"!
#%$'&)(*&+ ",.-/$
0.13241"57698;:=<?>A@CBD8FEG<96
HIKJMLNJOPJ7QMRNSKTNJMLVUDTWJXOYUZTN[JMOP\^]4UZLNJ7_a`DbdcXI4UDefRNJ7Thgi`DjlkmUDbfnlopLqRNJsrRtvuwyxqzibK`DR{IfJ7TA|DJ7T}\
SKLWJMjGSKO~TNJMjGJMTNJMbKQ7JPL`[
UDboL
RNJsrR+tvuwyq/IKPQI)L
SfLNJ7_Pb)UZRNIDF
GR{IfJUDeKeKOPJM_
UZR{IKJ U=R{QMLmLNJMSKJ7bKQMJDx
JDopTNJ[D`bK[+R{` bKJ7JM_)U/If`OJ]KSKbKQI)`DjXR{J7T PbK`OP`[D\LW` J`Djm/IfQIopOOVL}R{Qn
IKJMTNJDxqPTNL}R/LN` J^bK`DR{UZR{P`b+UDbK_+j3UDQMRNLjGTN` Q;UDOPQ7SKOPSKL7 ¢¡F UDb+`SfR£XUDTN_e9`Db
R{PbK[SKbK¤RbK`T UZO9|JMQMRN`T/_KJM¥4bfJ7_¦U=RJ;UDQIel`b
R/`bR{IfJ
n
]l`SKbK_KUDTW\
`Dj~U `T _K J7bfLN`Db4UDOCLN`OP_
/IKQI§PL¨SKLNSKUDOO¤\¦UDLWLNS J7_
∂Ω
2
3
Ω
RN`]lJ©UDb`e9JMb+Q7`bKbfJ7QMRNJ7_¦LWJMR//PRNIUDb+`TNPJ7b
RUD]fOJ^]l`SKbK_4UZTW\x
y
R{IKJm[DT{UD_KPJ7b
RA`Dj4R{IKJ~LNQ7UDOYUDTCjGSKbKQsR{`Db
`DjfR{IKJmLWe4UZRNYUDO
|ZUDTNUD]KOPJ7L7xAªx[fxPPb
5f
f
RNIKJ¨Q;UZLNJ«`ZjR{IKTWJ7J«LWe4UDQMJ¨_K J7bfLN`DbKL7
~J¨`]fR{UDb"R{IKJ¬|J7QsR{`T~|DUZOSKJM_jGSKbKQsR{`Db
D
E
∂f ∂f
5f = ∂f
,
,
= hfx , fy , fz i
3
∂x
R{IKJ§∂y_K¤|∂z
J7TW[J7bfQ7J`ZjU)|JMQMR{`DTjGSKbfQMR{P`b
/IKJMTNJ*J;UDQIQ7``TN_KPb4UZRNJ
5
F
_KJMe9· JMF
bK_KL`b®R{IKJ LNe4U=R{YUZO/|ZUDTWYUD]fOJ7LMxª/xp[Kx¤Pj
mR{IKJMb
F = hF, Gi
5·F =
x
∂F
∂G
+
=F +G
∂x jGSfbKQM∂y
G¯z°
RN`bPL¬x±s²³´³=yµ·¶PbU
_K` UDPb¤jL~I4UDLmQM`b
R{bSK`DSKLme4UDT}R{UDO¸_KJ7TWP|ZUZR{¤|JML
Pb UZOO?¤R{Lm|ZUDTNUD]KOPJ7LmPb+RNIKJ^_K` UDPbSKbK_KJMTiQM`bKLNP_KJ7TNUZR{P`b?x
y¹º©UDSKLWLi»iP|JMTN[JMbKQ7J%HIKJM`TNJ ¼ibf_KJ7T/LNSfPRUZ]KOJ^L ``DR{IfbKJ7LWL¬QM`bK_K¤R{P`bKL
Z
Z
5 · Fdx =
F · ndx.
Ω
∂Ω
½¬J7TNJo SKLNPbK[/R{IKJ¾Q7`Db|DJ7b
R{P`b¯RNI4UZR?¿
TWJ7eKTWJ7LNJMb
R{LARNIKJUDeKeKTW`eKTWYUZRNJV_KPÁ9J7T¢
dx
À
JMbRNYUDO.JMOJ J7b
R7x/¸`DTiJÂrKU eKOPJD¸Pj
Li=_K J7bKLW`bKUDOÃ
TNJ7efTNJ7LWJ7b
R{L«_KPÁ9J7T¢
JMbRNYUDOÄ|`OPS J/UZbK_dbdQ;UDOPQ7SKOPSKLhbK`ZF
RUZRN`b~J¬~`SKOP_dI4U;|dx
J/SKLNJM_RNIKTNJMJ¬Pb
R{J7[DT{UDO
LW[bKLMx
£bKQM_KJMbR{UDOO¤\fº%»iHPLmT{U=R{IKJMT `TNJi`TN_KPb4UDT}\"RNI4UDb\D`S P[I
R~RNIKbKn¸xcX`bfLN_fJ7TX/I4U=R
PR J;UZbKLb+`bKJ©_K JMbKLNP`b?x
ÅÂÆ ÇÂÆfÈ4ÉÊËsÌÍZÎ
ÏÄÐKÑYÉʧÏ9Ê9ÒÓfÏÔDÏlÊÈKÌÔZÏZÕ"Ë
Ö
01×?1د<96CÙ7Ú:´ÛKÜf8´E3<l6ÝÜ46C>ÞÜ4ßGÜ46CBÚáàÜ
âÙ
UD[PbKJ¨R{I4U=R¨UdeKI
\LNQ7UDO.S4UDb
R{¤R£\+L/_KL}R{TW]KSfRNJ7_`´|J7TiULWe4UZRNYUDO?TWJ7[P`b
xXHIKPL
TNJM[`DbáQ7`SKOP_§]9J¡M£_K J7bfLN`Db4UDOÃUDbK_*U[J7bfJ7TNPQae9`Db
R^L«TNJMjGJMTNJMbKQ7JM_]
\¦R{IfRJaOJMRWR{JMT
xãHISKL7
Q7`SKOP_TNJsjGJ7TNJMbKQ7J*UOP`ÄQ7UZR{P`b
`bR{IfJ¦TNJ7UDO¨OPbKJD/Uel`PbR
PbU
x
(x, y)
R£~`Z£_K JMbKLNxP`b4UDO9TNJM[`b+`T
Pb URNIKxTNJMJs£_f J7bKLW`b4UZOTNJM[`Db?x
(x,|ZUDy,TNz)
ä¬JsrR;LWSKeKel`LNJRNI4UZR"R{IKJ+
UD]KOPJ
PL"RN`TNJMeKTNJMLNJMbRUeKI
