Lampiran 1 (Titik Tetap)

LAMPIRAN

Lampiran 1 (Titik Tetap)

Hasil ini diperoleh dengan menggunakan fasilitas Dynpac yang dapat di download di www.wolframresearch.com kemudian dijalankan pada software Mathematica 5.1. In[328]:= sysid

Mathematica 5.1.0, DynPac 10.69, 5ê 24ê 2007

In[329]:= "Mencari titik keseimbangan "; In[330]:= intreset; plotreset;

In[331]:= setstate@8x, y, z
In[332]:= setparm@8 p, q, r, K, L, M, a, b, c, d, e, f, v, U
In[333]:= slopevec = : p x jj1 k

xy z

y y zy i i z - a x y - b x z , q y jj1 - zz +c x y - d y z - v U y, r z jj1 zz +e x z - f y z - w U z>; { k { k K L M{

In[334]:= eqstates = findpolyeq; In[335]:= E0 = eqstates@@1DD

Out[335]= 80, 0, 0<

In[336]:= E1 = FullSimplify @eqstates@@2DDD

Out[336]= 8K, 0, 0<

In[337]:= E2 = FullSimplify @eqstates@@3DDD Out[337]= :

K H p q +a L H-q +U vLL acKL+p q

,

L p Hc K +q - U vL acKL+p q

, 0>


K H p r + b M H- r +U wLL b eKM+p r

, 0,

M p He K + r - U wL b eKM+p r

>


K H r H-a L q + p q +a L U vL - d L M H f p - a r +a U wL + b M H-q r + f L Hq - U vL +q U wLL -L M Ha d e K + b c f K +d f pL + b e K M q +Ha c K L + p qL r

L H p r Hc K +q - U vL - d M p He K + r - U wL + b K M He q - c r - e U v +c U wLL -L M Ha d e K + b c f K +d f pL + b e K M q +Ha c K L + p qL r

,

M Ha e K L Hq - U vL +c K L H f p - a r +a U wL - p He K q - f L q +q r + f L U v - q U wLL M Ha d e K L +d f L p + b K Hc f L - e qLL - Ha c K L + p qL r

,

>

In[340]:= E5 = FullSimplify @eqstates@@6DDD Out[340]= :0, L -

LU v q

, 0>

In[341]:= E6 = FullSimplify @eqstates@@7DDD Out[341]= :0,

L H-q r + r U v +d M H r - U wLL d fLM-q r

,

M H f L Hq - U vL +q H-r +U wLL d fLM-q r

>

In[342]:= E7 = FullSimplify @eqstates@@8DDD Out[342]= :0, 0, M -

MU w r

>

31

Lampiran 2 (Nilai Eigen)



In[344]:= intreset; plotreset;

In[346]:= setparm@8 p, q, r, K, L, M, a, b, c, d, e, f, v, w

xy z

yy zy i i z - a x y - b x z , q y jj1 - zz +c x y - d y z - v U y, r z jj1 - zz +e x z - f y z - w U z>; k k L{ M{

K{

In[348]:= eqstates = findpolyeq; In[349]:= E0 = eqstates@@1DD;

In[350]:= E1 = FullSimplify @eqstates@@2DDD; In[351]:= E2 = FullSimplify @eqstates@@3DDD; In[352]:= E3 = FullSimplify @eqstates@@4DDD; In[353]:= E4 = FullSimplify @eqstates@@5DDD; In[354]:= E5 = FullSimplify @eqstates@@6DDD; In[355]:= E6 = FullSimplify @eqstates@@7DDD; In[356]:= E7 = FullSimplify @eqstates@@8DDD;

Konstruksi Matrik Jacobi In[357]:= jac = Simplify @88D@slopevec@@1DD, xD, D@slopevec@@1DD, yD, D@slopevec@@1DD, zD< , 8D@slopevec@@2DD, xD, D@slopevec@@2DD, yD, D@slopevec@@2DD, zD< , 8D@slopevec@@3DD, xD, D@slopevec@@3DD, yD, D@slopevec@@3DD, zD<
Out[358]//MatrixForm= ij p - 2 p x - a y - b z -a x jj K j jj jj c y jj jj jj e z k

yz zz zz 2q y zz - d z -d y q - U v +c x zz L zz z -f z r - U w +e x - f y - 2 r z zz M {

-b x

Pada titik tetapnya berlaku Kontruksi Matriks Jacobi untuk titik tetap E0

In[359]:= jac0 = jac ê. 8x Ø E0@@1DD, y Ø E0 @@2DD, z Ø E0@@3DD< ; In[360]:= "JHE0L=" MatrixForm @ jac0D

0 ij p 0 yz zz Out[360]= JHE0L= jjj 0 q - U v 0 z k0 0 r-U w{

32

Kontruksi Matriks Jacobi untuk titik tetap E1

In[361]:= jac1 = jac ê. 8x Ø E1@@1DD, y Ø E1 @@2DD, z Ø E1@@3DD< ; In[362]:= "JHE1L=" MatrixForm @ jac1D

-bK ij - p -a K yz zz Out[362]= JHE1L= jjj 0 c K +q - U v 0 z k0 0 eK+r-U w{


In[363]:= jac2 = FullSimplify @ jac ê. 8x Ø E2@@1DD, y Ø E2 @@2DD, z Ø E2@@3DD


In[367]:= jac4 = FullSimplify @ jac ê. 8x Ø x4, y Ø y4 , z Ø z4

i ij U v yz 0 yz 0 jj p +a L j-1 + zz q { k jj zz jj zz d L H- q+U vL jj c ijjL - L U v yzz zz -q +U v Out[370]= JHE5L= j zz jj k q { q zz jjj zz i y jj 0 0 r + f L j-1 + U v z - U w zz q { k k {

Kontruksi Matriks Jacobi untuk titik tetap E6 In[371]:= jac6 = FullSimplify @ jac ê. 8x Ø E6@@1DD, y Ø E6 @@2DD, z Ø E6@@3DD

ij p + b M J -1 + U w N 0 yz 0 jj zz r jj zz U w jj 0 zz q - U v +d M J -1 + N 0 Out[374]= JHE7L= j zz jj r zz jj z f M H- r+U wL jj e J M - MU w N - r +U w zz k { r r

33

Nilai Eigen di titik tetap E0

In[375]:= l0 = FullSimplify @Eigenvalues@ jac0DD;

