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DOI:10.24200/J30.2018.1349
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|=yxQ=w}O =@ T=tD QO p=}U RQt Q@ �|HQ=N p=tQv� Owta |xm} Q=OQ@ n 'u; QO xmp=}U =@ T=tD QO ?rY |xR=U ?=DW a Q=OQ@ w �1 pmW� CU= (�s) ?rY'|vwDU}B R=UGwt Qw=Ht RQt QO \ki 'lD Gwt Q=WDv= w O}rwD |xr�Ut QO "CU=R=UGwt p=OB |}=Hx@=H hQat �(t) Qo = T=U= u}= Q@ "CW=O Oy=wN OwHw ?=DWa = h0 0iT '=yxQ=w}O Q}=U QO a = h��(t) 0iT 'R=UGwt Qw=Ht RQt QO OW=@7 |x]@=Q CQwYx@ R}v (�F ) O=R; K]U |wQ Q@ |RQt \QW 'u}vJty "Ow@ Oy=wN
"�1 pmW� OwW|t xDiQo Q_v QOpj�F = 0 (7)
%CW=O s}y=wN =Q 8 |x]@=Q '3 |x]@=Q x@ xHwD =@ xmpH j�F = � g yj�F (8)
w ?rY |=yxQ=w}O \UwD xOWx]=L= p=}U |=yRQt w xr�Ut |xvt=O "1 pmW"R=UGwt
113
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pH = <"
NXi=1
1sie�ix+�iyV T
i
!R �PB
#(23)
"CU= xOW x@U=Lt Q}O=kt |k}kL CtUk Qou=}@ <[0] 'u; QO xm
R=UGwt p=OB |xOvvml} QLD |xrO=at "4Gwt O}rwD |OOa |R=Ux}@W w xrO=at u}}aD 'lD Gwt |wQq=@ |UQQ@ QO pw= sOkxO=iDU= 'x=oW}=tR; QO lD Gwt Q=WDv= w O}rwD pw=ODt |=yVwQ R= |m} "CU= lDC=YNWt =@lD Gwt O=H}= |=Q@ "CU= xOvvml} QLD sDU}Ul} =@ R=UGwt p=OB R=xv=}=Q \UwD p}rLD R= TB w O}; CUOx@ u; |[=} Q \@=wQ CU= sRq =OD@= 'Q_vOQwtOv}=Qi pw] QO lD Gwt pmW uOw@ Q=O}=B ZQi =@ "OwW p=ta= x=oW}=tR; VN@ x@[3]"CU= xOQm x�=Q= |vwDU}B R=UGwt p=OB CmQL u}}aD |=Q@ |rwtQi nv} Q)o 'O}rwDC}akwt |xrO=at nv} Q)o 'O}rwD Ov}=Qi pw] QO lD Gwt pmW uOw@ Q=O}=B ZQi =@
%Ovm|t u=}@ 24 |x]@=Q R= xO=iDU= =@ =Q R=UGwt p=OBd�dt
= �u(�; t) (24)
"CU= Gwt p=OB CaQU =@ Q@=Q@ �u w t u=tR Qy QO Gwt p=OB C}akwt �(t) 'u; QO xm%O};|t CUOx@ 26 w 25 |=yx]@=Q '24 |x]@=Q QO X = (ct� �) Q}eDt Q}}eD =@d�dt
= d�dX
dXdt
= d�dX
(c� d�dt
)
! �u = d�dX
(c� �u) (25)
d�dX
= �u(X(�))c� �u(X(�))
(26)
%OwW|t x@U=Lt 27 |x]@=Q CQwYx@ �u '25 |x]@=Q QOd�dt
= �u(�; t) = c�(X)h+ �(X)
(27)
w Gwt CaQU c 'p=OB CQw=Ht QO p=}U O=R; K]U s�=k u=mtQ}}eD � 'u; QO xm|iQat lD Gwt O}rwD |=Q@ nv} Q)o |SwrODt s=v x@ VwQ u}= "CU= ?; jta hlD Gwt O}rwD |=Q@ VwQ u} QD|rY= u=wvax@ u; x@ 'Q}N= |=yxyO QO w OwW|tlD Gwt O}rwD |=Q@ |SwrODt u}= R= R}v Q[=L VywSB QO 'u}=Q@=v@ &CU= xOW xHwD