\LNPQ;UDO¬S4UDb
R{¤R£\OPnZJ
N_KPLWR{TW]KSÄR{J7_`´|JMTiLWe4UDQ7J
UDbK_
el`eKSKOUZR{P`b? UDLWL7¸J7bKJMTN[D\D¸JMR{QZxP4RNI4UZR¬L/
U;\|ZUDT}\/PRNI
R{ JZxåHIfJ7bPRPLbKJMQ7J7LWL{UDT}\R{` J;UDLWSKTNJ¦RNIKJÝæ³=çæèsç4µyéWêDµ3ëó=çìG`T§íèsçf±së·µ3îf"`Dj©R{IKPL
LNSf]KLWR{UDbKQ7JZx+4`T^JsrfU eKOJZ.b§R{I4UZR%Q7UDLNJ"`DjU+LNPbK[OPJaLNe4UDQMJ_K JMbKLNP`bGRNIKJdLN eKOPJ7LWR
Q;UZLNJ´i~J~`SKO_áIKU´|DJR{I4U=R
UDbK_á¤jVR{IKJSfbKPRNL¨`Dj¾R{IKJ
eKI
\ÄLWQ7UDOhS4UDb
R{¤R£\
N
=
N
(x,
t)
UDTWJ
fRNIKJ7bR{IKJ©SKbfPR{L`Zj
UDTWJ%[DP|JMb+]\
Q
N
Q
[N ] = ,
L
/IKJMTNJ
PLOJMbK[DRNI?xk~Hmï
UZbK_
~`SKOP_®]lJQ7UDOOPJ7_ðÂñçlí
êD²èsç4µêDòñç4ë·µG±sxóÄ` J
L
Q
L
`DRNIKJ7TjGSKbK_KU JMb
RUDO^SKbfPR{L+UZTNJ UZLNL
UDbf_RN J
xôJá/OPO^UDLNLWS J§RNI4UZR+R{IfJ
M`Tq_KJ7L}R{TN`´ \DJ7_"TSKbKOJMLNLq~J/UDTWJm[P|DJ7 bdUDb
JsrÄeKOPQ7¤R
PLhbKJ7¤R{IKJMThQ7TWJ;UZRNJ7_a
eKI
\LNPQ;UDOKS4UZbRNPR£\
N
J7QI4UDbfL jG`T_K`PbK[aLN`K4OPnZJ©UdLW`SKTWQ7J%`T_KJ7Q7U;\jGSKbKQMRN`bx
ä¬`FõLWSKeKel`LNJ R{I4UZR
LULWSK]KTNJM[`DbÝ`Dj
/PRNIÝ]l`SKbK_4UZTW\
x°£bÝ~`TW_KL7/U
Ω
R
∂Ω
Q7`DbKLNJMTW|ZUZR{P`b+OYU;RUDnZJ7L/R{IKJ^jG`T H J^TNUZR{J©`DjAQI4UDbK[DJ%`ZjARN`DRUDO
bKLW_KJ
J7S4UDOPL
Q
Ω
z `SfbR¬`Dj
`´/PbK[d`SfR/`Dj
UDQMTN`LWL
e9JMT/SKbKPRRN J©÷
z `SfbRX`Dj Q]9JMö bK[¯LN`SKTWQ7JM_`DTXΩLNPbKnDJM_*3QM∂Ω
TNJ;U=R{J7_`DTX_KJMLWR{TW`´\J7_+PLX`bKJ«b
R{JMTNeKTWJMRUF
R{P`b¸~Pb
3`ZjqQMQ
`SKTNLWJD¸LN`DSKTNQMJ7_L/e9`DLNPRNP|DJ^UDbK_LNPbKnDJM_LbfJ7[
UZRNP|DJDx
zQsR{S4Ω
UDOPOP\DAR{IfJb4U JPL
ULNOP[I
R PLNbK` J7TMxHATNUD_K¤R{`Db4UDOO¤\?RNIKJOU´ô^IKU´|DJ%øWSKL}R
_KJMLNQ7TW]lJ7_L
UùêDò¤êDçæèò¤êDúVh/IKOPJUæ³=çf±´èséÂû=êDµyëó=çò¤êDúôL
U¦]4UDOUDbKQ7JOYU;ôb/IfQI
R{IfJ7TNJ©PLbK`LW`SKTWQ7J^R{JMT x
J¯bfJ7J7_`bKJ `DTNJ^R{JMT 4b4U J7OP\U¿ SÄr R{JMT
R{I4UZR¬TWJ7eKTWJ7LNJMb
R{L«RNIKJ
U `SfbR
`Dj
RNI4UZR `´/LUDQ7TW`LNLU^SKbK¤RVLNeKUZR{UDOKJ7OPJ ö J7b
R¾À `Dj
elΦJ7TVSKbfPRVR{ JZxhóSKQIU¨jGSKbKQsR{P`b
Φ]lJa|J7T}ö \§Q7` eKOPQ;U=R{J7_§UZbK__fJ7elJ7bK_bK`DR©`bKO¤\á∂Ω
Q;UZb
`b§LNeKUZR{UDOhQ7``TW_Kb4U=R{J7L©UZbK_*RN JD
]KSfRJM|JMb
UDbK_PRNL_KJMTNP|ZUZRNP|DJ7L7xåäi`ZR{QMJ RNI4UZR
LU|DJ7QsR{`TS4UDb
RNPR£\i]9JMQ;UDSfLNJ
N
Φ
`´/L¨I4U;|DJaU_KPTNJ7QsR{P`bUZLi~J7OOAUDL«U UZ[bKPRNSK_KJZxiAb4UDOPOP\DfXJ¯mUDb
R¨RN`UDOPO`´°jG`DTiR{IfJ
öel`LNLW]KPOPR£\ R{I4U=R¯RNIKJ
UZR{JMTNUDOVL©]lJ7bf[LN`DSKTNQMJ7_)`T%LNPbKnDJM_)UDQ7QM`TN_fbK[R{`¦U¿¢LW`SKTNQMJ À
3TNUZR{J;ijGSKbKQsR{`Db
/IK`LWJSfbKPRNL©UDTNJ
xaHIKL^jGSKbKQsR{P`b§Q7`SKOP_§_KJ7elJ7bK_`báR{ JZ
el`LN¤R{`Db¯UZbK_JM|DJ7fb
xAJDopOOLNSfeKeKTNJMLN[N
Lh¤R{]LA/T
UDTN[DS JMb
RqOPLWR7xCäi`´)XJZoTWJX[`DbK[/R{`iJsrÄeKTNJMLNL
PR/PbLW\ ]l`OLMxhJZN
oTWJ¯TWJ;UD_Ä\jG`TmR{IfJ%Pb
R{J7[DT{UDOjG`DT SKOUZR{P`b?