In[376]:= 8"l1="@l0@@1DDD, "l2 ="@l0@@2DDD, "l3="@l0@@3DDD<

Out[376]= 8l1=@ pD, l2 =@q - U vD, l3=@ r - U wD<



In[378]:= 8"l1="@l1@@1DDD, "l2 ="@l1@@2DDD, "l3="@l1@@3DDD<

Out[378]= 8l1=@- pD, l2=@c K +q - U vD, l3 =@e K + r - U wD<



In[380]:= 8"l1="@l2@@1DDD, "l2 ="@l2@@2DDD, "l3="@l2@@3DDD< Out[380]= :l1=B

1

2 Ha c K L + p qL

I-c K p q +a L p Hq - U vL - p q H p +q - U vL +

, I p I p Hq Hc K - a L + p +qL +Ha L - qL U vL2 - 4 Ha c K L + p qL Hc K +q - U vL H p q +a L H-q +U vLLMMMF ,

l2=B -

1

2 Ha c K L + p qL

Ic K p q - a L p Hq - U vL + p q H p +q - U vL +

, I p I p Hq Hc K - a L + p +qL +Ha L - qL U vL2 - 4 Ha c K L + p qL Hc K +q - U vL H p q +a L H-q +U vLLMMMF ,

l3=B r +

-c f K L p +a e K L H-q +U vL + p He K q - f L q + f L U vL

acKL+pq

- U wF>



In[382]:= 8"l1="@l3@@1DDD, "l2 ="@l3@@2DDD, "l3="@l3@@3DDD< Out[382]= :l1=B q + l2=B

p r Hc K - U vL - d M p He K + r - U wL - b K M Hc r +e U v - c U wL beKM+p r

1

2 H b e K M + p rL

F,

I-e K p r + b M p H r - U wL - p r H p + r - U wL +

, I p I p H r He K - b M + p + rL +H b M - rL U wL2 - 4 H b e K M + p rL He K + r - U wL H p r + b M H- r +U wLLMMMF ,

l3=B -

1

2 H b e K M + p rL

Ie K p r - b M p H r - U wL + p r H p + r - U wL +

, I p I p H r He K - b M + p + rL +H b M - rL U wL2 - 4 H b e K M + p rL He K + r - U wL H p r + b M H- r +U wLLMMMF>



In[384]:= 8"l1="@l4@@1DDD, "l2 ="@l4@@2DDD, "l3="@l4@@3DDD< ;

34



In[386]:= 8"l1="@l5@@1DDD, "l2 ="@l5@@2DDD, "l3="@l5@@3DDD< i

Out[386]= :l1=@-q +U vD, l2=B p +a L jjj-1 + k

U v yz

i U v yz zzF , l3=B r + f L jjj-1 + zz - U wF> q { q { k



In[388]:= 8"l1="@l6@@1DDD, "l2 ="@l6@@2DDD, "l3="@l6@@3DDD< Out[388]= :l1=B l2=B

r H-p q +a L Hq - U vLL +d L M H f p - a r +a U wL + b M H- f L q +q r + f L U v - q U wL d fLM-q r

1

F,

I- f L r Hq - U vL +d M q H-r +U wL +q r Hq + r - U H v + wLL +

, I4 Hd f L M - q rL H-q r + r U v +d M H r - U wLL H f L Hq - U vL +q H- r +U wLL +

2d fLM-2q r

H f L r Hq - U vL +q Hd M H r - U wL - r Hq + r - U H v + wLLLL MMF ,

2

l3=B

1

I- f L r Hq - U vL +d M q H-r +U wL +q r Hq + r - U H v + wLL -

, I4 Hd f L M - q rL H-q r + r U v +d M H r - U wLL H f L Hq - U vL +q H- r +U wLL +

2d fLM-2q r

H f L r Hq - U vL +q Hd M H r - U wL - r Hq + r - U H v + wLLLL2MMF>



In[390]:= 8"l1="@l7@@1DDD, "l2 ="@l7@@2DDD, "l3="@l7@@3DDD< i

Out[390]= :l1=@- r +U wD, l2=B p + b M jj-1 + k

Uwy

Uwy zzF , l =B q - U v +d M ijj-1 + zzF> 3 k r { r {

In[391]:= Unprotect@ p, q, r, K, L, M, a, b, c, d, e, f, v, wD; In[392]:= ClearAll @ p, q, r, K, L, M, a, b, c, d, e, f, v, wD

35

Lampiran 3 (Evaluasi Nilai-nilai Parameter)



In[394]:= intreset; plotreset;

In[396]:= setparm@8 p, q, r, K, L, M, a, b, c, d, e, f, v, w

xy z

yy zy i i zz +e x z - f y z - w U z>; z - a x y - b x z , q y jj1 - zz +c x y - d y z - v U y, r z jj1 k k K{ L{ M{

In[399]:= eqstates = findpolyeq; In[400]:= U =

H-d L M p He K + rL v +L H p Hc K +qL r + b K M He q - c rLL v +

M Hc K L H-f p +a rL +q H-a e K L + p He K - f L + rLLL wLë 2

I2 L H b e K M + p rL v - 2 L M H b c K +a e K +Hd + fL pL v w +2 M Ha c K L + p qL w M;

2

In[401]:= U = evalparm@UD Out[401]= 8159.37 In[402]:= E0 = evalparm@eqstates@@1DDD

Out[402]= 80, 0, 0<


In[403]:= l0 = FullSimplify @eigval@E0DD; 8"l1="@l0@@1DDD, "l2 ="@l0@@2DDD, "l3="@l0@@3DDD<

Out[403]= [email protected] D, l2 [email protected] D, [email protected] D< In[404]:= classify@E0D

unstable

In[405]:= E1 = evalparm@FullSimplify @eqstates@@2DDDD

Out[405]= 8350000, 0, 0<


In[406]:= l1 = eigval@E1D; 8"l1="@l1@@1DDD, "l2 ="@l1@@2DDD, "l3="@l1@@3DDD<

Out[406]= [email protected] D, [email protected] D, l3 [email protected] D< In[407]:= classify@E1D

unstable

36


Out[408]= 8291154., 168130., 0<



Out[409]= [email protected] +0.479572 ÂD, [email protected] - 0.479572 ÂD, [email protected] D< In[410]:= classify@E2D

unstable


Out[411]= 8337777., 0, 72753.7 <


Out[412]= [email protected] D, [email protected] D, [email protected] D< In[413]:= classify@E3D

unstable


Out[414]= 8301407., 126253., 26215.4 <



Out[415]= [email protected] +0.286408 ÂD, [email protected] - 0.286408 ÂD, [email protected] D< In[416]:= classify@E4D

strictly stable


Out[417]= 80, 88062.9, 0<



Out[418]= [email protected] D, l2 [email protected] D, l3 [email protected] D< In[419]:= classify@E5D

unstable

37


Out[420]= 80, 61748.9, 15433.5 <




unstable


Out[423]= 80, 0, 36273.7 <




unstable

In[426]:= Unprotect@ p, q, r, K, L, M, a, b, c, d, e, f, v, w, UD; In[427]:= ClearAll @ p, q, r, K, L, M, a, b, c, d, e, f, v, w, UD

38

Lampiran 4 (Kestabilan Global)

Hasil ini diperoleh dengan menggunakan software Mathematica 5.1.

