©dv=wy_9ed}H_ƒG}me9oq”o]€1qšx n }ƒHo3{ n a
n _oempqsr©dv•x]}ašŽš}Hx n fq)‰H¤u^`_adcdemf]c]hN¤]‹u™d›-‹dªu›‹«t]¬G¥œ
n _oempqsru­]v=w@|G} n ox n }ƒHo3{ n aye9adfžqgde_ox n }zƒHo3{ n a•e9o=odŽHt
n _oempqsru­]v•_]}mq]emcm|3} n oNœ€ n }f n aEœu“]cG{f
n _oempqsru®]v|3} n os‰Šcga n fdqxU‹¯xg}aN‰°t‹dr°¥¤l
n _oempqsrŒ±Hv|3} n os‰Šcga n fdqxU‹«qmm|N‰«cma n fdq1x)i¢¨zcma n fdqxH‘mt1’tG]
n _oempqsrŒ²Hv=w‚} n]nz³ x´a{g_9qsi]iHi«}mq9oN¤µxšx9q]q@og€1qˆj`emšqmc]c n c
n _oempqsrŒ²Hvšq3Ÿ„Ž¶a n c9jS‰lxg}aN‰¢t-‹dr°¥¤µ’9qmm|N‰Šcga n fdqx-i¢¨zcma n fdqxH‘mt1’tG¥‹µ{–a1^G
qGŸ´Ž=›siŠ›]›]›g§H›g©d·]·
n _oempqsru·]v=w@a n o … emfSiHi]i«}mq9oS¤¯x@oqxzo”og€`q¸9ƒ`ed}d{boH™ n ^ n ƒdc•x n }zƒHo3{ n a
n _oempqsru·]v=wyxH{–a3_mq‚x|1e9o3{e}qgc]c n c{gx¶¹)‰Šf]c`mt1¥‹º€1ed}p3{–ad“=og€1q‚xoq9|3xH{—dq
}mq9oN¤¯xžx9qHqsi]i
n _oempqsru·]v=wyxb€ n ƒ3}f»_bƒHo=og€1q=qmc]c n cžf n ˜ga … ™že …3n ƒHož©Si
_
o
m
e
p
s
q
u
r
]
·

v
š
a

Ž
©

›
u
œ
m
c
a
d
f
q

x
ž
Ž
}
–
{
3
a
b
x
1
|
1
e
9
_
)
q
‰
¡
Ÿ
¢
‹

t
‹
a
3
£
g
Ÿ
¥

]
¤
u
œ
m
c
a
d
f
q
‚
x
Ž@cga n fdqx¦‰¢tsra¦œ
n
n
n
n _oempqsrµŸ›Hv=™d›Ž=qmm|N‰«cma n fdq1x)i¢¨zcma n fdqxg‘gt1¥œ
n _oempqsrµŸ]Ÿzv•xg}a‚Žš}Hx n fdq)‰H¤Š^`_badcdegf]cghN¤]‹Š™d›‹dªu›-‹«tm¬G¥œ
n _oempqsrµŸtHv=w˜q¶˜ n aS¤uo …3n og€1qmc=˜3{og€”|3} n oGx)iHi
n _oempqsrµŸtHvšq]tŽ¶a n c9jE‰µxg}aN‰°t‹dr°¥¤l’qmm|N‰«cma n fdq`x)i¢¨zc]a n fdq`xg‘mt1’mt1¥‹µ{ad^G
q]t”Ž”±NiŠ®g§]²H·]q`’›]­
n _oempqsrµŸb§dvšqGŸ‘mq]t
ea3x¶Ž”§Siu·]·]tg±
“ n]n f
n _oempqsrµŸ–©v=w˜q}]}U‹8{bo=“3{bpqxe¶cqmfmƒ3_o{ n ay^de_zo n c n ^že …Gn ƒHoš©Si
q a n ƒd“m€i
n _oempqsrµŸ–©vž¸9ƒ{bo
¼Ÿ`S ” ‰[I
R…bK`DR{JRNI4UZRaPbR{IKJj3UZQ7J`Dj«UDb)SKbKnbK`´‡/bLN`OPSfR{P`b?†hI4UZOP|‰bK[LWR{JMeLWMJ