/IKPQI L}RUDT}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`ZQ;UDOPOJ7_®b
RNJ7[TNUDOmjG`T SKOYUZRN`b`Dj¨Q7`bKLWJ7T}|DUF
R{P`b¸ýZ]KUDOYUDbfQ7J
Z
Z
d
N (x, t)dx =
(− 5 ·Φ + f ) dx.
dt Ω
Ω
äi`´ãXJ UDnDJdUQMTNSKQMYUDOqUDLNLWS eÄR{`Db§UD]9`DSfR
bKU JMOP\R{I4U=R^PR^PL^L `Ä`ZR{I¦Pb§UDOOAPj
PRNL¬|ZUDTWYUD]fOJ7L¬UDbf_ ~JaUDLNLWS J¯R{IKUZR
LiU]l`SKNbK_KJM_¦TNJM[`Db?xHIKJ7b¦~J U;\RUDnZJ
R{IfJ
_KJMTNP|ZUZRNP|DJ^LN[DbPbKLNP_KJ©UDbK_`]fR{UDb Ω
Z Ω
∂N (x, t)
− (− 5 ·Φ + f ) dx.
∂t
ÅÂÆ ÖÂÆ4ËÂÑ·þmÿÔDÌ+ÈKÏËMÌÄË
ÑPÊËMÐDÏ9Ê?ÐDÏ9Ê?ÌÄÉËiË7É9ÍlÈKÌfË
k SfRXR{IKPLXL~R{TNSfJ«jG`TmUZOOlLNSK]KTWJ7[P`bKL
X
`Dj xVziLNLWS PbK[
RNIKJ¨b
RNJ7[TNUDbK_L~Q7`b
RNbSK`SKLM
Ω
R{IfJ%`DbKOP\mU;\jG`TmRNIKLmRN`"]lJ^R{TWSKJ©LmR{IKUZR R
Nt (x, t) + 5 · Φ = f,
/IKPQI PLmR{IKJ©LN`=£Q;UZOOJM_*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¬qQnZJDopL«OYU;©4R{IKUZRiUDLNLWJ7TWRNLiRNI4UZR¬R{IKJ%S4UDb
R{¤R£\R{J7bf_KLiRN` `´
jGTN` If[IKJMT¯QM`bKQMJ7b
R{TNUZR{P`bR{`O`´XJMTUZRUT{U=R{JefTN`el`TWRN`bKUDOhR{`RNIKJQM`bKQMJ7b
R{TNUZöR{P`b
[TNUD_KJMb
R;xV£bLW\ ]l`OL
Φ = −D 5 N.
£bR{IKJ `L}R[J7bfJ7T{UZO?Q;UDLWJ
L/U
R{JMbKLN`DTSKUDb
R{PR£\DKR{I4U=RLMKb`SfTmLNJsRNR{PbK[K4U UZRNTN¤r9x
`TWJ7`´|JMT7lb UDOOQ;UDLWJ7L7 DL¬UDLNLWS J7_ RN`]lJ¯U"L}\ " JMR{TWQ^el`LN¤R{¤|J©_KJM¥KbKPRNJ UZRNTN¤r9
LN`RNI4UZR^]
\LN` J`T}R{IK`[DD
`b4UDOCQI4UDbK[DJ`DjQ7``TN_fb4UZRNJ7L¬XJaQ7UDbá|ÄPJM
UDL^U_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
q
j
N
R
K
I
M
J
\
Z
U
N
T
%
J
D
U
O
O
7
J
4
S
D
U
y
O
K
X
¯
J
;
Q
D
U
+
b
RUDnZJ
R{`"]lJ
D
Uae9`DLNPRNP|DJ¨LNQ7UDOYUDTMÄ/IKPQI+Q7`SKOP_+]9J©U
jGSKbfQMR{P`b+`DjCe9`DLNPRN`bUDbK_+`DR{IfJ7Tm|ZUDTNUD]KOPJ7L7JM|DJ7b
PRNLNJMOPj¢x
N £b+R{IKPLQ;UDLWJ%RNIKJ©_KPÁ9J7TWJ7b
R{UDOjG`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¤ÁSfLN`Db¯JMSKUZR{P`b?xA.JsR;opLqLWe9JMQ7YUZOMJhR{`¬R{IKJQ;UDLWJ
/IKJMTNJ
L©UQM`bKLWR{UDb
R;CU|JMTW\áQM` " `b§UZbK_áSKLNJsjGSKOVQ;UZLNJDx
HIKJd_KPÁ9SKLNP`b¦JMSKUZR{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`DRNIKJ7TQ7``TN_KPb4UZRNJLW\L}R{J L+UDTWJ
eKTWJMjGJ7TNUD4
]KOPJ«R{`dQ;UZTWR{JMLNUDb?xV£b+QM\OPbK_KTWQ;UZOUDbf_LNeKIKJMTNPQ;UDO?QM`Ä`DTN_KPb4UZR{JML7fjG`DTmJsrfU eKOJZÄ~J
I4U;|J©R{IfJ©jG`DT SKOUDL
∂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
£bRNIKJ«Q7UDLNJi`Dj?LWeKIKJMTNQ7UDOlQ7``TN_KPb4UZRNJ7LV/PRNIjGSKbKQsR{P`bKL¾R{I4UZRXUDTWJiPbK_KJ7elJ7bf_KJ7b
Rm`Zj.UDbK[OPJD
R{IfJ¯.UDefOYUDQMYUDbLW eKO¤¥4J7LR{`
∂2N
2 ∂N
4N =
+
2
∂r
r ∂r
UDbK_bQs\Obf_KTNPQ;UDO?QM`Ä`DTN_KPb4UZR{JMLXR{`
4N =
∂2N
1 ∂N
.