* i p Hx - x L y i = jjHx - x* L jj- a H y - y* L - b Hz - z* Lzz + k { k dt K * i q Hy - y L y H y - y* L jj+c Hx - x* L - d Hz - z* Lzz + k { L * yy * i r Hz - z L Hz - z L jj+e Hx - x* L - f H y - y* Lzz zz; {{ k M

dV

In[328]:= X = 88x - x* < , 8 y - y* < , 8z - z* << ; In[329]:= X êê MatrixForm Out[329]//MatrixForm= * ij x - x yz jj * z jj y - y zzz * k z-z { In[330]:= A = ::

p a-c b-e a-c q d +f b -e d+f r , >, : >, : >>; , , , , , K 2 2 2 L 2 2 2 M

In[331]:= MatrixForm @AD Out[331]//MatrixForm= a- c ij p jj K 2 jj q jj a- c jj L jj 2 jj b- e d+ f k 2 2

b- e 2 d+f 2 r M

yz zz zz zz zz zz zz {

In[332]:= XT = Transpose @XD; In[333]:= MatrixForm @XT D

Out[333]//MatrixForm= H x - x* y - y*

z - z* L

39

Mensubstitusikan nilai a=c dan b=e ke dalammatriks A In[334]:= A = A ê. 8a Ø c, b Ø e< Out[334]= ::

p K

, 0, 0>, :0,

In[335]:= MatrixForm @AD Out[335]//MatrixForm= p ij 0 jj K jj q jjj 0 jj L jj d+ f j0 k 2

q L

,

d+f 2

>, :0,

d+f 2

,

r M

>>

yz zz z d+ f zz z 2 zzz r zzz M {

0

Mencari Nilai Eigen dari matriks A In[336]:= l = Eigenvalues@AD; In[337]:= "l1="@l@@1DDD

Out[337]= l1=B

p K

F

In[338]:= "l2="@l@@2DDD Out[338]= l2=B -

1

I-4 M q - 4 L r -

, IH4 M q +4 L rL2 +16 L M Id2 L M +2 d f L M + f2 L M - 4 q rMMMF

8LM

In[339]:= "l3="@l@@3DDD Out[339]= l3=B -

1

I-4 M q - 4 L r +

, IH4 M q +4 L rL2 +16 L M Id2 L M +2 d f L M + f2 L M - 4 q rMMMF

8LM

40

Lampiran 5 (Populasi Optimal)

Hasil ini diperoleh dengan menggunakan software Mathematica 5.1. In[328]:= Unprotect@x, y, z, p, ND; In[329]:=

In[330]:=

ClearAll@p, q, r, K, L, M, a, b, c, d, e, f, v, w, U4, x, y, z, x4, y4, z4, d, ans1, ans2, ans3, p, h1, h2, B, l1, l2, l3, M1, M2, M3, N, ans1, ans2, ans3D; H = Exp@- t dD Hh1 vy + h2wz - BL U + l1 Jl2 J-

p 2 x - axy - bxz + p xN +

K

q 2 r 2 y + cxy - dy z + y Hq - vULN + l3 Jz + exz - f y z + z Hr - wULN; L M i

p x4 y

In[331]:=

slopecostate1 = jjD -

In[332]:=

slopecostate2 = ax4 l1 - jjD -

In[333]:=

slopecostate3 = bx4 l1 - jjD -

k

zz l + cy4 l + ez4 l == 0; 1 2 3

K { i

qy4 y

i

r z4 y

k

k

zz l + f z4 l == U4h1 vExp@-d tD; 2 3

L {

zz l + dy4 l ã U4h2wExp@-d tD; 3 2

M {

Menyelesaikanpersamaan l1 denganmenggunakanoperator - D; In[334]:=

Simplify@Solve@Eliminate@8slopecostate1, slopecostate2, slopecostate3<, 8l2, l3
Out[334]= 99l1 Ø H‰

KU4 HeM HDh2Lw + dh1L vy4 - h2qwy4L z4 + cLy4 HDh1M v - h1r vz4 + f h2M wz4LLL ë 3 ID KLM + H bcf KLM + df LM p - beKM q - p qr + aKL HdeM - crLL x4y4 z4 - D2 HLM p x4 + KM qy4 + KLr z4L + D HacKLM x4y4 + M p qx4y4 + beKLM x4z4 + Lp r x4z4 - df KLM y4z4 + Kqr y4z4LM== -t d

Misalkan l1=

A1 B1

, dengan ;

In[335]:=

A1 = -t d H‰ KU4 HeM HDh2Lw + dh1L vy4 - h2qwy4L z4 + cLy4 HDh1M v - h1r vz4 + f h2M wz4LLL;

In[336]:=

B1 = 3 ID KLM + H bcf KLM + df LM p - beKM q - p qr + aKL HdeM - crLL x4y4z4 D2 HLM p x4 + KM qy4 + KLr z4L + D HacKLM x4y4 + M p qx4y4 + beKLM x4z4 + Lp r x4z4 df KLM y4z4 + Kqr y4z4LM;

Untuk penyederhanaan, kalikan A1 & B1 dengan Mengalikan A1 dengan

In[337]:= Out[337]=

A1 = ExpandBA1 *

1 K LM 1

KLM

1 K LM

;

;

F

cD ‰- t d h1U4 vy4 + De ‰- t d h2U4wz4 + de ‰- t d h1U4 vy4z4 c ‰- t d h1r U4 vy4z4 e ‰- t d h2qU4wy4z4 + c ‰- t d f h2U4wy4z4 M L

