11

Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

  • Upload
    others

  • View
    2

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

�1398

Q=y@�'h

}QWl}

v=mt|U

Ovyt11

3�103

"X'1|

xQ=tW'

35�3

|xQwO

Ori

gina

lArt

icle|oOvm =QB � |}=Hx@=H |xrO=at Twmat pL

xOv}q; `@vt |}=U=vW CyH =yxv=NOwQ QO�OWQ=|U=vWQ�|wHWv=O� |@=Wwr OtLt

�Q=}O=DU=� � |Qy=_t |Oyt�O=DU=� |v=t=U |rwOtLt p=tH

TQOt C}@ QD x=oWv=O '|RQw=Wm |xOmWv=O

x@ �|oOvm =QB � |}=Hx@=H� p=kDv= |=yOv}=Qi \UwD xv=NOwQ QO xOW x}rND |oOwr;"CU= C}ty= R�=L Q=}U@ xOv}q; `@vt QD`} QU xJQy uDi=} TB &OwW|t pkDvt CUOu}}=B

Cr=L QO |oOvm =QB � |}=Hx@=H |xrO=at Twmat pL R= xO=iDU= =@ Q=DWwv u}= QO|xOv}q; C_re |v=tR |QU |=yxO=O 'CN=wvm} Q}e w Q=oOv=t u=} QH \}=QW QO |Oa@l}

s=v x@ 'xOW x�=Q= Twmat pL VwQ T=U= "OvwW |t |R=UR=@ xv=NOwQ QO xOW x}rND�sQ=yJ jDWt sQD� |Q=O}=B sQD uOQm xi=[= =@ xm CU=v@t u}= Q@ '|Q}PB Twmat x@W VwQ

|=yMU=B 'OQ=O =Q |rY= Vkv |Q=O}=B ?} Q[ |Q=O}=B sQD QO xm QwmPt |xrO=at x@|v=tR |wor= xJQy xm OyO|t u=Wv G}=Dv "Ov=UQ|t sRq |}=Qosy x@ =Q xrO=at u}= |DWoQ@pOt \UwD 'u; |R=UR=@ '|oOvm =QB sQD QF= p}rO x@ OW=@ |QDW}@ |oO}J}B |=Q=O |Q=PoQ=@C}r@=k |Q}PB Twmat x@W VwQ u=vJty =t= 'CU= xH=wt |QDW}@ |Q=wWO =@ Twmat

"OQ=O xv=NOwQ QO xOv}q; C_re |Q=PoQ=@ |v=tR |wor= |R=UR=@ QO |@ wN

[email protected]@modares.ac.irsamani @modares.ac.ir

|v=tR |QU '|Q}PB Twmat x@W 'Twmat pOt 's}kDUt pOt %|O}rm u=oS=w"|Q=O}=B ?} Q[ '|oOvm =QB � |}=Hx@=H |xrO=at 'xOv}q; C_re

xtOkt "1|v=}=W ltm Ov=wD|t xOv}q; |R=U=yQ u=tR w u=mt QD`} QU xJQy uDi=} u}=Q@=v@|=yVwQ "OW=@ xDW=O |B QO =Q C=Q=UN |R=Uxv}tm w |oOwr;Vy=mCyH QO OW=@xOW CqmWt R= |Q=}U@ |=Wox=Q xOv}q; `@vt |}=U=vW QO 1Twmat |xr�UtxOv}q; `@vt |}=U=vW |=Q@ u=mt w u=tR QO Twmat CQwY x@ =yVwQ u}= &CU=

"Ovvm|t ptaCqO=at XwYN x@ hrDNt CqO=at |wQ Twmat p�=Ut |xar=]tl} "CU= xOw@ u=kkLt R= |Q=}U@ |xkqa OQwt 'G}=Dv PN= w |�RH p}Uv=Qi}'�?=wH |}=Dm} sOa� '�?=wH OwHw sOa� OQ=wt =@ CU= umtt Twmat |xr�Ut

[3]"OW=@ xH=wt =yu; R= |@}mQD =} �|Q=O}=B sOa�'Twmat pL |xv}tR QO xDiQo s=Hv= C=k}kLD Q@ |QwQt ut[ VN@ u}= QO(QR) |Q}PBTwmat x@WVwQ|xv}tR QO xDiQo s=Hv=C=k}kLD Q@|QwQt x@,=Y=YDN=

"s} R=OQB|t 2l} QO w |}x]kv |xOv}q; `@vt |}=U=vW |=Q@ =Q |r}rLD VwQ [4]w}r w nv=w|]=kv R= xOW OYQ C=aq]= OwHw OvtR=}v QwmPt |r}rLD VwQ "OvOQm x�=Q= Oa@

"CU= xv=NOwQ pw] QO R}=tDt=Q xDN=vW=v `@=vt |NQ@ =@ \@DQt |v=tR C}r=ai OwOL [5]u=Q=mty w |OtLOQwt =Q |Oa@ wO |]N |xDi=} xaUwD |oOvm =QB � |}=Hx@=H |xrO=at OQ@ Q=m =@

w CavY '|RQw=Wm VN@ QO u=w=Qi OQ@ Q=m p}rO x@ |L]U |=y?; `@=vt=yxv=NOwQ ?re= "OvQ=O Q=Qk |}=}t}W |=y|oOwr; ZQat QO |QyW hQ=Yt'?; |oOwr; R= Qw_vt "OvwW|t xOwr; =yu; h=Q]= QO xOW O=H}= `}=vY \UwD|=y?; w =yxv=NOwQ '=yxJ=} QO ?; O}=R O=wt =@ |oOwr; '|@ wQm}t '|}=}t}W |oOwr;=yxv=NOwQ x@ xOv}q; |R=U=yQ QO `}=vY uOQm OwOLt |=Q@ [1]"CU= |v}tRQ} RR=@ |OQ@ Q=m w syt u}v=wk `[w =@ |DL CU= |y}O@ =t= 'OQ=O OwHw |D=QQktl} hrND "CU= Q=mv= p@=kQ}e |Qt= |@; `@=vt x@ =yxOv}q; R=HtQ}e OwQw sys=Hv= CQwY xU x@ Ov=wD|t |@; `@=vt x@ =y?=UB |x}rND OQwt QO xOv}q; `@vt

[2]%OwW'OQ=Ov=DU= QO xOW u}}aD OL R= V}@ C_re =@ ?=UB |x}rND "1

'OQ=Ov=DU= QO xOW u}}aD |=yu=tR R= Q}e |}=yu=tR QO ?=UB |x}rND "2"OQ=Ov=DU= QO xOW wvtt O=wt |x}rND "3

pw�Ut xOvU} wv �"1397 1 28 VQ}PB '1397 1 14 x}LqY= '1396 10 20 Ci=} QO %M} Q=D

DOI:10.24200/j40.2018.50053.1455

Page 2: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

"""|oOvm

=QB�|}=H

x@=H|xr

O=atTw

matpLR= |Q}owrH QR VwQ |SD=QDU= "CU= |ivt |v=tR |=ys=o =@ |�RH |r}Uv=Qi}O

|rY= |xrO=at x@ l}ORv |}xrO=at pL u; |=H x@ w Q}PBTwmat Q}e |xrO=at pL[12]"CU= |ivt |v=tR s=o =@

|v} Ro}=H j} Q] R= Twmat u=tR =@ =Q 5|oO}WNB |xrO=at [11]uw}r w TDrC@Ft C@=F " u; QO xm "�2��� @

@t 'QR QwD=QB= =@ �� @@t

|oO}WNB QwD=QB="OvOQm pL 'CU= TqBq |xrO=at |=Q@ |}x}=B � w lJwm

xar=]t OQwt C=QP |oO}WNB |xrO=at |=Q@ =Q xDN=vW=v `@vt |}=U=vW [13]nv=}pL |=yMU=B |Q}oxR=Ov= |=yxO=O QO lJwm Q}}eD xm CiQo xH}Dv w= 'O=O Q=Qk

"Ow@ Oy=wN Q=PoQ}F-=D |oO}WNB |xrO=at Twmat?}=Q[ =@ |oOvm =QB � |}=Hx@=H |xrO=at |=yMU=B |Q=O}=B |=Q@ pL x=Q l}sQ=yJ jDWt |Q=O}=B sQD |Q=Po}=H =@ QR VwQ 'Twmat CQwY x@ 'C@=F=@ 7CQ=QL CiQty |xrO=at pL |=Q@ 6s}_vD VwQ l} QR VwQ [14]"CU=

[15]"CU= CiQty |xrO=at x@ s}_vD QDt=Q=B uOQm xi=[==tQo CiQty |Oa@xU |xrO=at |=Q@ =Q QR VwQ [12]u=Q=mty w xO=Rp}a=tU=Q=O}=B=v|v=tR TwmatCr=L QO =tQoCiQty |Oa@xU |xrO=at |=yMU=B "OvO=O x�=Q=

[12]"OwW|t xrO=at |=yMU=B |}=Qoty w |Q=O}=B Ea=@ QR VwQ TB 'CU=CkO Q@ |oOvm =QB sQD xm OvO=O u=Wv OwN G}=Dv QO [16]=HO=tD= w wroRDo@QF-wt Q=}U@ QR VwQ =@ |oOvm =QB � |}=Hx@=H |xrO=at Twmat pL |=yMU=B|R=UxO=}B Q=@m} =@\ki w xOvwW Q=QmD Q}e|tD} Qwor= |=Q=O QR VwQ OQm} wQ "CU=xr�Ut CqwyHt xm CQwY u}= x@ |�RH |r}Uv=Qi}O |xrO=at |=Q@ |OOa VwQ QO