`T¯`DR{IKJMT ” `Ä_fP¥4Q7UZR{P`bKL^I4U;|J+UZbK`DR{IfJ7TUZeKeKO‰PQ;UZRN‰`bƒ …Pj~\`S)IKU´|DJ_K`Sf]fR{LaUZ]9`SÄR…R{IfJ
|ZUDO‰P_KPR£\%`DjlR{IfJLN`OPSfR{P`b?†_KSfJmR{`©LN` ” J‰bfLWRˆUZ]K‰OPPR£\%‰bRNIKJbS ” JMTNPQ;UDOKLWQˆIKJ ” JD†I4UZOPjlLWRNJ7e
ÅsÆ ˆÆKþɃÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DɃͧËMÐ19Ò32
½
LN7JMLXUDbf_›LNJMJi¤j9\D`S›[DJMR~LN ” ‰O‰UDTqTWJ7LWSKOPRNL7xhœ‘jƒbK`DR;†\`DS›bKJMJ7_šRN`¯TWJMRNIK‰bKn\D`SKTŸbS ” JMTNPQ;UDO
e4UDTNU ” JMR{JMTNL`TRNIKJ©LNQˆIKJ ” J©PRNLNJMOPj¢x
QSRUT)VWGXHY8Zfx—¹ÄxvŽ\Ÿä¬JsrR;†lO‰JsR;opL¬RNTW\R{`"LN`DOP|J¯UšefTN`]KOPJ ”
‡ ¤R{IU"OPJMj·R SÄr™QM`bK_K¤R{‰`Db
/
“yUÄx„n¸x„Ufx›¿}äiJMS ” UDbKb­QM`bK_K¤R{P`b –Âx™óÄSKeKel`LWJšR{IKUZR…‡/I4U=R…PL¯I4UZeKe9JMbK‰bf[‰ö b*R£‡X`_K ” J7bĐ
LNP`b4UDO/LWe4UDQ7J ‰LdR{I4U=R"‡~J¦bK`´‡ À UDTWJ Pb=øWJ7QsR{PbK[­UZb U ” `SKb
R
“3elJ7T›SfbKPRdR{ ” J;–`Dj¨R{IfJ
q
UDOP`bK[…RNIKJ©]9`DSKbK_4UDT}\`DjqUaQ7PTNQ7OPJ
xs¾¬J J ]9JMTmR{I4U=RR{IKJ SrjGSKbKQ
” UZR{JMTNYUZO
r0‰bK[UD” Q7TW`” LNL%R{IKJš]9`SfbK_4UDT}ö \xœ‘jR{IfJ
R{P`b*‰bR£‡~N`¦_K ” JMbKLNP`bKL©TNJMeKTNJMLNJMbRNL ” U=R{J7TWYUDrO =`´‡/
R{`ZRˆUDO4U ” `SfbR `F‡/PbK[%UDQ7TW`LNLVRNIKJ/Q7‰TWQ7OPJ‰L
†ö RNIKJ7bdR{IKUZRVR{IKJ/QM`TNTWJ7QMRŸP_KJ7b
R{¤R£\aUZR¾RNIKJ
q(t)
]l`SKbK_4UZTW\PL ö
SÄr
q(t)
,
2πr0
LN`*R{I4U=R
xä¬`´‡ ” UDLWL{UD[DJ+RNIKJ ¥KO‰J^`_adcdemf]^`}ƒdghEijLN`*R{I4UZRšPR
Nr = q(t)/ (2πr0 D)
L{U=R{‰L}¥4J7L^RNIKJ7LWJšQM`bK_K¤R{‰`DbKL¨jG`T^`SKT¨RNJ7L}R¯efTN`]KOPJ ”
xšóÄPbKQ7J‡~JšnbK`´‡
2
N (r, t) = er /2−t
R{IKUZR
f
†
X
‡
©
J
W
L
7
J
^
J
{
R
4
I
=
U
R
{
R
K
I
^
J
M
Q
`
N
T
W
T
7
J
M
Q
/
R
ˆ
Q
f
I
`
‰

M
Q
©
J
G
j
D
`
T
‰

L
1/2−t
1/2−t x
N
q(t)
q(t)