"CU= xOW xO=iDU= xr�Ut pL OvwQ |xt=O= w|xrO=at Q@ |vD@t xOWO}rwD lD Gwt C=YNWt u}}aD |=Q@ swUQt |Qw�D l}|SwrODt |xOWx�=Q= pwtQi T=U=Q@ xm [4]'CU= lUv}Uw@ |Qw�D 'p=OB CmQLO=R; K]U |}=Hx@=H 'lUv}Uw@ |Qw�D QO "OwW|t p=ta= �24 |x]@=Q� nv} Q)o
%OwW|t u=}@ 28 |x]@=Q CQwYx@ p=}U� = HSech2(KX) (28)
"CU= X = (ct � �) w Gwt `=iDQ= H '|RQt |xOvy=m ?} Q[ K 'u; QO xm%OwW|t pY=L 29 |x]@=Q CQwYx@lUv}Uw@ |Qw�D |=Q@ |RQt |xOvy=m?} Q[K =
r 3H4h3 (29)
Q@ xOWxO=iDU= \=kv O=OaD m w p=}U |=yRQt |wQ Q@ \=kv pm O=OaD n 'u; QO xmCUOx@ 15 w 14 |=yx]@=Q R= (@pB)k w (pB)k u}vJty "CU= O=R; K]U |wQ
%O};|t(pB)k = [� g y]x=xk;y=yk
8(xk; yk) 2 �F ; k = 1; :::;m (14)
(@pB)k = �� [nx _ux + ny _uy]x=xk;y=yk
8(xk; yk) 2 �S ; k = m+ 1; :::; n (15)
lQLDt |=yxQ=w}O CQw=Ht QO \ki 15 |x]@=Q Q=Okt xm CW=O xHwD O}=@T=U=Q@ V i Q=OQ@ "CU= QiY hr=Nt �Gwt O}rwD p=v=m QO R=UGwt p=OB Q}_v�|=yx}=QO T=U=Q@ 'Qo}O CQ=@a x@ =} w |RQt \=kv R= l} Qy QO OwHwt |RQt \}=QWs)=i |x}=B @=@n O}=@ '�s x@ \w@ Qt \=kv QO u}=Q@=v@ "OwW|t h} QaD �PB Q=OQ@x]kv u; QO s)=i |x}=B Q=Okt R}v �F x@ \w@ Qt \=kv QO &OwW x@U=Lt x]kv u; QOx@ xHwD =@ 'u}=Q@=v@ "OwW|t xO=O Q=Qk V i Q=OQ@ QO u; Q}_v |x}=QO QO w x@U=Lt
%OQw; CUOx@ 16 |x]@=Q R= =Q V i u=wD|t xOWx�=Q= C=L}[wDV i = 1
sjf(pi)1 (pi)2 ::: (pi)mj
(@pi)(m+1) (@pi)m+2 ::: (@pi)ngT (16)
%O};|t CUOx@ 18 w 17 \@=wQ R= (@pi)k w (pi)k 'u; QO xm(pi)k = [e�ix+�iy]x=xk;y=yk
8(xk; yk) 2 �F ; k = 1; :::;m (17)
(@pi)k = [(�inx + �iny)e�ix+�iy]x=xk;y=yk
8(xk; yk) 2 �S ; k = m+ 1; :::; n (18)
19 |x]@=Q R= xm CU=yx}=B CmQ=Wt |=yQ=OQ@ uOQm xm} ?} Q[ si u}vJty[23]%O};|t CUOx@
si = maxl(jV li j); l = 1; :::;M (19)
"CU= 22|k=Lr=OwN pw] j0j CQ=@a w V i u=tr= u}t)=l Q@=Q@ V li 'u; QO xm%OwW|t xDWwv 20 |x]@=Q CQwYx@ '12 |x]@=Q uDiQo Q_vQO =@ PH |QUPH =
NXi=1
(V Ti R �PB)e�ix+�iy (20)
=Q 21 |x]@=Q 's}vm OQw;Q@ PH |QU \UwD =Q �PB Q=OQ@ |=yx}=QO Qo = p=L%CW=O s}y=wN