[17]"Ovm|t pL Q=QmD uwO@ w u}at CQwY x@ =QxOv}q; p=kDv= |xrO=at Twmat pL |=Q@ =Q QR VwQ [18]q=@=m w Ro =mU=OvDW=O O=kDa= =yv; "OvOQ@ Q=m x@ |oOwr; `@=vt |}=U=vW |=Q@ |v}tRQ} R |=y?; QOOQ@ Q=m p@=k |oOvm =QB � |}=Hx@=H |xrO=at Twmat pL QO QwmPt VwQ xm

[16w18]"CU==@ |v}tRQ} R |=y?; QO C@=F ?}=Q[ =@ =Q p=kDv= |xrO=at [11]uw}r w TDrCQwYx@ |oOvm =QB � |}=Hx@=H |xrO=at pL QO QwmPt VwQ "OvOQm pL QR VwQ|i}a[ OQmrta CU= u=} QH Q@ ?r=e |oOwr; xm |ak=wt QO '|v=tR Twmat

[16w19]"OQ=O`} RwD |x}rw= \}=QW u}}aD x@ Q=}U@ [15]u=Q=mty w nv=} S j}kLD QO QR VwQ

[15]"Ow@ |mDt |D=@U=Lt |xvt=O QO xOv}q; C_re|RQt\}=QW QO X=N |=yC}OwOLt uOQm xi=[= =@ OvOQm C=@F= [20]w}=Q w =QwO

"OQw; CUO x@ |QDj}kO ?D=Qt x@ |=yMU=B u=wD|tjDWt xOvvm Q=O}=B sQD uOQm xi=[= =@ xm CU= |WwQ |Q}PB Twmat x@W VwQQO QwmPt |xrO=at |=yMU=B |Q=O}=BEa=@ '|oOvm =QB �|}=Hx@=H |xrO=at x@ sQ=yJ|oOvm =QB �|}=Hx@=H|xrO=at u=wD|tCQwYu}O@ [21]&OwW|t|ivt|v=tR|=ys=o|QU |=yxO=O uOQw; CUO x@ hOy "OQm pL Twmat CQwY x@ Kq]Y= QO =Qu=tR u}vJty w xv=NOwQ QO �|[Qi COW `@=D� xOW =yQ |xOv}q; C_re |v=tR?} Q[ |Q=O}=B sQD QO xm 'OQm |R=UR=@ xOW x�=Q= pOt \UwD u=wD|t =Q |R=U=yQ

[22]"OQ=O =Q |rY= Vkv |Q=O}=Bxv=NOwQ QO =Q xOv}q; `@vt |}=U=vW u=} QH CqO=at QO |R=U xO=U ,=YNWt=Q xOv}q; `@vt |}=U=vW QwmPt CqO=at QO |R=UxO=U sOa Tma x@ w QDu=U;

[23w22]"Ovm|t QDQ=wWO|}=Hx@=H |xrO=at pL |=Q@ |Q}PBTwmat x@W VwQ Q[=L Q=DWwv QOQO xOW xDiQo Q_v QO xv=NOwQ 'OwQ|t Q=m x@ |ivt |v=tR |=ys=o =@ |oOvm =QB �

C_re C@F R= xO=iDU= =@ |}=U=vW VwQ l} |Q=QkQ@ =@ =yu; "OvO=O Q=Qk |}=U=vW|UQQ@ OQwt |xvt=O QO u=} QH |OwQw RQt Q@ Q=W C@F w u=} QH |HwQN RQt QOp=mW= |xQ=@ QO u}W}B C=aq]= xvwoI}y uwO@ =Q |v=tR C}r=ai OwOL OvDUv=wD

"Ovvm u}}aD Q}oQO |xDN=vW=v `@=vtp=kDv= |xrO=at R= xO=iDU= =@ =Q |}x]kv |xOv}q; `@vt l} |}=U=vW [6]|OtLuDiQo Q_v QO =@ =Q Q_v OQwt `@vt |w "O=O s=Hv= |]N |Oa@ wO |xDi=} xaUwDxvt=O R= |HwQN u=} QH RQt QO C_re |Q}oxR=Ov= =@ w xOvvm OwOLt C=}[Qiu=tR R= p@k `@vt C_re `@=D xm u}= ZQi =@ TBU "CU= xOQm |}=U=vW pQDvmQ@ |v@t K} QY x@W |}=U=vW VwQ l} |Q=QkQ@ =@ OwW|t s=tD |}=yv pQDvm|Q}oxR=Ov= =@ =Q Q_v OQwt `@vt |=yQDt=Q=B CUv=wD OwHwt |RQt pQDvm G}=Dv |NQ@CUO x@ u=} QH |OwQw RQt QO Q=W |Q}oxR=Ov= R= w |HwQN u=} QH RQt QO C_re

"OQw;|Oa@ l} CQwY x@ p=kDv= |rm |xrO=at R= xO=iDU= =@ [7]|yOwiyt w |OtLTQDUO QO R}v 'Vvm =w w CaQU '|oOvm =QB |v=mt Q}eDt ?}=Q[ uDiQo Q_v QO =@ w`@vt COW `@=D w u=mt OvDUv=wD xOv}q; `@vt OQwt QO |v}W}B C=aq]= uDW=OxOv}q; `@vt CUOu}}=B w CUOq=@ QO C_re C@F =@ xv=NOwQ QO =Q |}x]kv |xOv}q;

"Ovvm |}=U=vWQ}F-=D CLD |xk]vt |DWoQ@ p=tDL= pOt R= xO=iDU= =@ [8]Ov} Qi w uUrwt"OvOQm x�=Q= u=wN@; |Q}PB?}U; u=R}t u}}aD |=Q@ |WwQ w xOQm XNWt =Q x=JT=U= Q@ x=J Q}F-=D CLD |xk]vt u}}aD Qw_vt x@ u} Ro}=H |WwQ Q=DWwv u}= QOu=tR swyit R= xO=iDU= =@ 'Qo}O CQ=@a x@ "CU= xOW x�=Q= \UwDt Qta swyitw |Oa@ wO pOt "OwW|t XNWt u=wN@; |rm pmW 'xOv}q; V}=t}B |DWoQ@xDiQo Q=Qk xO=iDU= OQwt �=O=v=m � wrQD=w R} Q@; |x[wL QO |y=J |=Q@ u; |Oa@xU

"CU=|Q=t; u}tR |=yVwQ Q@ |vD@t pOt w |DWoQ@ p=tDL= pOt [9]u=Q=mty w q=B wm\}=QW CLD |y=oW}=tR; |=yxO=O R= Q}N= VywSB QO "OvOQm xU}=kt Qo}Om} =@ =QQO=k |Q=t; u}tR VwQ xm OyO|t u=Wv G}=Dv "CU= xOW xO=iDU= xOW pQDvm ,qt=m&OyO X}NWD |@ wN x@ =Q |R=U=yQ u=mt w xOv}q; |R=U=yQ |x}rw= `@=D CU=|R=U=yQ `@=D |=Q@ x}rw= |x}[Qi x@ R=}v x@ u=wD|t pOt u}= CqmWt xrtHR=p=tDL= VwQ "OQm xQ=W= xOv}q; `@vt x@ uwv_t |x]kv OvJ uOQm XNWt w|r=L QO 'OQ=O R=}v xOy=Wt |OwOat O=OaD =@ OYQ |x]kv l} x@ =yvD |DWoQ@"OQ=O R=}v xOy=Wt |O=} R O=OaD =@ OUQ |x]kv wO pk=OL x@ |Q=t; u}tR pOt xmu=tR =} u=mt |=yQDt=Q=B R= |m} |DWoQ@ p=tDL= VwQ QO xm CU= QmP x@ sRqxOv}q; |x}rw= u=mt R= |@ wN |v}@V}B pOt wO Qy "OW=@ XNWt O}=@ |R=U=yQ

"OyO|t x�=Q=TB 'OwW|t QWDvt xv=NOwQ QO 4|oOvm =QB 3|}=Hx@=H |=yOv}=Qi\UwD xOv}q;=@ QwmPt |xrO=at pL 'xOv}q; `@vt |Q=PoQ=@ |v=tR |wor= wv w u=tR uDi=} |=Q@O=H}= =Q s}NO@ |�RH |r}Uv=Qi}O |xrO=at xm 'O@=}|t CQwQ[ |ivt |v=tR |=ys=oxm OwW|t jq]= |DqO=at x@ s}NO@ |�RH |r}Uv=Qi}O CqO=at [10]"OQm Oy=wNpY= QO w OvDU}v pL p@=k |�RH |r}Uv=Qi}O CqO=at pL pw=ODt |=yVwQ =@OvUQ|tv sRq |}=Qoty x@ =yMU=B xmr@ 'OW Oy=wNv O=H}= |Q=O}=B MU=B =yv; |=Q@

[10]"OvwW|t =Qo =w w's}NO@ |�RH |r}Uv=Qi}O CqO=at pL ut[ u=wD@ =D OwW x�=Q= |WwQ O}=@ TB|Q}PB Twmat x@W VwQ u}=Q@=v@ "Ov=UQ sRq |}=Qoty x@ =Q CqO=at |=yMU=BCqO=at pL VwQ u}= hOy "CU= xOW x�=Q= [11]uw}r w TDr \UwD �QR�

104

Page 3: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

1|xQ=t

W'35

�3|xQ

wO'�13

98Q=y@

�h}QW

l}v=mt

|UOvyt %[24]CU= u}vJ |Oa@l} Cr=L QO xOv}q; p=kDv=

@(AC)@t

= �@(QC)@x

+ @@x

�AEx

@C@x

��AkC +AS (5)

'u=} QH |@O Q 'u=} QH `]kt K]U A 'u=tR Oa@ t 'u=mt Oa@ x 'u; QO xmk u}vJty "CU= xv=NOwQ pw] CyH QO |oOvm =QB ?} Q[ Ex 'xOv}q; C_re C|=yQDt=Q=B u}}aD |=Q@ '5 |xrO=at QO "CU= xOv}q; `@vt sQD S w Vvm =w sQD ?} Q[sRq "OwW|t xO=iDU= Cv=vw � CvU CqO=at pL R= |Oa@ l} Cr=L QO u=} QHu=} QH \QW =@ u=} QH |xrO=at w |oOvm =QB � |}=Hx@=H |xrO=at xm CU= QmP x@

%OvwW|t |U} wvR=@ 6 |x]@=Q j@=]t CN=wvm} Q}e w Q=oOv=t

A@C@t

= �@(QC)@x

+ @@x

�AEx

@C@x

��AkC +AS (6)

|R=Ux}@W Cr=L |=Q@ OwHwt |=ypOt |xty QO xDiQ Q=m x@ u=} QH |xrO=atCN=wvm} Q}e w Q=oOv=t u=} QH |=Q@ xmCU=Cv=vw �CvU u=} QH CqO=at '|Oa@l}

"CU= 8 w 7 CqO=at j@=]t@Q@x

= 0 (7)

@@x

�Q2

A

�+ gA@y

@x+ gA(Sf � S0) = 0 (8)

\N ?}W Sf 'u=} QH jta y 'u=} QH `]kt K]U A 'u=} QH |@O Q 'u; QO xm"CU= xv=NOwQ QDU@ ?}W S0 '|SQv=

CqO=at xDUO QO|ivt|v=tR |=ys=o CQwY x@|oOvm =QB �|}=Hx@=H |xrO=at|oOvm =QB � |}=Hx@=H |xrO=at |=yMU=B [16]"OQ}o|t Q=Qk s}NO@ |�RH |r}Uv=Qi}OsQD |Q}PBTwmat x@W VwQ "OvUQ|tv sRq |}=Qoty x@ |ivt |v=tR |=ys=o QOxi=[= xm Ovm|t xi=[= |oOvm =QB � |}=Hx@=H |xrO=at x@ =Q 8sQ=yJ jDWt Q=O}=B|ivt |v=tR |=ys=o Cr=L QO xrO=at |=yMU=B s}_vD ?Hwt |QwD=QB= u}vJ uOQmx@W VwQ j@] 6 |xrO=at TB [18]"Ovm|t |Q}owrH =yMU=B |}=Qo =w R= w xOW

"OwW|t |U} wvR=@ u}vJ |Q}PBTwmat

A@C@t

= �@(QC)@x

+ @@x

�AEx

@C@x

�+ "@

4C@x4 (9)

xOW XNWt sQ=yJ jDWt sQD x@ "OQ=O s=v |Q=O}=B ?} Q[ " 'u; QO xm|=ys=o =@ 9 |xrO=at |}=Qoty QO =Q |rY= Vkv xm OwW|t xDio |Q=O}=B sQD|xrO=at |=yMU=B |}=Qoty |=Q@ |Q=O}=B sQD w |Q=O}=B ?} Q[ "OQ=O |ivt |v=tR|=v@t Q@ |Q}PBTwmat x@W VwQ T=U= w x}=B xm OQ}o|t Q=Qk xO=iDU= OQwt QwmPtC=Q}}eD |w=UD AJ CtU QO 9 |x]@=Q |=ysQD u}vJty [11]"CU= |Q=O}=B sQDx@ AJ CtU R= |w=UD CU=Q CtU QO w 'OyO|t u=Wv =Q u=tR x@ C@Uv C_reu=} QH `]kt K]U Q@ (Q) |@O s}UkD =@ xm OQ=O s=v |}=Hx@=H sQD pw= sQD 'CU=Qx@ xm OQ=O s=v |oOvm =QB sQD 'swO sQD u}vJty "O};|t CUO x@ |}=Hx@=H ?} Q[x�=Q= ?} Q[ u}= |=Q@ sRq C=L}[wD xt=O= QO "OwW|t xDio |oOvm =QB ?} Q[ Ex

"OwW|tQ@=Q@ Vvm =w ?} Q[ w xOv}q; `@vt sQD Q[=L j}kLD QO xm CU= QmP x@ sRq

%R= CU= CQ=@a 9 |xrO=at |}=yDv= w |RQt \}=QW "OwW|t xDiQo Q_v QO QiY

|[Qi COW `@=D j@] j}kLD u}= QO "CU= CN=wvm} Q}e w Q=oOv=t u=} QH \}=QW|[Qi COW `@=D uDi=} |=Q@ &OwW|t =yQ |OwQw RQt QO |}xOv}q; xv=NOwQ QOC}OwOLt p}rO x@ QwmPt VwQ |rm Qw] x@ "OwQ|t Q=m x@ |Q}PBTwmat x@W VwQ\}Lt QO 'xv=NOwQ u=} QH QO OwHwt |ak=w \}=QW w u; |x}rw= \QW QO OwHwt=@ "CU= xOWv `k=w xO=iDU= OQwt u}W}B |=yj}kLD w C=ar=]t Qo}O QO xv=NOwQOQwt xv=NOwQ QO Q=DWwv u}= QO |Q}PB Twmat x@W VwQ |D}OwOLt u}vJ `iQ

"CU= xDiQo Q=Qk xO=iDU=|rw] p}iwQB� |x}rw= \}=QW C}OwOLt `iQ |=Q@ O}OH |WwQ Q=DWwv u}= QO|x�=Q= =@ u}vJty "CU= xOW xO=iDU= |Q}PB Twmat x@W VwQ �xOv}q; C_rex@W VwQ 'xv=NOwQ CN=wvm} Q}e w Q=oOv=t u=} QH \}=QW =@ ?U=vDt |tD} Qwor=xOW `k=w uwtR; OQwt xOv}q; `@vt |}=U=vW |=Q@ xv=NOwQ QO |Q}PB Twmat

"CU=C}OwOLt xm OQm xYqN xvwou}= u=wD|t =Q Q=DWwv u}= QO OwHwt |=y|Qw;wv|Q}PB Twmat x@W VwQ |x}rw= \QW |=Q@ xOv}q; C_re |rw] p}iwQB CW=OQ@xDiQo xQy@ xv=NOwQ QO QwmPt VwQ R= u}vJty "CU= xOW hQ]Q@ |WwQ \UwD

"CU= xOw@v QU}t Qt= u}= u}W}B C=k}kLD QO xm xOW|Q}PB Twmat x@W VwQ R= xO=iDU= Q[=L Q=DWwv |x�=Q= R= hOy |rm Qw] x@"CU= xv=NOwQ |OwQw RQt QO xOW =yQ |xOv}q; `@vt |}=U=vW |=Q@ xv=NOwQ QO

=yVwQ w |Q_v |v=@t "2QwD=QB= xm OW x�=Q= =tQo |xrO=at |=Q@ =OD@= |Q}PB Twmat x@W VwQ |rm Qw] x@

%R= CU= CQ=@a QwmPt |xrO=at QO =tQo�@@t� k�

�u = 0 (1)

u(x; T ) = X(x) (2)

u=tR pm T 'u=tR Oa@ t 'u=mt Oa@ x '=tQo C}=Oy ?} Q[ k 'u; QO xm|xrO=at s}kDUt pL R= xOt; CUO x@ `@=D w |}=yDv= \}=QW X(x) '|R=Ux}@W

"CU= QwmPt|=yVwQ x@ '1 |xrO=at s}kDUt pL R= xOt; CUO x@ |}=yDv= \QW 2 |x]@=Q2 |x]@=Q R= ,=YNWt 'OwW pL Twmat CQwY x@ 1 |xrO=at Qo = =t= "CU= pwtatl} ?}DQDu}O@ w 'OW Oy=wN xO=iDU= Twmat |xrO=at |}=yDv= \}=QW u=wva x@sRq |}=Qoty x@ u; |=yMU=B xm OwW|t O=H}= s}NO@ |�RH |r}Uv=Qi}O |xrO=atxi=[= 1 |xrO=at x@ 'Twmat |xrO=at pL Qw_vt x@ |Q=O}=B sQD TB "OUQ|tv

%OwW|t�@@t� k�� "�2

�u = 0 " > 0 (3)

'OQ=O s=v |Q=O}=B ?} Q[ " u; QO xm ("�2) |Q=O}=B sQD uOW xi=[= =@CQ=@a 3 |xrO=at |}=yDv= \}=QW "CU= pL p@=k Twmat CQwY x@ 1 |xrO=at

%R= CU=u(x; T ) = X(x) (4)

|xrO=at x@ |D=Q}}eD p=ta= =@ =Q 4 |xrO=at u=wD |t xOW xDio ?r=]t x@ xHwD =@|xrO=at =} |oOvm =QB � |}=Hx@=H |xrO=at |rm sQi "O=O s}taD |oOvm =QB � |}=Hx@=H