=
−2πe
r (1, t) = e
VO‰SK[^RNIK‰LVPbRN`^R{IKJ¥4OPJ/UDbK_"bK`´‡ÝR{JMLWRŸ`SKT¾TWJ7LNSfOPR{LV‡/¤R{I"UDb"‰b
R{JMT{UDQsR{P|DJ # QMRˆU;|DJiLWJ7LNLW‰`bƒ ö
n _oempqsrut]v=wyx]_zcG{b|HoEr
= −DNr =
cdegfG{ed}‡qƒgj`eaHai7j
n _oempqsrut]v=wog€1q‚x n }ƒdo3{ n a‚o n og€{]x„|dc n… }mqzj†{gxˆ‡-‰ŠcN‹Œo3zŽHqgm|N‰Šc`mtd‘mt1’bo
n _oempqsrut]v=w”“dq9o=og€1q•{–a{o3{ed}d{z—de9o3{ n a n ƒHo n ^o]€1q¶˜1em™
n _oempqsrut]vašŽ=tH›œucma n fdqxŽ}{–a3xb|1e1_9q)‰ Ÿ¡‹¢t‹a£3Ÿ]¥¤]œuc]a n fdq`xŽgcma n fdqxU‰ Ÿ¥ra¡¥œ
n _oempqsrŒ§Hv=w@a1qmgožog€`q‚{–a{bo3{ed}š_ n adfG{bo3{ n a
n _oempqsrŒ§Hv=™›Ž=qmm|N‰Šcga n fdqxi¢¨zcma n fdqxg‘mt`¥œ
n _oempqsr©dv=wy_9ed}H_ƒG}me9oq”o]€1qšx n }ƒHo3{ n a
n _oempqsr©dv•x]}ašŽš}Hx n fq)‰H¤u^`_adcdemf]^`}ƒdghN¤H‹u™d›‹dª«›‹«tm¬¥œ
n _oempqsru­]v=w@|G} n ox n }ƒHo3{ n aye9adfžqgde_ox n }zƒHo3{ n a•e9o=odŽHt
n _oempqsru­]v•_]}mq]emcm|3} n oNœ€ n }f n aEœu“]cG{f
n _oempqsru®]v|3} n os‰Šcga n fdqxU‹¯xg}aN‰°t‹dr°¥¤l
n _oempqsrŒ±Hv|3} n os‰Šcga n fdqxU‹«qmm|N‰«cma n fdq1x)i¢¨zcma n fdqxH‘mt1’tG]
n _oempqsrŒ²Hv=w@a n o‚exˆ“ n]n f‚ex´og€1q@~3{cG{g_€G}mq9ož|dc n… }mqzj‹ … ƒHo”o]€{gxž{gx´od™m|{g_9e}
n _oempqsrŒ²Hv=wy_b€1q_ ³ o]€1qOjGemG{ jHƒ]j•qgc]c n c
n _oempqsrŒ²Hvšq3Ÿ„Ž¶a n c9jS‰lxg}aN‰¢t-‹dr°¥¤µ’9qmm|N‰Šcga n fdqx-i¢¨zcma n fdqxH‘mt1’tG¥‹µ{–a1^G
qGŸ´Ž=›siŠ›]›]›]­m©H©H§]§
n _oempqsru·]v=w nz³ ‹¿}mq9oN¤µx´€`ed}zpqog€1q‚xoq9|3xH{—dq
n _oempqsru·]vašŽ©›œucma n fdqxŽ}{–a3xb|1e1_9q)‰ Ÿ¡‹¢t‹a£3Ÿ]¥¤]œuc]a n fdq`xŽgcma n fdqxU‰ Ÿ¥ra¡¥œ
n _oempqsrµŸ›Hv=™d›Ž=qmm|N‰«cma n fdq1x)i¢¨zcma n fdqxg‘gt1¥œ
n _oempqsrµŸ]Ÿzv•xg}a‚Žš}Hx n fdq)‰H¤Š^`_badcdegf]^1}ƒd]hN¤]‹u™d›-‹dªu›‹«t]¬G¥œ
n _oempqsrµŸtHv|G} n os‰«cma n fdq1x¦‹¯xg}aS‰¢t‹dr°¦¤µ