+
2
∂r
r ∂r
01
?1?EqßGڧدÜ4ÙMÚÄÙ%5M6CÙM8´Ü4698;Ü46CÚ<l@CÙ<9@.:=BÚÙ
ñé{è J^L}RUDT}R//PR{IRNIKLeKSfTNJ©
∂N
∂2N
= D 2 , −∞ < x < ∞, t > 0
∂t
∂x
N (x, 0) = f (x), −∞ < x < ∞.
ÅÂÆ ÖÂÆ4ËÂÑ·þmÿÔDÌ+ÈKÏËMÌÄË
ÑPÊËMÐDÏ9Ê?ÐDÏ9Ê?ÌÄÉËiË7É9ÍlÈKÌfË
½ JMTNJ
i
L%UDLNLWS J7_§R{`]9Ja~J7OOV]lJ7IKU´|DJ7_)UZbK_§_KJ7Q7U´\PbK[TNUDeK_fOP\LW`RNI4UZR%4`SKTWJMT
f
(x)
R{TNUDbKL}jG`T LQ;UDb]lJ¯UZeKeKOPJ7_?xh»«`LW`"UDbK_`]fRUZb
b
∂N
b,
= −Dξ 2 N
∂t
LN`dR{IKUZR/`bKJ©`]fR{UDbfL
−x
2 /(4Dt)
\
e
b = fbe
N
= fb √
4πDt
UDbK_+R{ISKLRNIKJ©Q7`b
|`DOSfRN`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{æè± 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)
/IKPQI [¤|JML¬LW`OSÄR{`Db
N (x, t) = √
HCT{UDbKLWe9`DTWR¬HAJMT L
£b
R{TW`Ä_fSKQ7J¯UR{T{UZbKLNel`TWR/RNJ7T
M
x2
e− 4Dt .
4πDt
b
R{`RNIKJ©eKTN`]fOJ KLW`dRNI4UZR/R{IfJ SÄrR{JMT
ö
PLbK`´
Φ = −D 5 N + v0 ,
/IKJMTNJ
L/UQ7`DbKLWR{UDb
RiLWe9JMJ7_ UZRm/IKPQIR{IKJ UZR{JMTNUDO9PLm]lJ7bf[R{T{UZbKLNel`TWRNJ7_+LNeKUZR{UDOO¤\x
3£báIf[vIK0JMT¨_K JMbKLNP`bKLiRNIKL«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°QIKUDbK[J¨`Zj|DUZTNYUZ]KOJML
OPJ;UD_KLR{`
RNIKJ¨JMSKUZR{P`b
UDbK_IfJ7bKQMJ^]\
z
=
x
−
v
t
N
=
DN
0
t
zz
/I4UZR/~J©I4U;|J¯UDOTWJ;UD_f\_K`bKJZ¸U"LW`OSÄR{`Db
Z
2
1
e−(x−v0 t−y) /(4Dt) f (y)dy.
4πDt
−∞
# jqQM`SKTWLNJD4RNIKL[¤|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=çèsç4µyë3êDò-,mé{³=úVµ·¶ £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`FRNIT{U=R{J3`T%U _fJ7Q;U;\T{UZRNJDx HA` [JMR
UjGJMJ7O¾jG`T^R{IKJ[TN`´R{I*R{JMT
UD[PbKaJRNI4UZRdR{IKJMTNJLdbK`*_KLWe9JMTNLW`b?]KSfRLN efOP\JÂrÄe9`bfJ7b
R{UDO[TN`´RNI UZRUá¥frÄJ7_
el`b
R;xhIKUZRXJ©R{IfJ7bI4U;|J%L/UDb # »¨ª
x~HIKPL/LSKLWJMjGSKOPbel`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_¦_fLNelJ7TWLNP`bPb
N (x, t) = √
ÅsÆ ÆKþÉÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DÉͧËMÐ19Ò32
4
U LNPbK[OPJ `Ä_KJMOÃxARQ;UDb+]lJ¨LNIf`F/bGLNJ7J"tvuwYR{IKUZR/U
LW`OSfRN`bR{`R{IfLmeKTW`]KOPJ
bfLWRUZbR{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
0165.1'E78Aß3ÚáØ¯Ü4Ù7ÚÙ9aد<96l8FEy6@C<l@CÙ:<l@C:=BÚÙ
HIKJMLNJaLW`SKTNQMJ7L UDnZJ¯R{IfJ7TiUDeKelJ;UZT{UDbKQMJ7Lib]l`SKbf_4UDTW\QM`bK_K¤R{`DbKLi`DTiQM`b
R{bSK`DSKL
b
RNJ7TNP`TmLW`SKTNQMJ7LMxJDopOO?L}R{PQnR{`aR{IfJ©jG`DT J7T7x
$/èæsµ3ë·ò ë·çèNêD;
é %©ë /ñ
±sëó=ç°úVë·µ·¶=
ê <~³=çf±sµÃêDç4µ?>¨³=ñçlí
êDésî@<~³=çlíDë·µ3ëó=ç ½iJ7TWJ*U®LW eKOJ
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`DOSfRN`bRN`"RNIKLeKTW`]KOPJ
∂2N
, 0 < x < ∞, t > 0
∂x2
= f (x), 0 < x < ∞
= N∗ .