�PB =NXi=1
(V Ti R �PB)V i = GR �PB ;
G =NXi=1
(V iV Ti ) (21)
|x@DQt CU= umtt xm |}=Hv; R= "CU= M �M uQ=kDt T} QD=t G 'u; QO xmx@U=Lt 22 |x]@=Q CQwYx@ R T} QD=t u}=Q@=v@ &OwW M R= QDtm Q}N= T} QD=t
%OwW|tR = G+ (22)
114
2/1|
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u=DU@=D�
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u=Qta|
UOvyt |v=tR s=o l} QO |RQt \=kv ?=DW an�1 = a(xn�1; tn�1) Qo = p=L
%OwW|t xDiQo Q_v QO 38 |x]@=Q CQwYx@ ?=DW 'OW=@ p@ka(�t) = an�1 + an � an�1
�t�t 0 � �t � 2�t (38)
'38 |x]@=Q R= |Q}op=QoDv= =@ p=L "CU= t = tn�1 =@ Q_=vDt �t = 0 'u; QO xm%OwW|t pY=L 39 |x]@=Q CQwYx@ CaQU
u( �T ) = c1 + an�1�t+ 12an � an�1
�t�t 2 (39)
x@U=Lt 40 |x]@=Q CQwYx@ u=tR |x}rw= Q=Okt T=U=Q@ c1 ?} Q[ 'u; QO xm%OwW|t
�t = 0! c1 = un�1 (40)
|RQt \=kv C}akwt '�39 |x]@=Q� CaQU |x]@=Q R= |Q}op=QoDv= =@ TBU%O};|t CUOx@ 41 |x]@=Q CQwYx@ p=}U
x(�t) = c2 + un�1�t+ an�1
2 �t 2 + 16an � an�1
�t�t 3 (41)
pY=L 42 |x]@=Q CQwYx@ u=tR |x}rw= Q=Okt T=U=Q@ c2 ?} Q[ 'R}v u; QO xm%OwW|t
�t = 0! c2 = xn�1 (42)
=D �t = �t |v=tR |xR=@ QO |RQt \=kv |}=Hx@=H |x@U=Lt |=Q@ p=L%CW=O s}y=wN =Q 44 w 43 |=yx]@=Q �t = tn+1 =D t = tn =@ Q_=vDt� �t = 2�t
�~xn = x(�t = 2�t)� x(�t = �t) (43)
�~xn = un�1�t+ 16 (2an�1 + 7an)�t2 (44)
%O};|t CUOx@ 45 |x]@=Q u}=Q@=v@~xn+1 = xn + �~xn (45)
%CW=O s}y=wN =Q 46 |x]@=Q C}=yvQO w
~xn+1 = xn + un�1�t+ 16 (2an�1 + 7an)�t2 (46)
x]U=w |xUOvy R= =Q} R [24]&OwW|t xO}t=v 24x]U=w |xUOvy ~xn+1 'u; QO xms=o u}= QO |}=yv |xUOvy TBU w xO=iDU= xR=@ |=yDv= QO pO=aD <=[Q= |=Q@tn+1 |x_Lr '|v=tR s=o |=yDv= QO |RQt \}=QW u}=Q@=v@ "OwW|t x@U=Lt |v=tRu=}@ 47 |x]@=Q CQwYx@ 13 |x]@=Q R= xO=iDU= =@ '~xn+1 x]U=w |xUOvy QO w
%OwW|t�P n+1B = �PB(~xn+1; tn+1) (47)
'~xn+1 |xUOvy QO w tn+1 |x_Lr QO 16 |x]@=Q R= xO=iDU= =@ R}v V i Q=OQ@%OwW|t u=}@ 48 |x]@=Q CQwYx@ w x@U=Lt
V n+1i = V i(~xn+1; tn+1) (48)
u=}@ 30 |x]@=Q CQwYx@ lUv}Uw@ |Qw�D QO R}v Gwt CaQU u}vJty%OwW|t
c =pg(h+H) (30)