n _oempqsrµŸb§dvšq]tŽ¶a n c9jE‰µxg}aN‰°t‹dr°¥¤l’qmm|N‰«cma n fdq`x)i¢¨zc]a n fdq`xg‘mt1’mt1¥‹µ{ad^G
q]t”Ž=›siŠ›]›]›GŸb§d®]tg±
n _oempqsrµŸ–©všqGŸ‘mq]t
ea3x¶Ž”§Siu·]·m©d­
n _oempqsrµŸ­Hv=w n a_9q”eg“de1{–aS‹Àe …3n ƒHož©Si
n _oempqsrµŸ­Hvž¸9ƒ{bo
QSRUT)VWGXHY:Zfxv¹x—`
’ \‚ä¬`F‡®LWSKeKel`LNJmRNI4UZRq‡~JX‡XUDb
RhR{` ” `_KJ7OÄU«el`eKSKO‰UZR{P`b©R{I4UZRq`]lJM\L
R{IfJ%[D`F|DJ7TWbK‰bK["J7•S4U=R{‰`DbKL
ÅsÆ ˆÆKþɃÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DɃͧËMÐ19Ò32
Á
2
∂ N
1 ∂N
∂N
= D
+
+ f (r, t, N ), 0 < r < ∞, t > 0
∂t
∂r 2
r ∂r
N (r, 0) = M δ(r), 0 < r < ∞.
“3HIfJ7TNJ%‰L¬UDbf`DR{IKJMT/ ” eKO‰PQ7PR~Q7`DbK_KPRN‰`bƒ hR{IKJ%LN`OPSfR{P`b+LNIK`DSKO‰_™TNJ ” UDPb]9`DSKbK_KJM_?xv–"HIKJ
” SKLWnT{UZR+el`eKSKO‰UZR{P`bÝJÂrfU ” efO‰J¦`Dj¯kmUZbKnL§t uK†ie?x ’’mÂZw«LNJMTW|JML™UDL+UDbJsrfU ” eKOPJ¦`Dj¯U
_K¤ÁƒSKLW‰`beKIKJ7bf` ” JMbK`b¦‡/¤R{IáJsrÄel`bKJMbRNYUDOA[TN`´‡RNI®“
†ƒ‡/IKJ7TWJ
‰L^Uel`LN¤R{¤|J
f = aN )
a
[TW`F‡RNI TNUZR{JZx
»«JMOPR{U›Q7`bf_KPRN‰`bKL«UDTNJ¯TNJ7UDO‰O¤\+_K P"Q7SfOPR¬R{`QM`Ä_KJ…SKeábS ” JMTNPQ;UDOPOP\]lJ7Q7UDSKLNJRNIKJM\UDTWJ
UaeKTNJsRNR£\LWJM|DJ7TNJ©jG`T ” `DjC_KPLNQ7`DbRN‰bSK¤R£\xh˜J^Q;UDb+¥4bKJMLNLR{IKPL‰b
R{`dU ” `TNJ«bS ” JMTN‰Q7UDO‰O¤\
U ” J7b4UZ]KO‰JmefTN`]KOPJ ” UDLVjG`O‰OP`F‡/LM CUDLNna`SfTNLNJMOP|DJ7LV‡/I4UZRŸI4UDefe9JMbKLŸU©L ” UZO‰OÄRN ” JUZj·RNJ7ThR{ ” J
xm˜J¯JÂrÄe9JMQMR«RNIKJ…PbKLWR{UDb
RˆUDbfJ7`SKL¬eKSKOPLNJ©R{`šJM|`DOP|J¯‰b
R{`šUšL ” ``DRNI¦º©UDSKLWLNYUZbĐ£OP‰nDJ
t_KP=
0
LWR{TW‰]KSÄR{‰`Db?xÝz [``Ä_®Q7OPSKJ+‰LdR{IKJ™JsrÄeKOP‰Q7¤RdjG`T ” RN`*RNIKJJÂrfel`bKJMb
R{YUZO¬[DTN`´‡R{IT{UZRNJ
eKTW`]KO‰J ” “}t uK†4e?x¾’’
¹=wY–
M at−r2 /(4Dt)
e
.