PL`]fR{UDbKJM_+]\.UDefOYUDQMJ«R{TNUDbKLWjG`DT LiUZbK_+L
= D
N = N∗ erfc √
/IKJMTNJ
x
4Dt
2
erfc z = 1 − erf z = 1 − √
π
01A?1CB
Z
z
2
e−ξ dξ
0
<>CÚßED%:F<GFqßGÚHÙJI<¸:;98F@A>LK
J¬/POOjG`ÄQMSKL¾`bTNUD_KUDOK_K¤ÁSKLW`bxCJDopTNJ¬bK`DR¾[`DbK[^R{`©QM`bKLNP_KJ7TU«¿¢el`b
RVLW`SKTWQ7J
UDbK_)~J U;\XUDb
RdR{`§`bKO¤\QM`bKLNP_KJ7T"U¦¥4bK¤R{J_K` UDPb?x4SKT}R{IKJMT `TNJZVXJXUDb
R"RN` À
UDOPO`´jG`TLN` J^Q7` eKOPJsrTNJ;UZQMR{P`bR{JMT L7x~óÄ`dIKJ7TWJ©LmR{IfJ%[DJ7bKJMT{UDOL}RUZRNJ 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`bKLUZR
UDbf_
xï½i`´XJs|JMT7
r
=
r
r
=
R
0
eKTW`]KOJ L^/¤R{I
`T
eKTNJMLNJ7b
RaLWe9JMQ7UDO¾eKTN`]fOJ L©PjXJb
R{JMbK_RN` QM`bÄ
LWRNTNSKQsR«bS J7TWQ7UDrO.0UD=
eKeK0TW`;rf RU=R{=
`DbK∞
LmR{`dR{IKJMLNJ¯LN`OPSfR{P`bKL7x~äi`ZR{J©R{I4UZRikmUZbKnL¬jG`Q7SKLWJ7Li`b
eKTW`]KOJ L%/IKJ7TWJR{IfJ7TNJ+UZTNJJÂrÄeKOPQ7PR
UDb4UZOP\R{PQ;UDO¾LN`DOSfRN`bKLMhJM|DJ7bPj¢CjG`TUZOOeKT{UDQsR{PQ;UDO
eKSKTWe9`DLNJ7LMARNIKJ7LWJLW`OSfRN`bfL©UDTNJaR{IfJ LNJ7O¤|JML%|DJ7TW\*_K6 P"Q7SKO¤R^R{`Q;UDOPQ7SKOUZR{JZxó`~JIKU´|DJ
R{`"¥4bKJMLNLiRNIKJ%LNelJ7Q7UDO.]9`DSKbK_4UDT}\+Q7`bf_KPRN`bKLM¸UDbK_R{IKJMTNJUZTNJ UDb
\XU;\ÄL¬R{I4U=Ri~J
Q7UDb
_K`aR{IfL7x
QSRUT)VWGXHY[Zfxv¹xP]
¡ \.J¬Q;UDb e9`LWJ¬»«TWQIKOPJMR]9`DSKbK_4UDT}\"Q7`bK_fPR{P`bKLM/IfQI J;UDbfL
R{IKUZR«~JLNelJ7QMPj·\+R{IfJ¯|ZUDOSfJ
`Dj
UZR«J;UDQIá]l`SKbK_KUDTW\DxiHIKJMLNJ|ZUDOPSKJ7LiQ;UDb¦]lJ
QM`bKL}RUDb
R
N
`T/JM|DJ7b¦|ZUDT}\/PR{I xmjhRNIKJ7LWJUDTWJ©`TN_KPb4UDT}\jGSKbKQsR{`DbKL74R{IfJ7LNJ¯eKTN`D]KOJ L¬UDTWJ¯I4UZbK_KOJM_
j3UDPTNOP\J;UZLNO¤\]
\R{IfJ t # QsRU;|J¯LNQ7TWef"
R ^`_badcdegf]cghEi7j§4`TJÂrKU eKOPJDKLWSKeKel`LNJ%XJ%I4UD_øWSKL}R
/TN¤RNRNJ7b¯RNIKJQ7`_KJXUZbK_¯XUDb
R{JM_
R{`/R{JMLWRh¤R;xCJMR7oLAeKSKOORNIKJLWRUZbK_4UDTW_
eKTW`[T{U J7TMoLR{TWQn
UDbK_¦RNTW\PR¨`DSfR^`b§UeKTW`]KOPJ /PR{I§UnbK`´/b§LN`DOSfRN`b?x%ó`LWRUZTWR¨/¤R{I¦R{IfJLN`OPSfR{P`b?x
ÅsÆ ÆKþÉÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DÉͧËMÐ19Ò32
k
ófU;\
UDbf_
x¨ziLNLNS J_fPÁ9SKLNP`b QM`ÄJlP"Q7JMb
R
x
2 /2−t
t)
=
eRNrIKJ©
1|ZUZ≤
rJML/
≤UD2bK_[JMRt/≥¤R{I
0 UdOPRWR{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[JRNIKJ¥4OPJ:^`_adcdemf]c]hEi7j®LW`¦R{I4UZR
PR
L{UZRNLW¥KJ7LR{IfJ7LNJQM`bK_K¤R{P`bKL7x)ÃcXIKJMQn
R{IfJOYUZRNJ7L}R«|JMTNLW`b¦Pb ;
\ VSK]KOPQ
_fTNJMQMRN`TW\DxvHIKJMb¦R£\e9JPb
R{J7TNUDQMRNP|DJ7OP\PbáUDb¦`QMR{U´|DJ
LNJMLNLW`bG/IKPQIk~Hm XTWJ7QM`TN_KJM_SKLWbK[aRNIKJ©_KYUDT}\jGJ7UZR{SKTWJ´Â n _oempqsrut]v=wyx]_zcG{b|HoEr
cdegfG{ed}z~{cG{g_b3}mq9oEi7j
n _oempqsrut]v=wog1qx n }do3{ n ao n og{]x|dc n
}mqzj{gx-cNo3zHqgm|Nc`mtdmt1bo
n _oempqsrut]v=wdq9o=og1q{a{o3{ed}d{zde9o3{ n a3x n Ho n ^og1q@em
n _oempqsrut]va=tHucma n fdqx}{a3xb|1e1_9q) ¡¢ta£3g¥¤]ucma n fdqx@cga n fdqx¦¢tsra¦
n _oempqsr§Hv=w@a1qmgoog`q{a{bo3{ed}_ n adfG{bo3{ n a
n _oempqsr§Hv==qmm|Ncga n fdqxi¢¨zcma n fdqxgmt`¥
n _oempqsr©dv=wy_9ed}H_G}me9oqo]1qx n }Ho3{ n a
n _oempqsr©dvx]}a}Hx n fq)H¤u^`_adcdemf]c]hN¤]ud-dªu«t]¬G¥
n _oempqsru]v=w@|G} n ox n }Ho3{ n aye9adfqgde_ox n }zHo3{ n ae9o=odHt