105

Page 4: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

"""|oOvm

=QB�|}=H

x@=H|xr

O=atTw

matpL

"|[Qi |xv=NOwQ "1 pmW

QO xv=NOwQ QO xOv}q; C_re |rw] p}iwQB CW=OQ@ '9 |xrO=at |x}rw= \QW |=Q@|Q}PBTwmat x@W VwQ\UwD xOW|U} wvR=@ |xrO=at =Q} R 'CU= |QwQ[ u=tRl}l} QO xOv}q; C_re CW=OQ@ C}ak=w QO xm OQ=O C_re |v=mt p}iwQB x@ R=}vO}=@ TB "CU= umttQ}e =} Q@xv} Ry 'pmWt Q=}U@ |Qt= xv=NOwQ QU=QU R= u=tR=@ sy w |OQ@ Q=m sy xm OwW s=Hv= |}xvwo x@ xv=NOwQ R= xOv}q; C_re CW=OQ@,qt=m xv=NOwQ Qy QO |Q}oxR=Ov= x=oDU}= l} OwHw =Q} R OW=@ umtt xv} Ry u} QDtmR= xO=iDU= C}OwOLt =@ CU= XNWt xm Qw] u=ty =t= "CU= |rta w |k]vt=} |v=mt `@=D CQwY x@ =Q xOv}q; C_re O}=@ 'OQwt u}= QO |Q}PBTwmat x@W VwQu}= QO "O}W}Ov= |}xQ=J pmWt u}= |=Q@ O}=@ TB "OQm CW=OQ@ |rw] p}iwQB u=tyCOW `@=D R= xO=iDU= =@ w =yu=tR |xty QO x]kvl} R= xO=OCW=OQ@ Q=mD@= R= j}kLDu}= u=wD@ =D CU= xOW xDiQo xQy@ |v=mt p}iwQB uOQw; CUO x@ |=Q@ 'xOv}q; C_reCW=OQ@ |=H x@ =D OR=U|t QO=k =Q =t,qta 'xOW x�=Q= VwQ "OQm hQ]Q@ R}v =Q pmWt\ki 'CU= Q@ xv} Ry Q=}U@ |Qt= xm 'XNWt u=tR l} QO O=} R |=yx]kv R= xO=O"s}vm |Q=OQ@xO=O VHvU x=oDU}= l} \ki j} Q] R= w u=tR Qy QO x]kv l} R=|x�=Q= =@ |Q}PBTwmat x@W |xrO=at |}=yDv= \QW C}OwOLt `iQ VwQ K}[wDx=oDU}= QO w 1 pmW j@=]t |}xv=NOwQ QO xm CU= CQwY u}= x@ xO=U |r=Ft

"CW=OQ@ u=tR Qy QO =Q xOv}q; C_re u=wD|t 'P xHvU12 'xOv}q; C_re |Q}oxR=Ov= x=oDUO xO=O CW=OQ@ |v=tR xR=@ Qo = p=Ft |=Q@|r= Qy_ 12%00 Ca=U R= xm O=O KQW u}vJ u=wD|t 'OwW xDiQo Q_v QO Ca=UxOv}q; C_re |a@=D j@] ,=[Qi "CU= xOQm |Q=OQ@ xO=O x=oDUO 'QYa 24%00R= 'XNWt |DOt |] xOv}q; C_re w CU= xOW CW=OQ@ Ca=U 12 pw] QOC_re w 'CU= xDiQ}PB u=}=B TBU w xDi=} V}=Ri= |OvwQ |] 'xOW wQW QiYxDWPo |Q}oxR=Ov= |x]kv R= xOv}q; pta QO w xOW QiY Q@=Q@ xv=NOwQ QO xOv}q;R= |DWoQ@ pOt QO OwHwt |=yxO=O R= xO=iDU= |=Q@ xm CU= xrLQt u}= QO "CU=CUO x@ xOv}q; C_re COW `@=D xm CQwY u}= x@ 'OwW|t xO=iDU= xOQ@t=v VwQpOt QO |OwQw |RQt \QW u=wva x@ u=wD|t 'P |x]kv QO xHvU x=oDU}= R= xOt;s}kDUt pOt |HwQN ,=Da}@] "OQm xO=iDU= L =D P xR=@ QO 'p=kDv= |xrO=at s}kDUtu=wD|t xm 'Ow@ Oy=wN xOv}q; C_re |rw] p}iwQB 'L =D P xR=@ |=Q@ xOW h} QaDxO=iDU= |Q}PBTwmat x@W VwQ \UwD xOW x�=Q= |DWoQ@ pOt QO p}iwQB u}ty R=|=Q@ VwQ u}= R= u=wD|t |Q}PBTwmat x@W VwQ QO |D}OwOLt u}vJ `iQ =@ "OQm

"OQm xO=iDU= |OQ@ Q=m |Dr=L QO w u}}=B Q=}U@ |}xv} Ry =@ xv=NOwQ QO xO=iDU=

|OOa pOt "2"2|v=mt C=kDWt |=Q@ |RmQt p[=iD ?} QkD VwQ R= 9 |xrO=at |R=UxDUUo |=Q@|=}=Rt xrtHR= "CU= xOW xO=iDU= |v=tR jDWt |=Q@ uwUrm}v � lv=Qm |wor= R= wQO |QDnQR@ |v=tR s=o u=wD|t =Pr w CU= u; \QW w O}k|@ |Q=O}=B 'VwQ u}=|=Q=O u=tR QO sy w u=mt QO sy wor= u}= u}vJty "OQm Q=}DN= |}xv=}=Q |xt=vQ@TB [11]"CU= QiY Q@=Q@ R}v u; QO |OOa |oOvm =QB u=R}t w CU= wO xHQO CkO

%Ot; Oy=wNQO sQD x@ sQD |R=UxDUUo CQwY x@ 9 |xrO=at

@C@x

(0; t) = 0 (10)

@C@x

(L; t) = 0 (11)

@2C(0; t)@x2 = 0 (12)

@2C(L; T )@x2 = 0 (13)

C(x; t) = �(x) (14)

w |}=yDv= \QW �(x) '|R=Ux}@W u=tR pm T 'xv=NOwQ pw] L '=yv; QO xmOwW|t xOy=Wt xm u=vJ "CU= xv=NOwQ pw] pm R= xOv}q; C_re |rw] p}iwQB\QW 9 |xrO=at x@ xm |vat u}= x@ &CU= xOW xOQ@ s=v |}=yDv= \}=QW R= q=@ QOQ}}eD =@ =t= "OyO s=Hv= =Q |R=UR=@ |DWoQ@ CQwY x@ Ov=wD@ =D OwW|t xO=O |}=yDv=Qo}O pw=ODt pmW Ovv=t =D OQm p}O@D |}=OD@= \QW x@ =Q u; u=wD|t xO=U |Q}eDt w

%x@ OwW|t p}O@D |}=yDv= \QW Q}eDt w Q}}eD "OwW |�RH |r}Uv=Qi}O CqO=at� = T � t (15)

%R= CU= CQ=@a q=@ Q}eDt w Q}}eD =@ xOW |U} wvR=@ |x]@=Q

A@C@�

= @(QC)@x

� @@x

�AEx

@C@x

�� "@4C

@x4 (16)

%Ow@ Oy=wN Q} R CQwY x@ 16 |xrO=at |}=yDv= w |RQt \}=QW@C@x

(0; �) = 0 (17)

@C@x

(L; �) = 0 (18)

@2C(0; �)@x2 = 0 (19)

@2C(L; �)@x2 = 0 (20)

C(x; 0) = �(x) (21)

\UwD xrO=atu}=|x�=Q= R=Oa@ xOW QmPC=mv x@ xHwD =@ 6|xrO=atCW=O xHwDO}=@s=v |Q=O}=B sQD " @4C

@x2 '9 |xrO=at QO "CU= xOW|U} wvR=@ |Q}PBTwmat x@W VwQ|oOvm =QB � |}=Hx@=H |xrO=at |Q=O}=BEa=@ O};|t Q@ u; s=v R= xm Qw] u=ty w OQ=Ow OQ=O s=v |Q=O}=B?} Q[ " xrO=at u}= QO u}vJty "OwW|t |ivt |v=tR |=ys=o QOQ=JO =yMU=B |Q=O}=B?} Q[ Q}}eD =@ [18]"OQ=O =Q|rY=Vkv |Q=O}=B sQD QO ,=YNWt=y?=wH w xOW Q=O}=B=v =yMU=B QDsm |Q=Okt R= |Q=O}=B ?} Q[ Qo = w OvwW|t Q}}eDQiY x@ =yMU=B OwW QDW}@ |OL R= |Q=O}=B?} Q[ Q=Okt Qo = u}vJty 'OvwW|t =Qo =wu}}aD |=Q@|YNWt Q=}at xD@r= [11]"O=O Oy=wNv =Q K}LY MU=B ,=Da}@] w Ovm|t p}t'|oOvm =QB � |}=Hx@=H ?}=Q[ Q=Okt 'xm CW=O xHwD O}=@ w [18]OQ=Ov OwHw " Q=Oktu}}aD QO |Q}PBTwmat x@WVwQ\UwD OwHwt\}=QW|R=UR=@ |=Q@ |R=Ux}@Wu=tR|QDtm CkO =@ =yxO=O xJQy xm CU= |y}O@ w [16]CU= QFwt |Q=O}=B ?} Q[ Q=Okt'QDpmWt|Q}PBTwmat x@WVwQ\UwD OwHwt\}=QW|R=UR=@ 'OvW=@ xOWCW=OQ@