N (r, t) =
óÄ`Š‡XJ§Q7UDbRNTNJ7UZR+R{IKPL‰bKL}RˆUDb
RˆUZbKJ7`SfL™4πDt
LW`SKTWQ7J*eKTN`]fO‰J ” UDL™U)eKTN`]fO‰J ” ‡/PR{IU
SÄr
ö U=R
Q7`DbK_KPRN‰`b™UZR¬R{IfJ¯OPJMj·R¬]9`DSKbK_4UDT}\†¸]KSfR«T{UZRNIKJ7TRNI4UDbIKU´|DJUDb ‰bKL}RˆUDb
R{UDbKJ7`DSKLiLW`SKTWQ7J
†¾‡~J™Q7UDb®LNeKTWJ;UD_®`SfRaRNIKJ+eKSKO‰LWJ+`F|DJ7T"UáLNIK`DTWRdR{ ” JPb®U§L ” ``DRNIj3UDLNIKP`bŠ]
\
t_KJs=
¥4bK0‰bf[
2
2
2M
q(t) = √ e−t /(2σ )
σ 2π
‡/IKJMTNJ
PL/U"L ” UZO‰O9|ZUDTWYUDbfQ7JDx¾ä¬`DR{J¨RNI4UZR
σ
Z ∞
q(t)dt = M.
0
ó `¦‡~J‡/‰OPOŸTWJ7eKO‰UDQ7JšR{IKJPbKLWR{UDb
RˆUDbKJM`SKLLW`SKTWQ7J]
\)U¦QM`b
R{‰bSK`DSKLLN`DSKTNQMJR{I4UZRJ ” PR{L
Ä
UD]l`SfRŸR{IKJ«L{U ” J¨U ” `SKb
R7†
†K`´|J7TmU¯TNJMOYUZRNP|DJ7OP\dLWIK`T}RXR{ ” J¬LNeKUDb]
\›SfLN‰bf[…R{IKJ«UD]l`F|DJ
jG`T ” SKOYU…jG`DT
‡/PRNI UaLWSKM
PR{UD]KO‰J^QˆIf`‰QMJ%`Zj x
q(t)
σ
QSR)Y3ÃÄLÅÇÆY;ZfxÈÂ`\CHAUDnZJ«U^Q7TWPRN‰Q;UZOfO‰``n…UZRVR{IKJ/JÂrKU ” eKOPJ`DjƒkmUDbKnLit uK†eƒxV’’mÂZwyxVó`OP|DJ
R{If‰L©eKTN`D]KO‰J ” bS ” JMTNPQ;UDOPOP\ SKLNPbK[™RNIKJdjGSKbKQMRN‰`bá¥4OPJajGQ7bKTNUD_ SÄr©
¾ x ” xɟ
¼ `DS ” ‰[I
R©SKLWJ
UDL/‰bR{IKJ%eKTWJM|‰`SfL¬JÂrfU ” efO‰J©UDbK_ UDb™‰b
R{JMTW|ZUDOƒjGTW` ” ö RN`šLW` ” JsR{IKPbK[dO‰PnDJ
†
M
=
1
1
150
UDOP`bK[‡/PRNI
R{` ”  ” ‰Qˆn™UDb*‰bKL}RˆUDb
RˆUZbKJ7`SfL%LW`SKTWQ7JDx"z«L%kmUZbKnL ” J7b
R{P`bKLM†?\`S
Q7`DSKO‰_§SKLWJ σ = 1 ýF\ÄT©UZbK_
xa€‚``n¦UZR^RNIKJd[T{UZeKIKL%UDbK_§LNJMJdPjŸR{IKJaTNUD_K‰SfL
a
=
2.65
D
=
12.2
`DjŸR{IKJdjGTN`DbR%TW`SK[IKO¤\¦QM`TNTWJ7LNel`bK_fL%RN`™RNIKJd_4UZRˆU+[¤|J7bƒx"˜I4U=R…UZTNJaR{IKJ"UDefeKTN`;rÄ ” UZR{J
el`eKSKO‰UZR{P`bKL/‰bRNIKJ…\DJ;UDTWL
UDbK_
‡/IKJMTNJ…‡~JQ7`DSKb
R
UDLi\J;UZ