n _oempqsru]v_]}mq]emcm|3} n oN n }f n aEu]cG{f
n _oempqsru®]v|3} n oscga n fdqxU¯xg}aN°tdr°¥¤l
n _oempqsr±Hv|3} n oscga n fdqxU«qmm|N«cma n fdq1x)i¢¨zcma n fdqxHmt1tG]
n _oempqsr²Hv=w} n]nz³ x´a{g_9qsi]iHi«}mq9oN¤µxx9q]q@og1qj`emqmc]c n c
n _oempqsr²Hvq3¶a n c9jSlxg}aN¢t-dr°¥¤µ9qmm|Ncga n fdqx-i¢¨zcma n fdqxHmt1tG¥µ{a1^G
qG´=si]]g§Hg©d·]·
n _oempqsru·]v=w@a n o
emfSiHi]i«}mq9oS¤¯x@oqxzoog`q¸9`ed}d{boH n ^ n dcx n }zHo3{ n a
n _oempqsru·]v=wyxH{a3_mqx|1e9o3{e}qgc]c n c{gx¶¹)f]c`mt1¥º1ed}p3{ad=og1qxoq9|3xH{dq
}mq9oN¤¯xx9qHqsi]i
n _oempqsru·]v=wyxb n 3}f»_bHo=og1q=qmc]c n cf n ga
e
3n Ho©Si
_
o
m
e
p
s
q
u
r
]
·
v
a
©
u
m
c
a
d
f
q
x
}
{
3
a
b
x
1
|
1
e
9
_
)
q
¡
¢
t
a
3
£
g
¥
]
¤
u
m
c
a
d
f
q
x
@cga n fdqx¦¢tsra¦
n
n
n
n _oempqsrµHv=d=qmm|N«cma n fdq1x)i¢¨zcma n fdqxggt1¥
n _oempqsrµ]zvxg}a}Hx n fdq)H¤^`_badcdegf]cghN¤]ddªu-«tm¬G¥
n _oempqsrµtHv=wq¶ n aS¤uo
3n og1qmc=3{og|3} n oGx)iHi
n _oempqsrµtHvq]t¶a n c9jEµxg}aN°tdr°¥¤lqmm|N«cma n fdq`x)i¢¨zc]a n fdq`xgmt1mt1¥µ{ad^G
q]t±Ni®g§]²H·]q`]
n _oempqsrµb§dvqGmq]t
ea3x¶§Siu·]·]tg±
n]n f
n _oempqsrµ©v=wq}]}U8{bo=3{bpqxe¶cqmfm3_o{ n ay^de_zo n c n ^e
Gn Ho©Si
q a n dmi
n _oempqsrµ©v¸9{bo
¼`S [I
R
bK`DR{JRNI4UZRaPbR{IKJj3UZQ7J`Dj«UDb)SKbKnbK`´/bLN`OPSfR{P`b?hI4UZOP|bK[LWR{JMeLWMJ
`T¯`DR{IKJMT `Ä_fP¥4Q7UZR{P`bKL^I4U;|J+UZbK`DR{IfJ7TUZeKeKOPQ;UZRN`b
Pj~\`S)IKU´|DJ_K`Sf]fR{LaUZ]9`SÄR
R{IfJ
|ZUDOP_KPR£\%`DjlR{IfJLN`OPSfR{P`b?_KSfJmR{`©LN` JbfLWRUZ]KOPPR£\%bRNIKJbS JMTNPQ;UDOKLWQIKJ JDI4UZOPjlLWRNJ7e
ÅsÆ ÆKþÉÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DÉͧËMÐ19Ò32
½
LN7JMLXUDbf_LNJMJi¤j9\D`S[DJMR~LN OUDTqTWJ7LWSKOPRNL7xhjbK`DR;\`DSbKJMJ7_RN`¯TWJMRNIKbKn\D`SKTbS JMTNPQ;UDO
e4UDTNU JMR{JMTNL`TRNIKJ©LNQIKJ J©PRNLNJMOPj¢x
QSRUT)VWGXHY8Zfx¹Äxv\ä¬JsrR;lOJsR;opL¬RNTW\R{`"LN`DOP|J¯UefTN`]KOPJ
¤R{IU"OPJMj·R SÄrQM`bK_K¤R{`Db
/
yUÄxn¸xUfx¿}äiJMS UDbKbQM`bK_K¤R{P`b ÂxóÄSKeKel`LWJR{IKUZR
/I4U=R
PL¯I4UZeKe9JMbKbf[ö b*R£X`_K J7bÄ
LNP`b4UDO/LWe4UDQ7J LdR{I4U=R"~J¦bK`´ À UDTWJ Pb=øWJ7QsR{PbK[UZb U `SKb
R
3elJ7TSfbKPRdR{ J;`Dj¨R{IfJ
q
UDOP`bK[
RNIKJ©]9`DSKbK_4UDT}\`DjqUaQ7PTNQ7OPJ
xs¾¬J J ]9JMTmR{I4U=RR{IKJ SrjGSKbKQÂ
UZR{JMTNYUZO
r0bK[UD Q7TW` LNL%R{IKJ]9`SfbK_4UDT}ö \xjR{IfJ
R{P`b*bR£~N`¦_K JMbKLNP`bKL©TNJMeKTNJMLNJMbRNL U=R{J7TWYUDrO =`´/
R{`ZRUDO4U `SfbR `F/PbK[%UDQ7TW`LNLVRNIKJ/Q7TWQ7OPJL
ö RNIKJ7bdR{IKUZRVR{IKJ/QM`TNTWJ7QMRP_KJ7b
R{¤R£\aUZR¾RNIKJ
q(t)
]l`SKbK_4UZTW\PL ö
SÄr
q(t)
,
2πr0
LN`*R{I4U=R
xä¬`´ UDLWL{UD[DJ+RNIKJ ¥KOJ^`_adcdemf]^`}dghEijLN`*R{I4UZRPR
Nr = q(t)/ (2πr0 D)
L{U=R{L}¥4J7L^RNIKJ7LWJQM`bK_K¤R{`DbKL¨jG`T^`SKT¨RNJ7L}R¯efTN`]KOPJ
xóÄPbKQ7J~JnbK`´
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
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
VOSK[^RNIKLVPbRN`^R{IKJ¥4OPJ/UDbK_"bK`´ÝR{JMLWR`SKT¾TWJ7LNSfOPR{LV/¤R{I"UDb"b
R{JMT{UDQsR{P|DJ # QMRU;|DJiLWJ7LNLW`b ö
n _oempqsrut]v=wyx]_zcG{b|HoEr
= −DNr =
cdegfG{ed}qgj`eaHai7j
n _oempqsrut]v=wog1qx n }do3{ n ao n og{]x|dc n
}mqzj{gx-cNo3zHqgm|Nc`mtdmt1bo
n _oempqsrut]v=wdq9o=og1q{a{o3{ed}d{zde9o3{ n a n Ho n ^o]1q¶1em