41

In[338]:= In[339]:= Out[339]=

"Mengevaluasi operator-D diperoleh:"; A1 = A1 ê. D Ø -d

c ‰- t d h1r U4 vy4z4

de ‰- t d h1U4 vy4z4 -

M

e ‰- t d h2qU4wy4z4 L In[340]:=

Out[340]=

C1 = M1;

In[342]:=

"Sehingga";

In[343]:=

A1 = M1 ‰- t d ;

Out[344]=

D3 -

D2 p x4 K

bDex4z4 +

-

FF

+ cf h2U4wy4z4 -

- ch1U4 vy4 d - eh2U4wz4 d

Mengalikan B1 dengan B1 = ExpandBB1 *

‰- t d

M

L In[341]:=

A1

ch1r U4 vy4z4

eh2qU4wy4z4

In[344]:=

- c ‰- t d h1U4 vy4 d - e ‰- t d h2U4wz4 d

M1 = ExpandBFullSimplifyB deh1U4 vy4z4 -

+ c ‰- t d f h2U4wy4z4 -

1 K LM 1

KLM

D2 qy4

L Dp r x4z4

;

F

+ acDx4y4 +

Dp qx4y4 KL Dqr y4z4

-

D2 r z4 M

+

- dDf y4z4 + + adex4y4z4 + KM LM df p x4y4z4 beqx4y4z4 acr x4y4z4 p qr x4y4z4 bcf x4y4z4 + K L M KLM

"Daftar koefisien persamaan B1 terhadap operator-D"; In[345]:=

Coeff = CoefficientList@B1, DD

Out[345]= :adex4y4z4 +

df p x4y4z4

In[347]:=

K

-

beqx4y4z4

L acr x4y4z4 p qr x4y4z4 p qx4y4 + bex4z4 + , acx4y4 + M KLM KL p r x4z4 qr y4z4 p x4 qy4 r z4 - df y4z4 + ,, 1> KM LM K L M

In[346]:=

bcf x4y4z4 +

"Misalkan,";

a0 = Collect@Coeff @@1DD, x4y4z4D

i Out[347]= jjade + k

bcf +

df p K

-

beq L

-

acr M

-

p qr y zz x4y4z4 KLM {

42

In[348]:= Out[348]=

In[349]:= Out[349]=

In[350]:= Out[350]= In[351]:= In[352]:=

a1 = Coeff @@2DD p qx4y4

acx4y4 +

KL

+ bex4z4 +

p r x4z4 KM

a2 = Coeff @@3DD -

p x4 K

-

qy4 L

a3 = Coeff @@4DD

-

- df y4z4 +

qr y4z4 LM

r z4 M

1

Unprotect@ ND;

N = a3 D3 + a2 D2 + a1 D + a0 ê. D Ø -d

i Out[352]= jjade +

p qr y zz x4y4z4 K L M KLM { k p qx4y4 p r x4z4 qr y4z4 y ij zz d + + bex4z4 + - df y4z4 + jacx4y4 + KL KM LM { k qy4 r z4 y 2 3 ij p x4 zz d - d jL M { k K bcf +

df p

-

beq

acr

-

-

"Persamaan diferensial nya dapat ditulis sebagai:" ; Ha3D +a2D +a1D+a0 Ll1= M1 ‰ 3

2

-t d

;

"Dengan menggunakan metode operator-D untuk kasus tersebut diperoleh:"; Auxiliary Equation adalah a3m3 +3a2m2 +3a1m+a0 =0, dengan akar-akar biHi=1,2,3L ; Complementary Function HC FL adalah A1‰b 1 t+A2 ‰b 2 t +A3‰b 3 t, dengan Ai Hi=1,2,3Ladalah konstanta sembarang;

Particular Integral HPIL adalah In[353]:=

M1 ‰- t d B1

;

"Solusi Umum l1 = C F + P I ";

l1 = A1 ‰b 1 t + A2 ‰b 2 t + A3 ‰b 3 t +

M1 ‰- t d N

Menyelesaikanpersamaan l2 denganmenggunakanoperator - D; In[354]:= In[355]:=

Unprotect@x, y, z, p, ND;

ClearAll@p, q, r, K, L, M, a, b, c, d, e, f, v, w, U, x, y, z, x4, y4, z4, d, ans1, ans2, ans3, p, h1, h2, B, l1, l2, l3, M1, M2, M3, N, ans1, ans2, ans3D;

43

In[356]:=

Out[356]= 99l2 Ø -I‰

LU4 ID2 h1KM v + H beh1KM v + h1p r v - h2M HaeK + f pL wL x4z4 + D Hf h2KM wz4 - h1 v HM p x4 + Kr z4LLMM ë 3 ID KLM + H bcf KLM + df LM p - beKM q - p qr + aKL HdeM - crLL x4y4z4 - D2 HLM p x4 + KM qy4 + KLr z4L + D HacKLM x4y4 + M p qx4y4 + beKLM x4z4 + Lp r x4z4 - df KLM y4z4 + Kqr y4z4LM== -t d

Misalkan l2=

A2 B2

, dengan ;

In[357]:=

A2 = -I‰- t d LU4 ID2 h1KM v + H beh1KM v + h1p r v - h2M HaeK + f pL wL x4z4 + D Hf h2KM wz4 - h1 v HM p x4 + Kr z4LLMM;

In[358]:=


Untuk penyederhanaan, kalikan A2 & B2 dengan Mengalikan A dengan

In[359]:=

Out[359]=

A2 = ExpandBA2 *

1 K LM 1

KLM

-D2 ‰- t d h1U4 v +

1 K LM

;

;

F

D ‰- t d h1p U4 vx4 K

+

D ‰- t d h1r U4 vz4

- D ‰- t d f h2U4wz4 - be ‰- t d h1U4 vx4z4 M ‰- t d h1p r U4 vx4z4 ‰- t d f h2p U4wx4z4 + ae ‰- t d h2U4wx4z4 + KM K