lD Gwt CWB QO |O}rwD |}xr=@vO G=wt= CUv=wD nv} Q)o xm CW=O xHwD O}=@"Ov=UQ@ xv}tm u=R}t x@ lUv}Uw@ |Qw�D \UwD =Q
xUOvy |R=Us=ovyx@ w pL |Sv=Qo q sD} Qwor= "5=D CU= sRq '|}=tv x}=B `@=wD \UwD Q=Wi TqBq |xrO=at pL |xwLv |x�=Q= R= TBKqY= T=U=Q@ Q_vOQwt sD} Qwor= '=Hv}= QO "OwW u=}@ u=tR QO p�=Ut pL |xwLv"CU= xOW xO=O xaUwD [24]'u=Q=mty w pw�Ut |xOvU} wv \UwD xOWu=}@ sD} Qwor==yRQt |wQ Q@ |RQt \=kv w h} QaD Q_vOQwt |xr�Ut |x}rw= |xUOvy '=OD@= QO|yOQ=Okt xr�Ut |x}rw= \}=QW T=U=Q@ R}v |RQt \=kv CaQU "OwW|t xO=O Q=Qk|x}rw= CaQU 'OvwW|t R=e; uwmU Cr=L R= xm |r�=Ut QO CU= |y}O@ &OwW|t|v=tR |=yxR=@ x@ pL u=tR pm s}UkD R=TB "Ow@ Oy=wN QiY Q@=Q@ |RQt \=kv |xtyQO |R=Ux}@W Qo = "OwW|t pL x@DQt wO 4 |xrO=at '|v=tR s=o Qy QO �tlJwm|va} 's)=tn s=o |=OD@= QO p=}U CaQU 'CU= (�tn = tn+1 � tn) s)=n s=o|RQt \}=QW '=OD@= "OwW|t xO=O u=Wv un = u(xn; tn) CQwY x@ tn |x_Lrj@=]t 13 |x]@=Q R= xO=iDU= =@ 'tn |x_Lr QO xn |xUOvy QO w tn |x_Lr QO
%OwW|t x@U=Lt 31 |x]@=Q�P nB = �PB(xn; tn) (31)
'xn |xUOvy QO w tn |x_Lr QO 16 |x]@=Q R= xO=iDU= =@ R}v V i Q=OQ@%OwW|t u=}@ 32 |x]@=Q CQwYx@ w x@U=Lt
V ni = V i(xn; tn) (32)
%OwW|t xDiQo Q_v QO 33 |x]@=Q CQwYx@ 4 |xrO=at MU=B ?}DQD u}O@pnH =
XCni e
�ix+�iy (33)
%Ot; Oy=wN CUOx@ 34 |x]@=Q R= Cni ?}=Q[ wCni = V nT
i Rn �P nB (34)
%OwW|t x@U=Lt 35 |x]@=Q R= Rn 'u; QO xmRn = (
X(V n
i VnTi ))�1 (35)
tn |x_Lr QO s)=n |v=tR s=o |=OD@= QO Q=Wi TqBq |xrO=at pL u}=Q@=v@a =@ =Q DuDt |Sv=Qo q ?=DW Qo = &O} Q}o@ Q_v QO =Q 5 |x]@=Q p=L "Ow@ Oy=wN pt=m
%O};|t CUOx@ 36 |x]@=Q R= a 's}yO u=Wva = �1
�rPH (36)
=Q tn |x_Lr QO w xn C}akwt QO |RQt \=kv ?=DW u=wD|t u}=Q@=v@ w%CWwv 37 |x]@=Q CQwYx@
an = a(xn; tn)
= �1�<"
NXi=1
1sni
(�i�i
)e�ix+�iyV nT
i
!Rn �P n
H
#(37)
115
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@W|RQt \=kv C}akwt '�56 |x]@=Q� CaQU |x]@=Q R= |Q}op=QoDv= =@ |iQ] R=%O};|t CUOx@ 60 |x]@=Q CQwYx@ p=}U
x(�t) = 124�t2 (12an�1�t 2�t2 � (6an�1 � 8an + 2an+1)�t 3�t
+(an�1 � 2an + an+1)�t 4) + c2 (60)
CUOx@ 61 |x]@=Q CQwYx@ u=tR |x}rw= Q=Okt T=U=Q@ c2 ?} Q[ 'u; QO xm%O};|t