"Ow@ Oy=wN |QDsm CkO =@ w QDW}@ C}OwOLt |=Q=O |Q=O}=B ?} Q[ R= xO=iDU=

|Q}PBTwmat x@W VwQ |}=yDv= \QW C}OwOLt `iQ "1"2'OW xQ=W= |DWoQ@ |xrO=at |x}rw= w |RQt \}=QW |x�=Q= CtUk QO xm Qw] u=ty

106

Page 5: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

1|xQ=t

W'35

�3|xQ

wO'�13

98Q=y@

�h}QW

l}v=mt

|UOvyt ��

� 4"2�x4 �

(AE)i+ 122�x2 + Qi+14�x2

�Cji+1+

���� 6"

2�x4 +(AE)i+ 122�x2 +

(AE)i� 122�x2

�Cji +

���Qi+14�x2 � Qi�1

4�x2 + Ai��

�Cji +

��� 4"2�x4 �

(AE)i� 122�x2 � Qi�14�x2

�Cji�1+

��� "

2�x4

�Cji�2 (24)

\}=QW u=ty w XNWt C_re uwDU w swrat Q=OQ@ C::::j 24 |x]@=Q QOu}vJty 'OwW|t OQ=w |}=yDv= \}=QW u=wva x@ |DWoQ@ |xrO=at QO xm CU= x}rw="O};|t CUO x@ |DWoQ@ pL R= xm CU= |D_re uwDU w CqwyHt Q=OQ@ Cj+1

::::

x@ xHwD =@ "Ov=xOW |iQat p@k \@=wQ w CqO=at QO R}v =yQDt=Q=B w T}Ov= Qo}OsQD x@ xHwD =@ w ?}=Q[ T} QD=t p}mWD |=Q@ xm OwW|t xOy=Wt 24 |x]@=QTB "s} Q=O pwyHt n O=OaD xm |r=L QO OQ=O OwHw xrO=at n� 4 'sQ=yJ jDWtO}=@ Qw_vt u}O@ "OwW CqwyHt Q@=Q@ CqO=at O=OaD =D CU= sRq Qo}O |xrO=at 4x@ xHwD =@ "OQm xO=iDU= 'Ovm|t CqO=at Q@=Q@ =Q CqwyHt O=OaD xm |RQt \QW 4 R=?}=Q[ T} QD=t QO xDiQ Q=m x@ \@=wQ '24 |x]@=Q w xOW xDiQo Q_v QO |RQt \}=QW

%R= Ow@ Oy=wN CQ=@aa11 = �1; a12 = 1::::a1n = 0a21 = 1; a22 = �2; a23 = 1;a24 = 0::::a2n = 0 a31 = ��

�"

2�x4 );

a32 = ���� 4"

2�x4 +(AE)3� 122�x2 � Q3�1

2�x2

�;

a33 = ��� 6"2�x4 �

(AE)3+ 122�x2 � (AE)3� 122�x2 �Q3+14�x2 + Q3�1

4�x2 + A3��

�;

a34 = ���� 4"

2�x4 +(AE)3+ 122�x2 � Q3+1

4�x2

�;

a35 = ���

"2�x4

�a41 = 0; a42 = ��

�+ "

2�x4

�;

a43 = ���� 4"

2�x4 +(AE)4+ 122�x2 � Q4�1

4�x2

�;

a44 = ��� 6"2�x4 �

(AE)4+ 122�x2 � (AE)4� 124�x2 �Q4+14�x2 + Q4�1

4�x2 + A4��

�;

a45 = ���� 4"

2�x4 +(AE)4+ 122�x2 � Q4+1

4�x2

�;

a46 = ���

"2�x4

�; a51 = 0:::: (25)

x�=Q= \@=wQ x@ xHwD =@ "OQm pta u=wD|t x@=Wt Qw] x@ R}v ?}=Q[ Qo}O |=Q@

A@C@��= Aji

�Cj+1i � Cji

��

�@(QC)@x

�=(QC)j+

12i+ 12� (QC)j+

12i� 12

�x

@@x

�AEx

@C@x

��=

�(AEx)i+1C

j+ 12i+1

��x2

�2�(AEx)iC

j+ 12i

��x2 +

�(AEx)i�1C

j+ 12i�1

��x2

"@4C@x4 �=

"�Cj+ 12

i+2 � 4Cj+ 12i+1 + 6Cj+ 12

i � 4Cj+ 12i�1 + Cj+

12i�2

�x4

�(22)

|xOvQ=tW j w u=mt |xOvQ=tW i u}vJty w '|v=mt s=o �x '|v=tR s=o ��

%Ow@ Oy=wN 23 |x]@=Q j@=]t xrO=at u}= |rm |R=UxDUUo "CU= u=tR

Aji

�Cj+1i � Cji

��

�=

(QC)j+12

i+ 12� (QC)j+

12i� 12

�x

��(AEx)i+1C

j+ 12i+1

��x2 +

2�(AEx)iC

j+ 12i

��x

��(AEx)i�1C

j+ 12i�1

��x2

�"�Cj+ 12

i+2 � 4Cj+ 12i+1 + 6Cj+ 12

i � 4Cj+ 12i�1 + Cj+

12i�2

�x4

�(23)

22 |x]@=Q pmW x@ |R=UxDUUo QO xOW x�=Q= |=y|wor= x@ xHwD =@CqwyHt Q=OQ@ |x@U=Lt |=Q@ xOt; CUO x@ \@=wQ w ?}=Q[ T} QD=t |x�=Q==@ TB "Ovm|t xDiQ Q=m x@ |OOa |wor= swyit QDy@ xJQy lQO x@ |v=}=W ltm|=Q@ 'xOW x�=Q= 22 |x]@=Q QO xm 9 |xrO=at |wQ xDiQo s=Hv= |R=UxDUUo x@ xHwD|R=UxDUUo QO xmCU= QmP u=}=W 'CU= R=}v 24\@=wQ x@?}=Q[T} QD=t uDN=Uu=tR QO xOv}q; C_re uwDU w ?}=Q[ AJ CtU QO xOW xDUUo CqO=at 'Q} RC_re uwDU?}=Q[CU=QCtU QO u}vJty "CU= CqwyHt R= xm Ow@ Oy=wN p@k"CU= |Q}PBTwmat x@W VwQ \UwD |D=@U=Lt u}vJty w CU= swrat xOv}q;

���

+ "2�x4

�Cj+1i+2 +

���� 4"

2�x4 �(AE)i+ 122�x2 � Qi+1

4�x2

�Cj+1i+1 +

��� 6"2�x4 �

(AE)i+ 122�x2 � (AE)i� 122�x2

�Cj+1i +

���� Qi+1

4�x2 + Qi�14�x2 + Ai

��

�Cj+1i +

���� 4"

2�x4 +(AE)i� 122�x2 + Qi�1

4�x2

�Cj+1i�1 +

���

+ "2�x4

�Cj+1

1�2 =

���� "

2�x4

�Cj1+2+

107

Page 6: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

"""|oOvm

=QB�|}=H

x@=H|xr

O=atTw

matpL"|[Qi xv=NOwQ C=YNWt "1 pwOH

=yQDt=Q=B xv=NOwQ C=YNWtQDtwr}m 24 xv=NOwQ pw]QDt 100 |v=mt |=ys=o0 035 nv}v=t |Q@ R ?} Q[

x}v=F Q@ ?amt QDt 100 C@=F |@O CUOq=@ QO |RQt \QWQDt QO QDt 0 030 u=} QH \N ?}W CUO u}}=B QO |RQt \QW

Ca=U 12 |R=Ux}@W u=tR

umtt ZQi u}= "CU= CN=wvm} `]kt QO C_re `} RwD xm CU= u}= Q@ ZQiO}=@ xD@r= "OW=@v jO=Y 'xv=NOwQ uOw@ Z} Qa Cr=L QO ,qFt '|]}=QW CLD CU=x@ C@Uv |Oa@l} Cr=L QO CqO=at |oO}J}B w |D=@U=Lt Q=@ xm CW=O u=}@QO |[Qi |=yp=Ft |=Q@ [24]"CU= QDpyU w QDsm |Oa@xU w |Oa@ wO |=yCr=LQ_v QO s_vt=v |[Qa `]=kt =@ w QDtwr}m 24 pw] x@ |}xv=NOwQ '|Oa@ l} Cr=L

"�1 pwOH�OwW|t xDiQou=} QH wv R= Q_v OQwt |xv=NOwQ QO xOt; OwHw x@ u=} QH '2 pmW x@ xHwD =@|=yQ}eDt '|Oa@l} Cr=L QO u=} QH CqO=at pL =@ "CU= CN=wvm} Q}e w Q=oOv=tQ=oOv=t Q}O=kt ?}DQD x@ pmW u}= QO "O};|t CUO x@ p=Ft u}= |=Q@ u=} QH =@ \@DQtxOW x@U=Lt |oOvm =QB ?} Q[ u}vJty w `]=kt K]U |rw] p}iwQB 'CaQU '|@OxOW xO=O u=Wv xv=NOwQ pw] QO �28 |x]@=Q� [25]u=Q=mty w nvO |x]@=Q \UwD

%CU=

DL = 0 158"t0

U 2B 53

H23U�

(28)

"t0 = 0 145 + 13520

�UU�

��BH

�1 38(29)