"Mengevaluasi operator-D diperoleh:"; In[360]:= Out[360]=

A2 = A2 ê. D Ø -d

- be ‰- t d h1U4 vx4z4 -

ae ‰- t d h2U4wx4z4 + ‰- t d h1r U4 vz4 d

M

‰- t d h1p r U4 vx4z4

KM t d ‰ f h2p U4wx4z4 K

+ -

‰- t d h1p U4 vx4 d

K

-

+ ‰- t d f h2U4wz4 d - ‰- t d h1U4 v d2

44

In[361]:=

Out[361]=

M2 = ExpandBFullSimplifyB - beh1U4 vx4z4 -

h1p U4 vx4 d

In[362]:=

h1p r U4 vx4z4

KM h1r U4 vz4 d

-

K

FF

A2 ‰- t d

M

+ aeh2U4wx4z4 +

f h2p U4wx4z4 K

-

+ f h2U4wz4 d - h1U4 v d2

C2 = M2;

"Sehingga"; In[363]:=

A2 = M2 ‰- t d ;

Mengalikan B2 dengan

In[364]:=

Out[364]=

B2 = ExpandBB2 * D3 -

D2 p x4

-

K

bDex4z4 +

1 K LM 1

KLM

D2 qy4

L Dp r x4z4

;

F

+ acDx4y4 +


-

D2 r z4 M

+


"Daftar koefisien persamaan B2 terhadap operator-D"; In[365]:=

Coeff2 = CoefficientList@B2, DD

Out[365]= :adex4y4z4 +

bcf x4y4z4 +

df p x4y4z4

beqx4y4z4

L acr x4y4z4 p qr x4y4z4 p qx4y4 + bex4z4 + , acx4y4 + M KLM KL p r x4z4 qr y4z4 p x4 qy4 r z4 - df y4z4 + ,, 1> KM LM K L M

K

-

"Misalkan,"; In[366]:=

a0 = Collect@Coeff2@@1DD, x4y4z4D

i Out[366]= jjade + k

In[367]:= Out[367]=

bcf +

df p

a1 = Coeff2@@2DD acx4y4 +

K

p qx4y4 KL

-

beq L

-

acr M

+ bex4z4 +

-

p qr y z

z x4y4z4

KLM {

p r x4z4 KM

- df y4z4 +

qr y4z4 LM

45

In[368]:= Out[368]=

In[369]:= Out[369]=

a2 = Coeff2@@3DD -

p x4 K

-

qy4 L

-

a3 = Coeff2@@4DD

r z4 M

1


2

-t d

;

"Dengan menggunakan metode operator-D untuk kasus tersebut diperoleh:"; Auxiliary Equation adalah a3m3 +3a2m2 +3a1m+a0 =0, dengan akar-akar biHi=1,2,3L ; Complementary Function HC FL adalah A1‰b 1 t+A2 ‰b 2 t +A3‰b 3 t, dengan Ai Hi=1,2,3Ladalah arbitrary constants ; Particular Integral HPIL adalah In[370]:=

M2 ‰- t d

Ha3 D3 + a2 D2 + a1 D + a0L

;

N = a3 D3 + a2 D2 + a1 D + a0 ê. D Ø -d;

Solusi Umum l2= C F + P I ; l2 = A1 ‰b 1 t + A2 ‰b 2 t + A3 ‰b 3 t +

M2 ‰- t d N

Menyelesaikanpersamaan l3 denganmenggunakanoperator - D; In[371]:=

In[372]:=

ClearAll@p, q, r, K, L, M, a, b, c, d, e, f, v, w, U, x, y, z, x4, y4, z4, d, ans1, ans2, ans3, p, h1, h2, B, l1, l2, l3, M1, M2, M3, N, ans1, ans2, ans3D;

Out[372]= 99l3 Ø

-I‰- t d M U4 ID2 h2KLw + H- bch1KL v - dh1Lp v + ach2KLw + h2p qwL x4

y4 + D Hdh1KL vy4 - h2w HLp x4 + Kqy4LLMM ë ID3 KLM + H bcf KLM + df LM p - beKM q - p qr + aKL HdeM - crLL x4y4z4 D2 HLM p x4 + KM qy4 + KLr z4L + D HacKLM x4y4 + M p qx4y4 + beKLM x4z4 + Lp r x4z4 - df KLM y4z4 + Kqr y4z4LM==

Misalkan l3= In[373]:=

A3 B3

, dengan ;

A3 = -I‰- t d M U4 2 ID h2KLw + H- bch1KL v - dh1Lp v + ach2KLw + h2p qwL x4y4 + D Hdh1KL vy4 - h2w HLp x4 + Kqy4LLMM;

46

In[374]:=


Untuk penyederhanaan, kalikan A & B dengan Mengalikan A3 dengan

In[375]:=

Out[375]=

A3 = ExpandBA3 *

1 K LM 1

KLM

-D2 ‰- t d h2U4w +

1 K LM

;

;

F

D ‰- t d h2p U4wx4

dD ‰- t d h1U4 vy4 +

K D ‰- t d h2qU4wy4

L

d ‰- t d h1p U4 vx4y4

+ bc ‰- t d h1U4 vx4y4 +

- ac ‰- t d h2U4wx4y4 -

K

‰- t d h2p qU4wx4y4

KL

"Mengevaluasi operator-D diperoleh:"; In[376]:=

A3 = A3 ê. D Ø -d;

In[377]:=

M3 = ExpandBFullSimplifyB

Out[377]=

bch1U4 vx4y4 + h2p U4wx4 d K

In[378]:=

A3 ‰- t d

FF

dh1p U4 vx4y4 K

+ dh1U4 vy4 d -

- ach2U4wx4y4 -

h2qU4wy4 d L

h2p qU4wx4y4 KL

-

- h2U4w d2

C3 = M3;

"Sehingga"; A3 = M3 ‰- t d ; Mengalikan B3 dengan

In[379]:=

Out[379]=

B3 = ExpandBB3 * D3 -

D2 p x4 K

bDex4z4 +

-

1 K LM 1

KLM

D2 qy4

L Dp r x4z4

;

F

+ acDx4y4 +


-

D2 r z4 M

+


47

"Daftar koefisien persamaan B3 terhadap operator-D"; In[380]:= Out[380]=

Coeff3 = CoefficientList@B3, DD

:a d e x4 y4 z4 + b c f x4 y4 z4 +

b e q x4 y4 z4

-

d f p x4 y4 z4

a c r x4 y4 z4

-

K p q r x4 y4 z4

, KL M p q x4 y4 p r x4 z4 q r y4 z4 + b e x4 z4 + - d f y4 z4 + , a c x4 y4 + KL KM LM p x4 q y4 r z4 , 1> K L M L