�t = 0! c2 = xn�1 (61)
=D �t = �t |v=tR |xR=@ QO |RQt \=kv |}=Hx@=H |x@U=Lt |=Q@ p=L%CW=O s}y=wN =Q 63 w 62 |=yx]@=Q �t = tn+1 =D t = tn =@ Q_=vDt� �t = 2�t
�xn = x(�t = 2�t)� x(�t = �t) (62)
�xn = un�1�t+ 124 (9an�1 + 26an + an+1)�t2 (63)
%CWwv =Q 64 |x]@=Q u=wD|t p=Lxn+1 = xn + �xn (64)
(tn+1 � tn) s)=n s=o |=yDv= QO xn+1 25|}=yv |xUOvy 'C}=yv QO w%OwW|t x@U=Lt 65 |x]@=Q CQwYx@
xn+1 = xn + un�1�t
+ 124 (9an�1 + 26an + an+1)�t2 (65)
CQwYx@ '59 |x]@=Q x@ xHwD =@ |}=yv |xUOvy QO |RQt \=kv |}=yv CaQU%OwW|t u} Ro}=H |Oa@ |v=tR s=o |x}rw= CaQU u=wvax@ w x@U=Lt 66 |x]@=Q
un+1 = u(xn+1; tn) + 112 (�an�1(xn+1; tn�1)
+8an(xn+1; tn) + 5an+1(xn+1; tn+1))�t (66)
Q=O?}W Kw]U |wQ Q@ lD Gwt |wQq=@ |R=Ux}@W "6Gwt |wQq=@ |R=Ux}@W QO Q[=L |OOa VwQ G}=Dv CkO =OD@= '|vwvm VN@ QOxOW O}}-=D w |UQQ@ |y=oW}=tR; |xvwtv l} =@ xU}=kt QO Q=O?}W Kw]U QO lDQ}O=kt =@ w x@U=Lt Gwt |wQq=@ |xv}W}@ Q=Okt 'G}=Dv CkO R= u=v}t]= R= TB "CU=
"CU= xOW xU}=kt |y=oW}=tR;
|OOa |R=Ux}@W |HvUCLY "1"6Gwt |wQq=@ |xv}tR QO xOQDUo w syt |}xar=]t '1986 p=U QO T}m qwv}Uu; QO QwmPt |xar=]t C}ty= xm [7w6]'CU= xO=O s=Hv= pL=wU ?}W |wQ Q@ lDpwYL QO q=@ CkO Q@ xwqa 'xOWs=Hv= |=yV}=tR; w |OOa |R=Ux}@W xm CU=x@ &CU= xOQm |UQQ@ =Q QwmPt |xO}OB |ak=w C}a[w '?U=vt T=}kt =@ 'G}=DvV}=tR; G}=Dv R= R}v xv}tR u}= QO OwHwt C=ar=]t R= |}xOQDUo |xvt=O xm |Qw]lD Gwt 'QwmPt V}=tR; QO "Ov=xOQm xO=iDU= |rY= `HQt l} u=wvax@ xOWQmP
CQwYx@ 4 |xrO=at MU=B w Cn+1i ?}=Q[ 'Rn+1 T} QD=t ?}DQD u}= x@
%Ow@ Ovy=wN 51 |r= 49 |=yx]@=QRn+1 = (
X(V n+1
i V n+1Ti ))�1 (49)
Cn+1i = V n+1T
i Rn+1 �P n+1B (50)
Pn+1H =
XCn+1i e�ix+�iy (51)
QO w tn+1 |x_Lr '|v=tR |xR=@ |=yDv= QO |RQt \=kv |Sv=Qo q ?=DW w%OwW|t x@U=Lt 52 |x]@=Q CQwYx@ 36 |x]@=Q x@ xHwD =@ ' ~xn+1 C}akwtan+1 = a(~xn+1; tn+1) =
1�<�� NX
i=1
1sn+1i
(�i�i
)e�ix+�iyV n+1T
i
�Rn+1 �P n+1
H
�(52)
C@=F pL pw] QO w xOWQmP \@=wQ |t=tD QO �i w �i ?}=Q[ xm OwW xHwD&OQw; CUOx@ ?=DW `} RwD R= |QDj}kO OQw;Q@ u=wD|t p=L "OvwW|t xDiQo Q_v QOu=}@ �53 |x]@=Q� |wtyU |x]@=Q CQwYx@ |Sv=Qo q ?=DW Cr=L u}= QO u}=Q@=v@