\UwDt ZQa B '\UwDt CaQU U '|rw] |oOvm =QB ?} Q[DL 'u; QO xm|oOvm =QB ?} Q[ "CU= |WQ@ CaQU R}v U� w `]kt \UwDt jta H '`]kt&OwW|t x@U=Lt u=} QH CqO=at pL R= xOt; CUO x@ |=yQDt=Q=B R= xO=iDU= =@|=yVwQ u}}=B CkO x@ u=wD|t |oOvm =QB ?} Q[ |x@U=Lt |=yC}OwOLt R='xDiQo Q=Qk xO=iDU= OQwt OOaDt Cq=kt QO xm|}=yVwQ xrtHR= "OQm xQ=W= |D=@U=LtxOw@ u}kkLt R= |Q=}U@ O=tDa= OQwt xm OQm xQ=W= u=Q=mty wnvO |x]@=Q x@ u=wD|t|oOvm =QB ?} Q[ |x@U=Lt |=Q@ u=Q=mty w nvO VwQ R= R}v Q=DWwv u}= QO "CU=xO=iDU= R}v |Q=t; |=yVwQ Q}_v Qo}O |=yVwQ R= Qw_vt u}O@ "CU= xOW xO=iDU=pwtQi x@ u=wD|t =ypwtQi u}= xrtHR= "CU= xOt; CUO x@ R}v |}=ypwtQi w xOWC=@U=Lt x@ xHwD =@ xm [26]OQm xQ=W= Qvwmr=i w QwB|iW=m |xar=]t QO xOW x�=Q="CU= xDW=O |rw@k p@=k |oDU@ty ?} Q[ |ak=w |=yxO=O =@ xU}=kt w xOW s=Hv=|QDq=@ CkO ?D=Qt x@ |}=}t}W |=yVwQ \UwD xOt; CUO x@ |oOvm =QB ?} Q[QDq=@ ?D=Qt x@ |}=}t}W |=yVwQ |xv} Ry =t= 'OQ=O OwHwt |=ypwtQi x@ C@Uv=@ \ki K[=w Qw] x@ |D=@U=Lt |=yVwQ "OQ=O |QDW}@ C=v=mt= x@ R=}v w CU=

"CU= QU}t u=} QH |=yQDt=Q=B |NQ@ uDW=O Q=}DN= QO|=yxO}OB QF= QO Ov=wD|t p=}U l} QO xOv}q; |xO=t l} C=QP uOW xOQDUo'|}=Hx@=H "OW=@ |oOvm =QB w s]qDt |oO}WNB '|rwmrwt |oO}WNB '|}=Hx@=HQO |rwmrwt |oO}WNB "CU= p=}U u=} QH CmQL QF= QO xOv}q; |xO=t C=QP p=kDv=R= xO=iDU= =@ u; |[=} Q u=}@ w OwW|t s=Hv= p=}U |=ypwmrwt |iO=YD C=mQL QF=|oO}WNB QF= QO |tQH Q=W 'uwv=k u}= T=U= Q@ &OQ}PB|t CQwY l} R}i pw= uwv=k

%x@ OwW|t p}O@D C=twrat Q=OQ@ w CqwyHt Q=OQ@ w ?}=Q[ T} QD=t 'xOW266666666666666666666666664

a11 a12 : : : 0a21 a22 a23 : : : 0a31 a32 a33 a34 a35 0 : : : 00 a42 a43 a44 a45 a46 0 : : : : : : 0...

. . .............

0 : : : an�1;n ann

377777777777777777777777775(26)

%Ot; Oy=wNQO Q} R CQwY x@ C=twrat w CqwyHt Q=OQ@266666666666664

C1C2.........

Cn

377777777777775

266666666666664

b1b2.........

bn

377777777777775(27)

u}@ xOW O=H}= |x]@=Q wv CU= XNWt 25 |=yT} QD=t QO xm Qw]u=tyCqwyHt Q=OQ@ x C=twrat Q=OQ@ b xm CU= ax = b |xrO=at CQwYx@ =yT} QD=txO=O u=Wv 26 |x]@=Q QO xm CU= ?}=Q[ T} QD=t a w CU= C1; C2:::Cn xmCUO x@ t1; t2:::tn |=yu=tR QO C1; C2:::Cn |rm CQwY x@ w CU= xOW

"O};|t

EL@ w G}=Dv "3l} =OD@= '|Oa@l} Cr=LQO p=kDv= |xrO=at Twmat pOt |HvUCLY |=Q@pL =@ TBU "OwW|t xDiQo Q_v QO s_vt=v |[Qa `]=kt =@ |[Qi |xv=NOwQ=yjta pt=W |=yQDt=Q=B 'swrat |RQt \}=QW =@ |Oa@l} Cr=L QO u=} QH CqO=atw OwW|t pL ,=t}kDUt p=kDv= |xrO=at TBU "O};|t CUO x@ : : : w =yCaQU w|xrO=at |=Q@ Twmat pL |OwQw u=wva x@ p=kDv= |xrO=at s}kDUt pOt |HwQN

"OQ}o|t Q=Qk |UQQ@ OQwt |Oa@l} Cr=L QO p=kDv=Q=m x@ |Oa@ l} CQwY x@ xv=NOwQ QO p=kDv= w u=} QH CqO=at xm |DkwZQi u=Um} xv=NOwQ |[Qa `]kt QO C_re w CaQU |=yC}tm `} RwD 'OwQ|t|Q}o p=QoDv= R= |a]kt u}ov=}t CQwY x@ |Oa@l} CqO=at Ck}kL QO "OwW|tQF-wt pt=wa u} QDsyt "O};|t CUO x@ xv=NOwQ |[Qa `]kt QO |Oa@xU CqO=atCU= QDU@ |Q@ R \}=QW w |i=QowB wD 'xUOvy 'xv=NOwQ QO u=} QH uOw@ |Oa@l} QO\}=QW w u=} QH \}=QW 'xOv}q; p=kDv= uOw@ |Oa@l} Q@ =Q Q}F-=D u} QDW}@ u}vJty w'OwW|t xDiQo Q_v QO |Oa@l} CQwYx@ p=kDv= |xrO=at xm |Dkw "OQ=O xOv}q; `@vt

108

Page 7: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

1|xQ=t

W'35

�3|xQ

wO'�13

98Q=y@

�h}QW

l}v=mt

|UOvyt

|OwQw RQt QO �|[Qi COW `@=D� xOv}q; C_re |v=tR |QU |vLvt "3 pmW"�pw= p=Ft�

=@ "CU= CN=wvm} Q}e w Q=oOv=t u=} QH =Q} R OwW|t xOy=Wt u=mt x@ C@Uv u=} QH?} Q[ ,=YNWt u=} QH wv T=U=Q@ 'p@k CtUk QO xOW xO=O C=L}[wD x@ xHwD

"CU= Q}eDt u=mt x@ C@Uv R}v |oOvm =QB

�pw= p=Ft� xv=NOwQ QO |Q}PBTwmat x@W VwQ OQ@ Q=m "1"3VwQ |x}rw= \QW |=[Q= |=Q@ xv=NOwQ QO xOv}q; C_re |v=mt p}iwQB CW=OQ@|=Q@ O}=@ TB "CU= s=Hv= p@=kQ}e ,qwtat w Q@xv} Ry Q=}U@ |Qt= |Q}PBTwmat x@W

"�3 pmW�O}W}Ov= |}xQ=J |rmWt u}vJ `iQOwHw |v=Uv= xJ w |y=oDUO xJ '=]N uwO@ |Q=OQ@xO=O xvwoI}y ,=YNWthrDNt |=yxO=O |Q}oxR=Ov= R= xOt; CUO x@ G}=Dv w C=@U=Lt |=Q@ xW}ty w OQ=Ov|=yx=oDUO x@ xHwD =@ � OQ=O X=YDN= xOv}q;C_reCW=OQ@ x@ j}kLD u}= QO xm �uOW QDl}ORv xJQy |=Q@ TB "OvQ}o|t Q_v QO =Q |}=]N u=R}t |Q}oxR=Ov= hrDNtu=ty =} xOW |Q}oxR=Ov= |=yxO=O x@ =]N |Q=Okt 'C}ak=w x@ Q_v OQwt |xr�UtxOQ=w |=]N u=R}t xD@r= [18]"OwW|t OQ=w s}kDUt pL R= xOt; CUO x@ |=yxO=O|k]vt "OvW=@ C}ak=w x@ l}ORv |Di=} QO |=yMU=B =D OW=@ |k]vt OL QO O}=@|=]N u=R}t =Q} R OW=@v QDlJwm |OL R= =} QDW}@ |OL R= |va} =]N u=R}t uOw@"CU= pw@k p@=k |OL =D \ki |v=Uv= |=]N =} |Q}oxR=Ov= |=yx=oDUO QO OwHwtx@ "CU= |O=} R Q=Okt xm xOW xDiQo Q_v QO OYQO 15 =]N u=R}t j}kLD u}= QOpL |=yMU=B Q@ |Q}F-=D xJ 'OW=@ O=} R QOkQy =]N xm OwW|t xOy=Wt ?}DQD u}=

"CW=Po Oy=wN Twmat=]N R= |Q=a w =]N uwO@ |}xO=O CW=OQ@ I}y 'CU= XNWt xm Qw] u=ty's}kDUt pL R= xOt; CUO x@ Q=Owtv x@ C}ak=w x@ xr�Ut uOW QDl}ORv |=Q@ "CU}vxOv}q; C_re |v=mt p}iwQB "�? 4 pmW� CU= xOW p=ta= OYQO 15 |=]N