M

"Misalkan,"; In[381]:= Out[381]=

a0 = Coeff3@@1DD a d e x4 y4 z4 + b c f x4 y4 z4 + b e q x4 y4 z4 L

In[382]:= In[383]:= Out[383]=

In[384]:= In[385]:= Out[385]=

In[386]:= In[387]:= Out[387]= In[388]:=

-

d f p x4 y4 z4

a c r x4 y4 z4 M

-

K p q r x4 y4 z4

KL M

A0 = a0;

a1 = Coeff3@@2DD a c x4 y4 +

p q x4 y4 KL

+ b e x4 z4 +

p r x4 z4 KM

- d f y4 z4 +

q r y4 z4 LM

A1 = a1;

a2 = Coeff3@@3DD -

p x4 K

-

q y4 L

-

r z4 M

A2 = a2;

a3 = Coeff3@@4DD 1 A3 = a3;


2

-t d

;

"Dengan menggunakan metode operator-D untuk kasus tersebut diperoleh:"; Auxiliary Equation adalah a3m3 +3 a2m2 +3a1m+a0 =0, dengan akar-akar biHi=1,2,3L ;

48

Complementary Function HC FL adalah A1‰ b1 t+A2 b t b t ‰ 2 +A3‰ 3 , dengan Ai Hi=1,2,3Ladalah arbitrary constants ;

Particular Integral HPIL adalah

Ha3 D3+a2 D2+a1 D+a0L

M2 −t δ

;

Solusi Umum l3= C F + P I ; l3 = A1 ‰ b1 t + A2 ‰ b2 t + A3 ‰ b3 t + In[389]:= In[390]:=

M3 ‰-t d

ClearAll@M1, M2, M3, ND

N

H = Exp@- t d D Hh1 v y + h2 w z - BL U + l1 Jl2 Jl3 J-

p K

y2 + c x y - d y z + y Hq - v ULN +

q L r M

‰-t d M1

z2 + e x z - f y z + z Hr - w ULN;

In[391]:=

l1 =

In[392]:=

l2 =

In[393]:=

l3 =

In[394]:=

ans2 = Flatten@Solve@v y l2 + w z l3 == 0, yDD

Out[394]=

In[395]:=

N ‰-t d M2

N ‰-t d M3

N

:y Ø -

; ; ;

M3 w z M2 v

>

L21 = FullSimplify@-∑x HD ê. ans2 ‰-t d I2 M1 p x - K Ie M3 z -

Out[395]=

In[396]:=

In[397]:=

Out[398]=

+ M1 Ip - b z +

a M3 w z MMM M2 v

L22 = FullSimplify@-∑ y HD ê. ans2 LN

J‰-t d J-

2 M3 q w z v

+

L H-h1 N U v + a M1 x + f M3 z + M2 H- q + U v - c x + d zLLNN

L23 = FullSimplify@-∑z HD ê. ans2

‰- t d J2M3r z + M J- h2 NUw + bM1x - d M3 w z + M3 J-r + Uw - ex - f M3 w z NNN v

Out[397]=

In[398]:=

c M3 w z v

KN

1 Out[396]=

x2 - a x y - b x z + p xN +

M2 v

MN Tl1 = ∑t l1 -

‰- t d M1 d

N

49

In[399]:= Out[399]=

In[400]:= Out[400]=

Tl2 = ∑t l2 -

‰- t d M2 d

N

Tl3 = ∑t l3 -

‰- t d M3 d

N

In[401]:=

ansU = Flatten@FullSimplify@Solve@Eliminate@8Tl1 == L21, Tl2 == L22, Tl3 == L23<, 8x, z
In[402]:=

ans1 = Flatten@FullSimplify@Solve@Eliminate@8Tl1 == L21, Tl2 == L22, Tl3 == L23<, 8U, z
In[403]:=

ans3 = Flatten@FullSimplify@Solve@Eliminate@8Tl1 == L21, Tl2 == L22, Tl3 == L23<, 8U, x
In[404]:=

p = 2.5;

In[405]:=

q = 2;

In[406]:=

r = 2.5;

In[407]:=

K = 350000;

In[408]:=

L = 220000;

In[409]:=

M = 225000;

In[410]:=

a = 2.5 * 10-6 ;

In[411]:=

b = 1.2 * 10-6 ;

In[412]:=

c = 2.5 * 10-6 ;

In[413]:=

d = 1.55 * 10-5 ;

50

In[414]:=

e = 1.2 * 10-6 ;

In[415]:=

f = 3.75 * 10-6 ;

In[416]:=

v = 0.000147;

In[417]:=

w = 0.000257;

In[418]:=

B = 2500;

In[419]:=

h1 = 700;

In[420]:=

h2 = 500; 4r q

In[421]:= Out[421]=

LM True cr

In[422]:= Out[422]=

Out[423]= In[424]:=

Out[424]= In[425]:=

Out[425]= In[426]:=

Out[426]= In[427]:=

Out[427]=

>M

de True eq

In[423]:=

> Hd + f L2

cf

>L

True U4 = H-dLM p HeK + rL v + L Hp HcK + qL r + bKM Heq - crLL v + M HcKL H-f p + arL + q H- aeKL + p HeK - f L + rLLL wL ë 2 2 I2L H beKM + p rL v - 2LM H bcK + aeK + Hd + f L pL vw + 2M HacKL + p qL w M 8159.37 x4 = HK Hr H-aLq + p q + aLU4 vL - dLM Hf p - ar + aU4wL + bM H-qr + f L Hq - U4 vL + qU4wLLL ê H-LM HadeK + bcf K + df pL + beKM q + HacKL + p qL rL 301407. y4 = HL Hp r HcK + q - U4 vL - dM p HeK + r - U4wL + bKM Heq - cr - eU4 v + cU4wLLL ê H-LM HadeK + bcf K + df pL + beKM q + HacKL + p qL rL 126253. z4 = HM HaeKL Hq - U4 vL + cKL Hf p - ar + aU4wL p HeKq - f Lq + qr + f LU4 v - qU4wLLL ê HM HadeKL + df Lp + bK Hcf L - eqLL - HacKL + p qL rL 26215.4

51

In[428]:=

d = 0.1;

In[429]:=

M1 = C1

Out[429]=

-60.6264

In[430]:=

M2 = C2

Out[430]=

-492.611

In[431]:=

M3 = C3

Out[431]=

600.723

In[432]:=

Unprotect@ ND;