%OwW|ta(�t) = A�1 +A0�t+A1�t 2; 0 � �t � 2�t (53)
xrO=at 3 x=oDUO l} 'Cr=L u}= QO "CU= t = tn�1 =@ Q_=vDt �t = 0 'u; QO xm%CW=O s}y=wN 54 |x]@=Q CQwYx@ pwyHt 3 =@0BB@1 0 0
1 �t �t2
1 2�t 4�t2
1CCA2664A�1A0A1
3775 =
2664an�1
an
an+1
3775 (54)
CUOx@ 55 |x]@=Q CQwYx@ ?=DW ?}=Q[ 'xOWQmP x=oDUO pL =@ p=L%Ov};|t
a(�t) = an�1 + (�3an�1
2�t+ 2an
�t� an+1
2�t)�t
+(an�1
2�t2 �an
�t2 + an+1
2�t2 )�t 2 (55)
CUOx@ 56 |x]@=Q CQwYx@ CaQU '55 |x]@=Q R= |Q}op=QoDv= =@ p=L%O};|t
u(�t) = c1 + an�1�t� (3an�1 � 4an + an+1)�t 2
4�t
+(an�1 � 2an + an+1)�t 3
6�t2 (56)
x@U=Lt 57 |x]@=Q CQwYx@ u=tR |x}rw= Q=Okt T=U=Q@ c1 ?} Q[ 'u; QO xm%OwW|t
�t = 0! c1 = un�1 (57)
=D �t = �t |v=tR |xR=@ QO |RQt \=kv CaQU QO Q}}eD |x@U=Lt |=Q@ p=L%CW=O s}y=wN =Q 59 w 58 |x]@=Q �t = tn+1 =D t = tn =@ Q_=vDt� �t = 2�t
�un = u(�t = 2�t)� u(�t = �t) (58)
�un = 112 (�an�1 + 8an + 5an+1)�t (59)
116
2/1|
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u=DU@=D�
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UOvyt Oa@|@ 67 |x]@=Q R= xO=iDU= =@ 5 pmW QO OwHwt Q} w=YD Q} R QO xOWxOy=Wt u=tR
%CU= xOWT = tp
h=g(67)
QO x0 u}vJty &CU= Q[=L |OOa VwQ \UwD |D=@U=Lt u=tR t 'u; QO xmG}=Dv |xU}=kt =@ "CU= 3 pmW x@ xHwD =@ C=YDNt -=O@t '6 w 5 |=ypmWxOy=Wt |OOa VwQ w |y=oW}=tR; G}=Dv u}@ |rw@k p@=k CkO '6 w 5 |=ypmW
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"OwW|tQO xOWu=}@V}=tR; OQwt wO R= 'Q[=L |OOa VwQ |HvUCLY s=Hv=CyHH=h = 0 019 pw= OQwt C=YNWt "CU= xOW xO=iDU= [7]'T}m qwv}U Q=DWwv"CU= h = 0 2855m =@ H=h = 0 04 swO OQwt w h = 0 3097m =@w C=YNWt 'Gwt |wQq=@ Q@ xwqa xm CU= Cra u}= x@ xOWQmP OQwt wO ?=NDv=|xU}=kt [7]"CU= OwHwt 'T}m qwv}U Q=DWwv QO =yu; |=Q@ p=}U O=R; K]U C=Q}}eD|OOa VwQ u}@ Gwt p=v=m QO hrDNt u=tR 6 |=Q@ p=}U O=R; K]U C=Q}}eDQw_vt x@ "OwW|t xOy=Wt 6 w 5 |=ypmW QO |y=oW}=tR; C=aq]= w Q[=L'CU= xOW u=}@ [7]'T}m qwv}U Q=DWwv QO xm xJv; =@ xr�Ut C=YNWt |R=Uu=Um}