"Ow@ Oy=wN 1 pmW j@=]t L =D P xR=@ QOx@ xOv}q; C_re |v=mt p}iwQB |DWoQ@ OvwQ xm OwW|t xOy=Wt 5 pmW QO|v=tR |QU Q=Owtv QO xm Qw]u=ty "CU= s=Hv= p=L QO |@ wN x@ |OwQw RQt CtUQO |DWoQ@ |=yQ=Owtv 'CU= uQ=kDt=v |FrFt CQwY x@ xOv}q; |Q=PoQ=@ |OwQw RQt

"OvQw;|t CUO x@ uQ=kDt=v Cr=L u=tR wO QyC_re |v=tR |QU Q=Owtv |R=UR=@ OvwQ 'OwW|t xOy=Wt 6 pmW QO xm u=vJ=@ Cr=L wO Qy QO |DWoQ@ pOt \UwD |rw@k p@=k Qw] x@ |OwQw RQt QO xOv}q;'xrk Gw= |x]kv |R=UR=@ QO |k]vt Q=Okt =D \ki w xDiQo s=Hv= OYQO 15 |=]N|=Q@ OwW QDW}@ =]N Q=Okt xJQy xm CW=O xHwD O}=@ "CU= xOW pmWt Q=JO pOt|}=Qo =w R= =D OQm xO=iDU= |QDW}@ |Q=O}=B ?} Q[ Q=Okt R= O}=@ pOt |R=UR=@ Ow@y@

"Oa@ |@ Cr=L QO u=} QH pOt |HwQN "2 pmW

?} Q[ '?U=vD u}= ?} Q[ &CU= ?U=vDt C_re |v=mt u=}O=Qo =@ |rwmrwt[4]"OQ=O s=v |rwmrwt |oO}WNB

Q}F-=D "OQ=O |oDU@ p=}U C=}YwYN x@ \ki |rwmrwt |oO}WNB ?} Q[OQ}o|t CQwY s]qD |xO}OB p}rO x@ p=}U C=QP C=mQL QF= Q@ xm s]qDt|oO}WNBu=} QH C=}YwYN x@ s]qDt |oO}WNB "CU= |rwmrwt |oO}WNB R= QDnQR@ Q=}U@[27]"OQ=O x]@=Q xO}OB |xYNWt pw] swU Q=yJ u=wD =@ ,=@} QkD w OQ=O |oDU@ p=}Uj}kLD u}= QO xOW x�=Q= |=yQ=Owtv |t=tD xm CU= xHwD p@=k xDmv u}= QmP

"CU= xOW x�=Q= Oa@|@ CQwY x@ G}=Dv QD|twyit w QDy@ xJQy |x�=Q= |=Q@u=} QH p}rO x@ xm xOW xO=O u=Wv u=mt x@ C@Uv |@O C=Q}}eD hr= 2 pmW QOC=Q}}eD ? 2 pmW QO u}vJty 'CU= C@=F u=tR w u=mt x@ C@Uv |@O Q=oOv=t`]kt K]U C=Q}}eD G 2 pmW j@=]t &OwW|t xOy=Wt u=mt x@ C@Uv CaQU

109

Page 8: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

"""|oOvm

=QB�|}=H

x@=H|xr

O=atTw

matpL

" xL = 0 21 QO s}kDUt pOt R= xOv}q; C_re |v=tR |QU Q=Owtv �pw= p=Ft� "4 pmW

"=]N OYQO 15 p=ta= =@ xOv}q; C_re |DWoQ@ |v=mt p}iwQB �pw= p=Ft� "5 pmW

=]N C=a@ Qt pk=OL Q}O=kt 2 pwOH |x�=Q= =@ "OwW |Q}owrH =yMU=B uOW CQB w'|Q=O}=B ?} Q[ hrDNt Q}O=kt w =]N hrDNt Cq=L QO |@Uv |=]N OYQO w

"CU= xOW VqD |DWoQ@ pOt R= xrY=L G}=Dv QDy@ lQO |=Q@

�swO p=Ft� xv=NOwQ QO |Q}PBTwmat x@W VwQ OQ@ Q=m 2"3pmW QO xOv}q; C_re |v=tR |QU |wor= p=Ft u}= |=Q@ xOW xDiQo Q_v QO |wor=p=kDv= |xrO=at Twmat pL |xv}tR |=Q@ u}kkLt R= |Q=}U@ \UwD xm CU= 7C_re |v=tR |QU s}kDUt pL |=yQ=Owtv ?}DQD x@ "CU= xOW xDiQo Q=m x@ xOv}q;|v=tR w |v=mt p}iwQB |DWoQ@ |=yQ=Owtv w =]N uwO@ w =]N hrDNt =@ x=Qty

"OW Ovy=wN x�=Q= �9w8 pmW�T-=Q u}tND QO |Q}PBTwmat x@W |DWoQ@ pOt CU= XNWt Q=Owtv QOQo}O T-=Q |@ wN x@ xDUv=wDv w xOW pqDN= Q=JO |UwULt OL =D QDlJwm

OYQO 15 w �\N� xOv}q; C_re |v=tR |QU Q=Owtv xU}=kt �pw= p=Ft� "6 pmW"(" = 2 2� 106) �u}J \N� =]N

110

Page 9: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

1|xQ=t

W'35

�3|xQ

wO'�13

98Q=y@

�h}QW

l}v=mt

|UOvyt "�pw= p=Ft� hrDNt |=yCr=L QO =]N |=yXN=W "2 pwOH

"2 "1 XN=W�15 �5 �0 �15 �5 �0 QDt=Q=B

99 75 99 82 99 91 99 69 99 84 99 93 R2���9 03 5 08 3 02 9 12 5 02 3 01 9 MRE ���mean relative error9

"�swO p=Ft� hrDNt |=yCr=L QO =]N |=yXN=W "3 pwOH"2 "1 XN=W

�15 �5 �0 �15 �5 �0 QDt=Q=B99 43 99 35 99 74 99 35 99 41 99 79 R2���9 77 7 61 5 89 9 63 7 78 5 98 MRE ���

OwW|t \=@vDU= xvwo u}= CU= xDiQo OwN x@ |QDsQv pmW w OvR@ u}tND =Q u;=@ w xDW=O |Wy=m TBU w |W}=Ri= OvwQ |tm Q=}U@ u=tR QO C_re uwJ xmxDUv=wDv |DWoQ@ pOt '|R=Ux}@W QO xOW xDiQo Q_v QO |v=tR |=ys=o x@ xHwDsOa QO |oOvm =QB ?} Q[ |Q=PoQF= u}vJty "Ovm |R=UR=@ =Q OvwQ |@ wN x@Twmat pOt |=Q@ |DqmWt w xOw@ TwULt ,qt=m lJwm xrk pt=m |R=UR=@pOt |=yMU=B OvwQ QO |oOvm =QB ?} Q[ |=RU x@ QF= Qou=Wv xm xOQm O=H}=sQD 'xv=NOwQ QO xOv}q; VNB s=ovy =Q} R CU= |Q}PBTwmat x@W VwQ |DWoQ@lJwm T-=Q OwW|t Qw;O=} "OwW|t xOv}q; |oOvm =QB w uOWR=@ ?@U |oOvm =QBO=H}= |rqDN= Twmat pOt \UwD |DWoQ@ |=yxO=O |R=UR=@ s=ovy w xOW R=@|v=tR |wor= QO lJwm |=yT-=Q pt=m |R=UR=@ x@ QO=k Twmat pOt w OwW|t'xO=O CW=OQ@ |v=tR |=ys=o QO C}OwOLt x@ xHwD =@ w Ow@ Oy=wNv |Q=PoQ=@QO xOv}q; |Q=PoQ=@ |v=tR |wor= QO Qo = xm OyO|t u=Wv Qt= u}= C}ak=w QOx@ |v=tR xR=@ QO lJwm |U-=Q Kq]Y= QO w |Wy=m w |W}=Ri= OvwQ 'xv=NOwQ"Ovm |R=UR=@ =Qu; |@ wN x@ Ov=wD|tv |Q}PBTwmat x@W |DWoQ@ pOt O}; OwHwQO TwULt CQwY x@ QwmPt T-=Q R= |}=yxv=Wv xm OwW|t xOy=Wt sy R=@ =t=xm OwW|t xOy=Wt u}vJty "CU= xOWv s=Hv= ,qt=m |R=UR=@ w CU=O}B Q=OwtvxJQy =Q} R CU= QDW}@ swO p=Ft QO |Q=O}=B ?} Q[ |=Q@ xOW xDiQo Q_v QO ?}=Q[|NU=B x@ uO}UQ |=Q@ O}=@ OW=@ QDxO}J}B xv=NOwQ QO |Q=PoQ=@ |v=tR |wor= wvpwOH� O=O V}=Ri= =Q |Q=O}=B ?} Q[ Q=Okt =yMU=B |}=Qo =w R= |Q}owrH w QD?wr]t