n _oempqsrut]va=tHucma n fdqx}{a3xb|1e1_9q) ¡¢ta£3]¥¤]uc]a n fdq`xgcma n fdqxU ¥ra¡¥
n _oempqsr§Hv=w@a1qmgoog`q{a{bo3{ed}_ n adfG{bo3{ n a
n _oempqsr§Hv==qmm|Ncga n fdqxi¢¨zcma n fdqxgmt`¥
n _oempqsr©dv=wy_9ed}H_G}me9oqo]1qx n }Ho3{ n a
n _oempqsr©dvx]}a}Hx n fq)H¤u^`_adcdemf]^`}dghN¤Huddª««tm¬¥
n _oempqsru]v=w@|G} n ox n }Ho3{ n aye9adfqgde_ox n }zHo3{ n ae9o=odHt
n _oempqsru]v_]}mq]emcm|3} n oN n }f n aEu]cG{f
n _oempqsru®]v|3} n oscga n fdqxU¯xg}aN°tdr°¥¤l
n _oempqsr±Hv|3} n oscga n fdqxU«qmm|N«cma n fdq1x)i¢¨zcma n fdqxHmt1tG]
n _oempqsr²Hv=w@a n oex n]n fex´og1q@~3{cG{g_G}mq9o|dc n
}mqzj
Hoo]{gx{gx´odm|{g_9e}
n _oempqsr²Hv=wy_b1q_ ³ o]1qOjGemG{ jH]jqgc]c n c
n _oempqsr²Hvq3¶a n c9jSlxg}aN¢t-dr°¥¤µ9qmm|Ncga n fdqx-i¢¨zcma n fdqxHmt1tG¥µ{a1^G
qG´=si]]]m©H©H§]§
n _oempqsru·]v=w nz³ ¿}mq9oN¤µx´`ed}zpqog1qxoq9|3xH{dq
n _oempqsru·]va©ucma n fdqx}{a3xb|1e1_9q) ¡¢ta£3]¥¤]uc]a n fdq`xgcma n fdqxU ¥ra¡¥
n _oempqsrµHv=d=qmm|N«cma n fdq1x)i¢¨zcma n fdqxggt1¥
n _oempqsrµ]zvxg}a}Hx n fdq)H¤^`_badcdegf]^1}d]hN¤]ud-dªu«t]¬G¥
n _oempqsrµtHv|G} n os«cma n fdq1x¦¯xg}aS¢tdr°¦¤µ
n _oempqsrµb§dvq]t¶a n c9jEµxg}aN°tdr°¥¤lqmm|N«cma n fdq`x)i¢¨zc]a n fdq`xgmt1mt1¥µ{ad^G
q]t=si]]Gb§d®]tg±
n _oempqsrµ©vqGmq]t
ea3x¶§Siu·]·m©d
n _oempqsrµHv=w n a_9qegde1{aSÀe
3n Ho©Si
n _oempqsrµHv¸9{bo
QSRUT)VWGXHY:Zfxv¹x`
\ä¬`F®LWSKeKel`LNJmRNI4UZRq~JXXUDb
RhR{` `_KJ7OÄU«el`eKSKOUZR{P`b©R{I4UZRq`]lJM\L
R{IfJ%[D`F|DJ7TWbKbK["J7S4U=R{`DbKL
ÅsÆ ÆKþÉÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DÉͧËMÐ19Ò32
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/ eKOPQ7PR~Q7`DbK_KPRN`b hR{IKJ%LN`OPSfR{P`b+LNIK`DSKO_TNJ UDPb]9`DSKbK_KJM_?xv"HIKJ
SKLWnT{UZR+el`eKSKOUZR{P`bÝJÂrfU efOJ¦`Dj¯kmUZbKnL§t uKie?x mÂZw«LNJMTW|JMLUDL+UDbJsrfU eKOPJ¦`Dj¯U
_K¤ÁSKLW`beKIKJ7bf` JMbK`b¦/¤R{IáJsrÄel`bKJMbRNYUDOA[TN`´RNI®
/IKJ7TWJ
L^Uel`LN¤R{¤|J
f = aN )
a
[TW`FRNI TNUZR{JZx
»«JMOPR{UQ7`bf_KPRN`bKL«UDTNJ¯TNJ7UDOO¤\+_K P"Q7SfOPR¬R{`QM`Ä_KJ
SKeábS JMTNPQ;UDOPOP\]lJ7Q7UDSKLNJRNIKJM\UDTWJ
UaeKTNJsRNR£\LWJM|DJ7TNJ©jG`T `DjC_KPLNQ7`DbRNbSK¤R£\xhJ^Q;UDb+¥4bKJMLNLR{IKPLb
R{`dU `TNJ«bS JMTNQ7UDOO¤\
U J7b4UZ]KOJmefTN`]KOPJ UDLVjG`OOP`F/LM CUDLNna`SfTNLNJMOP|DJ7LV/I4UZRI4UDefe9JMbKLU©L UZOOÄRN JUZj·RNJ7ThR{ J
xmJ¯JÂrÄe9JMQMR«RNIKJ
PbKLWR{UDb
RUDbfJ7`SKL¬eKSKOPLNJ©R{`JM|`DOP|J¯b
R{`UL ``DRNI¦º©UDSKLWLNYUZbÄ£OPnDJ
t_KP=
0
LWR{TW]KSÄR{`Db?xÝz [``Ä_®Q7OPSKJ+LdR{IKJJsrÄeKOPQ7¤RdjG`T RN`*RNIKJJÂrfel`bKJMb
R{YUZO¬[DTN`´R{IT{UZRNJ
eKTW`]KOJ }t uK4e?x¾
¹=wY
M at−r2 /(4Dt)
e
.
N (r, t) =
óÄ`XJ§Q7UDbRNTNJ7UZR+R{IKPLbKL}RUDb
RUZbKJ7`SfL4πDt
LW`SKTWQ7J*eKTN`]fOJ UDLU)eKTN`]fOJ /PR{IU
SÄr
ö U=R
Q7`DbK_KPRN`bUZR¬R{IfJ¯OPJMj·R¬]9`DSKbK_4UDT}\¸]KSfR«T{UZRNIKJ7TRNI4UDbIKU´|DJUDb bKL}RUDb
R{UDbKJ7`DSKLiLW`SKTWQ7J
¾~JQ7UDb®LNeKTWJ;UD_®`SfRaRNIKJ+eKSKOLWJ+`F|DJ7T"UáLNIK`DTWRdR{ JPb®U§L ``DRNIj3UDLNIKP`b]
\
t_KJs=
¥4bK0bf[
2
2
2M
q(t) = √ e−t /(2σ )
σ 2π
/IKJMTNJ
PL/U"L UZOO9|ZUDTWYUDbfQ7JDx¾ä¬`DR{J¨RNI4UZR
σ
Z ∞
q(t)dt = M.