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Q@=Q@ OvJ |OOa C=@U=Lt Ov}=Qi xm CU= xOW Ea=@ C} Rt u}= "Ow@ Oy=wNv R=}v"OwW swUQt |=yVwQ R= QDx=Dwm
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=yCWwv=B1. generation2. propagation3. runup
4. boussinesq5. trajectory6. Non-breaking7. linear and nonlinear theory8. boundary integral equation method (BIEM)
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9. nonlinear shallow water wave equation (NLSW)10. �nite-analytic method11. synolakis12. rayleigh13. pure14. amplitude15. tailing waves16. Goring17. shallow water �rst order (SWFO)18. shallow water second order (SWSO)19. fenton20. exponential basis function21. validation22. hermitian length23. pseudo inverse24. intermediate geometry25. �nal geometry26. CPU:Corei7 2600k3.8GHz ,RAM:8GB DDR3,
VGA:nVidia560 GTX1GB GDDR527. periodic waves28. compound or multilinear-slopes
�References� `@=vt1. Wazwaz A.M., Partial Di�erential Equations and Soli-
tary Wave Theory, Higher Education Press, Springer(2009).
2. Hall, J.V. and Watts, J.W. \Laboratory investigationof the vertical rise of solitary waves on impermeableslopes", Technical Report, Beach Erosion Board, USArmy Corps of Engineering, 33 (1953).
3. Goring, D.G. \Tsunami: The propagation of long wavesonto a shel", Ph.D. Thesis, California Institute of Tech-nology (1979).
4. Boussinesq, M.J. \Theorie de lintumescence liquide, ap-pelee onde solitaire ou de translation, se propageant dansun canal rectangulaire", C.R. Acad. Sci., Paris, 72, pp.755-759 (1871).
5. Pedersen, G. and Gjevik, B. \Run-up of solitary waves",Journal of Fluid Mechanics, 135, pp. 283-299 (1983).
6. Synolakis, C.E. \The run-up of long waves", Ph.D. The-sis, California Institute of Technology (1986).
7. Synolakis, C.E. \The runup of solitary waves", Journalof Fluid Mechanics, 185, pp. 523-545 (1987).
8. Zelt, J.A. and Raichlen, F. \A Lagrangian model forwater-induced harbor oscillations", Journal of Fluid Me-chanics, 213, pp. 203-225 (1990).