"�3

|Q}oxH}Dv "4Q=oOv=t u=} QH \}=QW CLD 'xv=NOwQ QO |Q}PBTwmat x@W VwQ Q[=L j}kLD QO"OW x�=Q= Oa@|@ CQwY x@ =yQ=Owtv w CiQo Q=Qk uwtR; OQwt CN=wvm} Q}e wQO xOv}q; C_re |Q=PoQ=@ |v=tR |wor= wv |=Q@ p=Ft wO ZQi =@ G}=Dv u}vJtyVwQ |=yQ=Owtv xU}=kt w xOv}q; |DWoQ@ u=mt p}iwQB |=yQ=Owtv CLD xv=NOwQxJQy xm OyO|t u=Wv xrY=L G}=Dv "OW x�=Q= =]N OYQO 15 =@ |Q}PBTwmat x@Wp}rO x@ 'OW=@ QDmJwm |U-=Q pmW |=Q=O w QDxO}J}B xOv}q; |Q=PoQ=@ |v=tR |wor=xJQy u}vJty "OwW|t Xkv Q=JO |DWoQ@ pOt |oOvm =QB sQD |oOvvm VNB QF=

|OwQw RQt QO �|[Qi COW `@=D� xOv}q; C_re |v=tR |QU |vLvt "7 pmW"�swO p=Ft�

" xL = 0=21 QO s}kDUtpOt R= xOv}q;C_re|v=tR|QUQ=Owtv �swO p=Ft� "8pmW

111

Page 10: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

"""|oOvm

=QB�|}=H

x@=H|xr

O=atTw

matpL

"=]N OYQO 15 p=ta= =@ xOv}q; C_re |DWoQ@ |v=mt p}iwQB �swO p=Ft� "9 pmWQ=Okt |Q=O}=B ?} Q[ |=Q@ O}=@ OW=@ |QDW}@ |oO}J}B |=Q=O |Q=PoQ=@ |v=tR |wor=

"CiQo Q_v QO |QDuwRi=x@W VwQ |xrO=at |x}rw= \QW C}OwOLt `iQ |=Q@ |WwQ Q=DWwv u}= QOVwQ u}= QO "OQ@ xQy@ xv=NOwQ QO QwmPt VwQ R= u=wD@ =D OW x�=Q= |Q}PBTwmatxv=NOwQ QO x]kv l} R= u=tR |] QO xOv}q; C_re |=yxO=O CW=OQ@ =@ \kiQR VwQ R= xO=iDU= u}vJty "Ci=} CUO QR |xrO=at |x}rw= \QW x@ u=wD|t|=yC}OwOLt R= |m} xm xv=NOwQ QO CN=wvm} Q}e w Q=oOv=t u=} QH \}=QW QO'Ow@ u=kkLt |=Q@ �CqO=at ?}=Q[ Q}O=kt uOw@ QDW}@ p}rO x@� u}W}B |=yj}kLDQO |Q}PBTwmat x@W VwQ |rm Cr=L QO "CiQo Q=Qk uwtR; OQwt Q=DWwv u}= QOQ=oOv=t u=} QH \}=QW =@ xv=NOwQ QO xOv}q; C_re |Q=PoQ=@ |v=tR |wor= |R=UR=@

"CU= xOQm pta jiwt CN=wvm} Q}e wOYQO 15 w �\N� xOv}q; C_re |v=tR |QU Q=Owtv xU}=kt �swO p=Ft� "10 pmW

"(" = 2=4� 106) �u}J \N� =]N

=yCWwv=B1. inverse problem2. quasi reversibility method3. advection4. dispersion5. di�usion equation6. regularization method7. heat conduction8. forth-order stabilization term

�References� `@=vt1. Chapra, S.C. Surface water quality modeling, New York:

McGraw-Hill, pp.389-564 (1997).

2. Neupauer, R.M., Borchers, B. and Wilson, J.L. \Com-parison of inverse methods for reconstructing the releasehistory of a groundwater contamination source", WaterResources Research, 36(9), pp. 2469-2475 (2000).

3. Atmadja, J. and Bagtzoglou, A.C. \Pollution sourceidenti�cation in heterogeneous porous media", WaterResources Research., 37(8), pp. 2113-2125 (2001).

4. Wang, Z. and Liu, J. \Identi�cation of the pollutionsource from one-dimensional parabolic eqation mod-els", Applied Mathematics and Computation, 219(8), pp.3403-3413 (2012).

5. Hamdi, A., Mahfoudhi, I. and Rejaiba, A. \Identi�cationof time active limit with lower and upper bounds of to-tal amount loaded by unknown sources in 2D transport

112

Page 11: Ovy oOvm=QB Hx H |xrO=at Twmat pL Ov U W CyH NOwQ QOsjme.journals.sharif.edu/article_20836_f09284a381283e... · 2020-06-14 · Q =y ' h} Q W l} v =m t | U Ovy t |xQ =t |xQwO Article

1|xQ=t

W'35

�3|xQ

wO'�13

98Q=y@

�h}QW

l}v=mt

|UOvyt equations", Journal of Engineering Mathematics, 97(1),

pp. 101-117 (2015).6. Hamdi, A. \Inverse source problem in a 2D linear evolu-

tion transport equation: detection of pollution source",Inverse Problems in Science and Engineering, 20(3), pp.401-421 (2012).

7. Hamdi, A. and Mahfoudhi, I. \Inverse source problemin a one-dimensional evolution linear transport equationwith spatially varying coe�cients: application to surfacewater pollution", Inverse Problems in Science and En-gineering, 21(6), pp. 1007-1031 (2013).

8. Molson, J.W. and Frind, E.O. \On the use of meangroundwater age, life expectancy and capture probabil-ity for de�ning aquifer vulnerability and time-of-travelzones for water protection", Journal of Contaminant Hy-drology, 127(1-4), pp. 76-87 (2012).

9. Cupola, F. and Tanda, M. and Zanini, A. \Laboratorysandbox validation of pollutant source location meth-ods", Stochastic Enviromental Research and Risk Assess-ment, 29(1), pp. 169-182 (2015).

10. Denche, M. and Bessila, K. \A modi�ed quasi-boundaryvalue method for ill-posed problems", Journal of Mathe-matical Analysis and Applications, 301(2), pp. 419-426.(2005).

11. Lattes, R., and Lions, J., The Method of Quasi-Reversibility: Applications to Partial Di�erential Equa-tions, Elsevier Sci., New York, 16(2), pp.236-247 (1969).

12. Ismail-Zadeh, A.T., Korotkil, I.A. and Tsepeler, A.I.\Three-dimensional numerical simulation of the in-verse problem of thermal convection using the quasi-reversibilitiy method", Water Rsources, 46(12), pp.2176-2186 (2006).

13. Yang, F., Fu, C. and Li, X.\Identifying an unknownsource in space-fractional di�usion equation", ActaMathematica Scientia, 34(4), pp. 1012-1024 (2014).

14. Zhang, T. and Chen, Q. \Identi�cation of contaminantsources in enclosed environments by inverse CFD mod-eling", Indoor Air, 17(3), pp. 77-167 In press. (2007).

15. Xiong, X.-T., Fu, C.-L. and Qian, Z. \Two numericalmethods for solving a backward heat conduction prob-lem", Applied Mathematics and Computation, 179(1),pp. 370-377 (2006).

16. Bagtzoglou, A.C. and Atmadja, J. \Marching-jury back-ward beam equation and quasi-reversibility methodsfor hydrologic inversion: application to contaminant

plume spatial distribution recovery", Water ResourcesResearch, 39(2), pp.156-164 (2003).

17. Bagtzoglou, A.C. \Application of particle methods of re-liable Identi�cation of groundwater polloution sources",Water Resources Management, 6(1), pp. 15-23 (1992).

18. Skaggs, T.H. and Kabala, Z. J. \Recovering the historyof a groundwater contaminant plum: Method of quasi-reversibilitiy", Water Resources., 31(16), pp. 2669-2673.(1995).

19. Wilson, J.L. and Liu, J. \Backward tracking to �nd thesource of pollution", Water Manage. Risk Remed, 1, pp.181-199 (1994).

20. Dorroh, J. R. and Ru, Xeuping. \The application ofthe method of quasi-reversibility to the sideways heatequation", Journal of Mathematical Analysis and Appli-cations, 236(6), pp. 503-519 (1998).

21. Zhang, T. and Wang, Li. H. \Identi�cation of particu-late contaminant source locations in enclosed spaces withinverse CFD modelling", 12th International Conferenceon Indoor Air Quality and Climate 2011, 1, pp. 667-672(2011).

22. Ghane, A., Mazaheri, M. and Mohammad Vali Samani,J. \Location and release time identi�cation of pollutionpoint source in river networks based on the backwardprobability Method", J Environ Manage, 180, pp. 164-171 (2016).

23. Qian, A. and Mao, J. \quasi-reversibility regularizationmethod for solving a backward heat conduction prob-lem", American Journal of Computational Mathematics,01(03), pp. 159-162 (2011).

24. Mazaheri, M., Mohammad Vali Samani, J. and Samani,H.M.V. \Mathematical model for pollution source iden-ti�cation in rivers", Environmental Forensics, 16, pp.310-321 (2015).

25. Deng, Z. Q., Singh, V. P. and Bengtsson, L. \Longitudi-nal dispersion coe�cient in straight rivers", Journal ofHydraulic Engineering, 127(11), pp. 927 -919 (2001).

26. Kashe�pour, S. M. and Falconer, R.A. \Longitudinaldispersion coe�cients in natural channels", Water Re-search, 36(6), pp. 1608 -1596 (2002).

27. Lewis, F.R. \Atmospheric di�usion shown on a distance-neighbour graph", Proceedings of the Royal Society ofLondon. Series A, Containing Papers of a Mathematicaland Physical Character, 110(756), pp. 709-737 (1926).

113