0
ó `¦~J/OPOTWJ7eKOUDQ7JR{IKJPbKLWR{UDb
RUDbKJM`SKLLW`SKTWQ7J]
\)U¦QM`b
R{bSK`DSKLLN`DSKTNQMJR{I4UZRJ PR{L
Ä
UD]l`SfRR{IKJ«L{U J¨U `SKb
R7
K`´|J7TmU¯TNJMOYUZRNP|DJ7OP\dLWIK`T}RXR{ J¬LNeKUDb]
\SfLNbf[
R{IKJ«UD]l`F|DJ
jG`T SKOYU
jG`DT
/PRNI UaLWSKM
PR{UD]KOJ^QIf`QMJ%`Zj x
q(t)
σ
QSR)Y3ÃÄLÅÇÆY;ZfxÈÂ`\CHAUDnZJ«U^Q7TWPRNQ;UZOfO``n
UZRVR{IKJ/JÂrKU eKOPJ`DjkmUDbKnLit uKexVmÂZwyxVó`OP|DJ
R{IfL©eKTN`D]KOJ bS JMTNPQ;UDOPOP\ SKLNPbK[RNIKJdjGSKbKQMRN`bá¥4OPJajGQ7bKTNUD_ SÄr©
¾ x xÉ
¼ `DS [I
R©SKLWJ
UDL/bR{IKJ%eKTWJM|`SfL¬JÂrfU efOJ©UDbK_ UDbb
R{JMTW|ZUDOjGTW` ö RN`LW` JsR{IKPbK[dOPnDJ
M
=
1
1
150
UDOP`bK[/PRNI
R{` QnUDb*bKL}RUDb
RUZbKJ7`SfL%LW`SKTWQ7JDx"z«L%kmUZbKnL J7b
R{P`bKLM?\`S
Q7`DSKO_§SKLWJ σ = 1 ýF\ÄT©UZbK_
xa``n¦UZR^RNIKJd[T{UZeKIKL%UDbK_§LNJMJdPjR{IKJaTNUD_KSfL
a
=
2.65
D
=
12.2
`DjR{IKJdjGTN`DbR%TW`SK[IKO¤\¦QM`TNTWJ7LNel`bK_fL%RN`RNIKJd_4UZRU+[¤|J7bx"I4U=R
UZTNJaR{IKJ"UDefeKTN`;rÄ UZR{J
el`eKSKOUZR{P`bKL/bRNIKJ
\DJ;UDTWL
UDbK_
/IKJMTNJ
~JQ7`DSKb
R
UDLi\J;UZ
T 7JMTN]
` Ê®»«`
2,
4,
6,
8
10
1905
\`DS ]lJ7OPJM|DJ^R{IKPL `_KJ7ËO Ê
QSR)Y3ÃÄLÅÇÆYZÄxv1
g \.óÄSfeKe9`DLNJmR{IKUZRq~J `_KJMO?GQ7TN Pb4UDOY_KTNSK[«L{UDOPJ7LVUDLqT{UD_KUZR{PbK[/jGTN`
U^LNPbK[OJXLWR{J7UD_f\aLW`SKTWQ7J//IK`LNJ/OP`ÄQ7UZR{P`b
~J¬nbK`´ UDbK_"`]9Js\bK[©U^T{UZ_KYUDOÄ_KPÁ9SKLW`baOU´©x
äi`´
TNJMeKTNJMLNJMbRNLaR{IKJ_KJMbKLN¤R£\`Dj¬_KTWSK[ábfQ7_fJ7b
R{LMx½¬J7TNJ+UZTNJPLNLNSfJ7LR{`¦RNIKbfn
NUD(r,
UD]l`SfR¬
bf_t)~J^/OO9_KLWQ7SKLWL¬RNIKJ b+LNJ Pb4UDT7 ¡xq½i`´®Q;UDb~JLNJsRVSKeRNIKJXjGSKbKQsR{P`b¯¥4OPJ7LqRN`¨I4UDbf_KOJXU¨Q7`b
R{PbSK`SKLhQM`bKL}RUDb
RVLW`SKTNQMJ
`DjlT{UZ_KYUDO_K¤ÁSKLW`bD[¤|J7b"U¨QM`bKLWR{UDb
R¾_KPÁ9SKLW`bQM`ÄlJ P"Q7JMb
RUDbK_R{IfJLN MJm`Dj¸R{IfJLN`SKTWQ7JZx
I4UZRmUDTNJ/R{IfJ¨UDeKefTN`eKTWYUZRNJ/]9`SfbK_4UDT}\QM`bK_K¤R{P`bK¯L Êy»«`%R{IKPLjG`DTXU
QM`b
R{PbSf`SKL `_KJ7O
UDL/bªqrfU eKO0
J Zfx¹ÄxLN`dR{I4U=Ri¤RLTWJ;UD_Ä\R{`"[`ajG`TmR{IfJ%TWJ7L}Ri`ZjARNIKLeKTW`]KOPJ xv
ÅsÆ ÆKþÉÒ?ÌÄÔ"ÿÍlÉ?ÓfÔDÌÄþË0/DÉͧËMÐ19Ò32
ÌÅ
x¾ä¬`´LNSKefe9`LWJ«R{IKUZR~~J¨UZTNJ¬[P|DJ7b_fTNSK[¯Q7TN J¬_4UZR{U%`DbU¯[DTN_ UDe`DjRNIKJ«QMPR£\
Ä
TNJ7UDOO¤\fsxV½¬`F°Q;UDb+~J%RNT{UDbKLWOYUZRNJ«R{IfL/b
R{`aPbfjG`T UZRN`bUD]9`DSfR
Ê
NLN(r,
t)R{`J7L}R{ UZRNJdRNIKJ
fxº©¤|JMbR{IfL©bfjG`DT UZR{P`báUD]l`SfR
qIK`´ìQ7UDb*~JSf
J"PR©
N (r, t)
_K¤ÁSKLW`bQM`ÄlJ P"Q7JMb
Ri`ZjqR{IfJ `_KJ7ËO Ê
Kxi3½«UDTW_KJ7+T Ív
I4UZR¬¤jC~J%_K`b?o R/nbK`´ R{IKJ©QMJ7b
R{JMT7¸]KSfR¬]lJ7OPJM|DJ¨R{IKJMTNJ©L`bfbJ Ê
ÎÏÐÑuÏuÒsÓ-Ô1Õ-Ö×EØ
Ì¢Ì
© Copyright 2026 Paperzz