T 7JMTN]
` Ê®»«`
2,
4,
6,
8
10
1905
\`DS ]lJ7OP‰JM|DJ^R{IKPL ” `_KJ7ËO Ê
QSR)Y3ÃÄLÅÇÆYZÄxv1
g \.óÄSfeKe9`DLNJmR{IKUZRq‡~J ” `_KJMO?“GQ7TN ” Pb4UDOY–ƒ_KTNSK[«L{UDOPJ7LVUDLqT{UD_K‰UZR{PbK[/jGTN` ”
U^LNPbK[O‰JXLWR{J7UD_f\aLW`SKTWQ7J/‡/IK`LNJ/OP`ÄQ7UZR{P`b…‡~J¬nbK`´‡ UDbK_"`]9Js\‰bK[©U^T{UZ_KYUDOÄ_KPÁ9SKLW‰`baO‰U´‡©x
äi`´‡
TNJMeKTNJMLNJMbRNLaR{IKJ_KJMbKLN¤R£\`Dj¬_KTWSK[á‰bfQ7‰_fJ7b
R{LMx­½¬J7TNJ+UZTNJPLNLNSfJ7LR{`¦RNIK‰bfn
NUD(r,
UD]l`SfR¬
bf_™t)‡~J^‡/‰O‰O9_K‰LWQ7SKLWL¬RNIKJ ” ‰b+LNJ ” Pb4UDT7 ¡xq½i`´‡®Q;UDb‡~JLNJsRVSKeRNIKJXjGSKbKQsR{P`b¯¥4OPJ7LqRN`¨I4UDbf_KO‰JXU¨Q7`b
R{PbSK`SKLhQM`bKL}RˆUDb
RVLW`SKTNQMJ
`DjlT{UZ_KYUDO_K¤ÁƒSKLW‰`bƒ†D[¤|J7b"U¨QM`bKLWR{UDb
R¾_KPÁ9SKLW‰`bQM`ÄlJ P"Q7‰JMb
RŸUDbK_R{IfJLN MJm`Dj¸R{IfJLN`SKTWQ7JZx
˜I4UZRmUDTNJ/R{IfJ¨UDeKefTN`eKTWYUZRNJ/]9`SfbK_4UDT}\šQM`bK_K¤R{P`bK¯L ʓy»«`%R{IKPLŸjG`DTXU…QM`b
R{PbSf`SKL ” `_KJ7O
UDL/‰b™ªqrfU ” eKO‰0
J Zfx—¹Äx„’LN`dR{I4U=Ri¤R‰LTWJ;UD_Ä\R{`"[`ajG`TmR{IfJ%TWJ7L}Ri`ZjARNIK‰LeKTW`]KOPJ ” xv–
ÅsÆ ˆÆKþɃÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DɃͧËMÐ19Ò32
̑Å
Ž x¾ä¬`´‡LNSKefe9`LWJ«R{IKUZR~‡~J¨UZTNJ¬[P|DJ7b›_fTNSK[¯Q7TN ” J¬_4UZR{U%`DbU¯[DTN‰_“ ” UDeš`DjƒRNIKJ«QMPR£\†
Ä
TNJ7UDO‰O¤\f–sxV½¬`F‡°Q;UDb+‡~J%RNT{UDbKLWOYUZRNJ«R{If‰L/‰b
R{`aPbfjG`T ” UZRN‰`bUD]9`DSfR
Ê
NLN(r,
t)R{`™J7L}R{ UZRNJdRNIKJ
’fxº©¤|JMbR{If‰L©‰bfjG`DT ” UZR{P`báUD]l`SfR
†qIK`´‡ìQ7UDb*‡~J›Sf
J"PR©
”
N (r, t)
_K¤ÁƒSKLW‰`bQM`ÄlJ P"Q7‰JMb
Ri`ZjqR{IfJ ” `_KJ7ËO Ê
Kxi“3½«UDTW_KJ7+T Ív–…˜I4UZR¬¤jC‡~J%_K`b?o R/nbK`´‡ R{IKJ©QMJ7b
R{JMT7†¸]KSfR¬]lJ7O‰PJM|DJ¨R{IKJMTNJ©‰L`bfbJ Ê
ÎόÐÑuÏuÒsÓ-Ô1Õ-Ö×EØ
Ì¢Ì