9. Liu, P.L.-F. and Cho, Y.-S. \Integral equation modelfor wave propagation with bottom frictions", Journalof Waterway, Port, Coastal, and Ocean Engineering,ASCE, 120(6), pp. 594-608 (1994).
10. Titov, V.V. and Synolakis, C.E. \Modeling of breakingand non-breaking long-wave evolution and run-up us-ing VTCS-2", Journal of Waterway, Port, Coastal, andOcean Engineering, ASCE, 121(6), pp. 308-461 (1995).
11. Li, Y. and Raichlen, F. \Solitary wave runup on planeslopes", Journal of Waterway, Port, Coastal and OceanEngineering, ASCE, 127(1), pp. 33-44 (2001).
12. Guizien, K. and Barthelemy, E. \Short waves modula-tions by large free surface solitary waves. Experimentsand models", Physics of Fluids, 13(12), pp. 3624-3635(2002).
13. Rayleigh, L. \On waves", Phil. Mag., 1, pp. 257-279(1876).
14. Clamond, D. and Germain, J.-P. \Interaction between astokes wave packet and a solitary wave", Eur. J. Mech.B/Fluids, 18(1), pp. 67-91 (1999).
15. Temperville, A. \Contribution a letude des ondes degravite en eau peu profonde", These d Etat, UniversiteJoseph Fourier - Grenoble I (1985).
16. Mahdavi, A. and Talebbeydokhti, N. \Numerical mod-eling of shallow water equation for simulation of longwaves runup", 27-2(2), pp. 107-115 (2012).
17. Rezaei, H. and Ketabdari, J. \Estimation of break-ing solitary wave pressure on moored pontoon oatingbreakwaters using VOF method", Sharif Journal CivilEngineering, 27-2(4), pp. 111-122 (2012).
18. Li, J., Liu, H., Gong, K., Tan, S.K. and Shao, S. \SPHmodeling of solitary wave �ssions over uneven bottoms",Coastal Engineering, 60, pp. 261-275 (2012).
19. Wu, N.J., Tsay, T.K. and Chen, Y. \Generation of sta-ble solitary waves by a piston-type wave maker", WaveMotion, 51(2), pp. 240-255 (2014).
20. Grimshaw, R.H.J. \The solitary wave in water of vari-able depth: Part 2", Journal Fluid Mechanics, 46(03),pp. 611-622 (1971).
21. Fenton, J. \A ninth-order solution for the solitary wave",Journal Fluid Mechanics, 53(2), pp. 257-271 (1972).
22. Han, S.H.T. and Cho, Y.-S. \Laboratory experiments onrun-up and force of solitary waves", Journal of Hydro-Environment, 9(4), pp. 582-591 (2015).
23. Boroomand, B., Soghrati, S. and Movahedian, B. \Ex-ponential basis functions in solution of static and timeharmonic elastic problems in a meshless style", Inter-national Journal for Numerical Methods in Engineering,81(8), pp. 971-1018 (2010).
24. Zandi, S.M., Boroomand, B. and Soghrati, S. \Expo-nential basis functions in solution of incompressible uidproblems with moving free surfaces", Journal of Compu-tational Physics, 231(2), pp. 505-527 (2012a).
25. Zandi, S.M., Boroomand, B. and Soghrati, S. \Exponen-tial basis functions in problems with fully incompressiblematerials: A mesh-free method", Journal of Computa-tional Physics, 231(2), pp. 7255-7273 (2012b).
26. Grilli, S.T., Fully Nonlinear Potential Flow Models Usedfor Long Wave Run-up Prediction. Long-wave Run-upModels, H. Yeh, P. Liu, and C. Synolakis, eds., WorldScienti�c, Singapore, pp. 116-180 (1997).
121
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