In[433]:=

a0 = A0;

In[434]:=

a1 = A1;

In[435]:=

a2 = A2;

In[436]:=

a3 = A3;

In[437]:= Out[437]= In[438]:= Out[438]= In[439]:= Out[439]= In[440]:= Out[440]= In[441]:= Out[441]= In[442]:= Out[442]= In[443]:= Out[443]= In[444]:= Out[444]=

N = a3 D3 + a2 D2 + a1 D + a0 ê. D Ø -d

-0.716155

FullSimplify@Tl1 == L21D

‰-0.1 t H- 168000. + 1. x - 1.74214zL ã 0


‰-0.1 t H732063. + 1. U + 0.844693x- 19.1456zL ã 0


‰-0.1 t H- 5850.92 + 1. U - 0.0032207x + 0.00760269zL ã 0

x = x ê. ans1 254768. x4 301407.

z = z ê. ans3 49805.6 z4 26215.4

52

z4 In[445]:= Out[445]= In[446]:= Out[446]= In[447]:= Out[447]=

M

11.6513

y = y ê. ans2 106185. y4 126253. y4

In[448]:= Out[448]= In[449]:= Out[449]= In[450]:= Out[450]= In[451]:= Out[451]= In[452]:= Out[452]= In[453]:= Out[453]= In[454]:= Out[454]= In[455]:=

100

L

100

57.3876 T = vy4U4 + wz4U4 206404.

U = U ê. ansU 6292.8 T1 = vy U 98225.7 T2 = wz U 80548.1 Td = T1 + T2 178774. U4 8159.37 U

Out[455]=

6292.8

In[456]:=

U4 - U

Out[456]=

1866.58

In[457]:= In[458]:= Out[458]= In[459]:=

Unprotect@pD;

p = Hh1 vy + h2wz - BL U

9.33 μ 107

ClearAll@p, q, r, K, L, M, a, b, c, d, e, f, v, w, U, x, y, z, x4, y4, z4, d, ans1, ans2, ans3, p, h1, h2, BD

53

Lampiran 6 (Kurva MSY)

Hasil ini diperoleh dengan menggunakan software Mathematica 5.1. In[328]:=

p = 2.5;

In[329]:=

q = 2;

In[330]:=

r = 2.5;

In[331]:=

K = 350000;

In[332]:=

L = 220000;

In[333]:=

M = 225000;

In[334]:=

a = 2.5 * 10-6 ;

In[335]:=

b = 1.2 * 10-6 ;

In[336]:=

c = 2.5 * 10-6 ;

In[337]:=

d = 1.55 * 10-5 ;

In[338]:=

e = 1.2 * 10-6 ;

In[339]:=

f = 3.75 * 10-6 ;

In[340]:=

v = 0.000147;

In[341]:=

w = 0.000257;

In[342]:=

In[343]:=

In[344]:=

y4 = HL Hp r Hc K + q - U vL - d M p He K + r - U w L + b K M He q - c r - e U v + c U w LLL ê H-L M Ha d e K + b c f K + d f pL + b e K M q + Ha c K L + p qL rL; z4 = HM Ha e K L Hq - U vL + c K L Hf p - a r + a U w L p He K q - f L q + q r + f L U v - q U w LLL ê HM Ha d e K L + d f L p + b K Hc f L - e qLL - Ha c K L + p qL rL; T = v U y4 + w U z4;

54

Solusi Pemanenan Maksimum In[345]:=

sol1 = FullSimplify@Solve@∑U T ã 0, UDD

In[346]:=

T = T ê. sol1

In[347]:=

ClearAll@TD

Out[345]= 88U →

8159.37<<

Out[346]= 8206404.<

In[348]:=

Plot@v U y4 + w U z4, 8U, 0 , 16500<, AxesLabel Ø 8U, T<, TextStyle Ø 8FontFamily Ø "Times", FontSize Ø 10
200000 150000 100000 50000

2500 Out[348]=

5000

7500

10000 12500 15000

U

Graphics

55

Lampiran 7 (Dinamika Sistem)

Hasil ini diperoleh dengan menggunakan software Mathematica 5.1. In[349]:= Unprotect@x, y, z, p , N, p, q, r, K, L, M, a, b, c, d, e, f, v, w D ; In[350]:=

ClearAll@parmval, x, y, z, p , N, p, q, r, K, L, M, a, b, c, d, e, f, v, w D;

In[351]:=

sysid

In[352]:=

intreset;

In[353]:=

plotreset;

In[354]:=

setstate[{x,y,z}];

In[355]:=

setparm[{p,q,r,K,L,M,a,b,c,d,e,f,v,w}];

In[356]:=

Mathematica 5.1.0, DynPac 10.69, 5ê24ê2007

slopevec = :p x J1 q y J1 -

y L

x K

N- ax y -b x z,

N + c x y - d y z - v U y, r z J1 -

z M

N + e x z - f y z - w U z>;

In[357]:=

eqstates = findpolyeq;

In[358]:=

parmval = 82.5, 2, 2.5, 350000, 220000, 225000, 2.5 * 10-6 , 1.2 * 10-6, 2.5 * 10-6, 1.55 * 10-5 , 1.2 * 10-6, 3.75 * 10-6 , 0.000147, 0.000257<;

In[359]:=

In[360]:=

E4 = FullSimplify@eqstates@@5DDD;

U = H- d L M p He K + rL v + L Hp Hc K + qL r + b K M He q - c rLL v + M Hc K L H-f p + a rL + q H- a e K L + p He K - f L + rLLL w L ê 2 H2 L Hb e K M + p rL v - 2 L M Hb c K + a e K + Hd + f L pL v w + 2 M Ha c K L + p qL w 2L;

In[361]:=

E4 = eqstateval@E4 D

In[362]:=

eigval@E4D

In[363]:=

classify@E4D;

Out[361]= 8301407., 126253., 26215.4< Out[362]= 8−1.74304 + 0.286408

, − 1.74304 − 0.286408 , −0.105861<

strictly stable

56

In[364]:=

evalparm@slopevec D

In[365]:=

<< Graphics`ParametricPlot3D`

In[366]:=

solution = NDSolveB:

Out[364]= :2.5 J1 −

x N x − 2.5 × 10−6 x y − 1.2 × 10−6 x z, 350000 y −6 −1.19943 y + 2.5 × 10 x y + 2 J1 − N y − 0.0000155 y z, 220000 z −6 −6 −2.09696 z + 1.2 × 10 x z − 3.75 × 10 y z + 2.5 J1 − N z> 225000

y£@tD y@tD z£@tD z@tD

x£@tD x@tD

== 2.5` J1 -

x@tD 350000

N - 2.5*^-6 y@tD - 1.2`*^-6 z@tD,

== -1.20648 + 2.5*^-6 x@tD + 2 J1 -

y@tD

220000

N - 0.0000155` z@tD,

== -2.10313 + 1.2`*^-6 x@tD - 3.75*^-6 y@tD + 2.5` J1 -

x@0D == 255297, y@0D == 107165, z@0D == 48695>,

z@tD

225000

N ,

8x, y, z< , 8t, 0, 150< , Method Ø Automatic F;

ParametricPlot3D@Evaluate@8 x@tD, y@tD, z@tD< ê. solutionD, 8 t, 0, 150<, PlotPoints Ø 1000, BoxRatios Ø 81, 1, 1<, AxesLabel Ø 8 "x", "y", "z"<, PlotRange Ø AllD; y 120000 110000 100000 00000

45000 40000 z 35000 30000 25000 260000 280000 x

In[367]:=

300000

intreset;

In[368]:= In[369]:=

plotreset;

57

In[370]:=

8X, Y, Z< = 8 x@tD, y@tD, z@tD< ê. Flatten@solutionD

Out[370]= 8InterpolatingFunction@880., 150.<<, <>D@tD,

InterpolatingFunction@880., 150.<<, <>D@tD, InterpolatingFunction@880., 150.<<, <>D@tD<

In[371]:=

Plot@8 X<, 8 t, 0, 25< , PlotRange Ø 8 250000, 320000< , AxesLabel Ø 8 "t", "x"< , PlotStyle Ø 8 RGBColor@0, 0, 1D
Out[371]= In[372]:=

10

15

20

25

Graphics

Plot@8 Y<, 8 t, 0, 50< , PlotRange Ø 8 80000, 130000<, AxesLabel Ø 8 "t", "y"< , PlotStyle Ø 8 RGBColor@0, 0, 1D
Out[372]= In[373]:=

20

30

40

50

Graphics

Plot@8 Z<, 8 t, 0, 50< , PlotRange Ø 8 20000, 50000< , AxesLabel Ø 8 "t", "z"<, PlotStyle Ø 8RGBColor@0, 0, 1D
Out[373]=

20

30

40

50

Graphics

58

Lampiran 8 (Syarat Teorema 7)

à Misalkan f0 = ‰-t d U Hv y h1 + w z h2 L, f1 = In[386]:=

f0 = Exp@-d tD Hh1 v y + h2 w zL U;

In[387]:=

f1 = p x J1 -

In[388]:=

f2 = q y J1 -

In[389]:=

f3 = r z J1 -

dx dy dz . , f2 = , f3 = dt dt dt

x N - a x y - b x z; K

y N + c x y - d y z - v U y; L

z N + e x z - f y z - w U z; M

dengan (x,y,z) adalah variabel state (autonomous) , t adalah waktu, dan sisanya adalah parameter. In[390]:= ∑t f0

Out[390]= - I‰-t d H- d L M p He K + rL v + L Hp Hc K + qL r + b K M He q - c rLL v +

M Hc K L H- f p + a rL + q H- a e K L + p He K - f L + rLLL wL d Hv y h1 + w z h2 LM ë I2 L Hb e K M + p rL v2 2 L M Hb c K + a e K + Hd + f L pL v w + 2 M Ha c K L + p qL w2M

Turunan parsial f1 terhadap x In[391]:= ∑x f1 Out[391]= -

px x + p J1 - N - a y - b z K K

Turunan parsial f2 terhadap y In[392]:= ∑y f2

Out[392]= - Hv H- d L M p He K + rL v + L Hp Hc K + qL r + b K M He q - c rLL v +

M Hc K L H- f p + a rL + q H- a e K L + p He K - f L + rLLL wLL ë

I2 L Hb e K M + p rL v2 - 2 L M Hb c K + a e K + Hd + f L pL v w + qy y 2 M Ha c K L + p qL w2M + c x + q J1 - N - d z L L

59

Turunan parsial f3 terhadap z In[393]:= ∑z f3

Out[393]= - Hw H- d L M p He K + rL v + L Hp Hc K + qL r + b K M He q - c rLL v +

M Hc K L H- f p + a rL + q H- a e K L + p He K - f L + rLLL wLL ë

I2 L Hb e K M + p rL v2 - 2 L M Hb c K + a e K + Hd + f L pL v w + rz z 2 M Ha c K L + p qL w2M + e x - f y + r J1 N M M

Terlihat bahwa turunan parsial dari f0, f1 , f2 , f3 kontinu In[382]:=

sehingga U dapat kontinu bagian demi bagian. H2.6 .3L

à ‡ f0 „ t ¶

0

Jika d > 0 maka Int = ‡ Exp@-d tD Hh1 v y + h2 w zL U „ t T

In[394]:=

0

Out[394]=

II1 - ‰-T dM H- d L M p He K + rL v + L Hp Hc K + qL r + b K M He q - c rLL v +

M Hc K L H- f p + a rL + q H- a e K L + p He K - f L + rLLL wL Hv y h1 + w z h2 LM ë II2 L Hb e K M + p rL v2 -

2 L M Hb c K + a e K + Hd + f L pL v w + 2 M Ha c K L + p qL w2M dM

In[395]:= Out[395]=

Limit@Int, T Ø ¶, Assumptions Ø d > 0D

HHc K L p r v + L p q r v - d L M p He K + rL v + b K L M He q - c rL v c f K L Mpw - a e K L Mqw +e K Mpqwf L M p q w + a c K L M r w + M p q r wL Hv y h1 + w z h2 LL ë I2 IM p q w2 + b K L M v He v - c wL +

L Ha K M w H- e v + c wL + p v Hr v - Hd + f L M wLLM dM

dari Definisi 18 akibatnya

integral pada (2.6.3) konvergen.

60

In[384]:=

Karena ‡ Exp@-d tD Hh1 v y + h2 w zL U „ t ada untuk d > 0, maka

In[384]:=

‡ Abs @Exp@-d Ht + sLD Hh1 v y + h2 w zL UD „ HtL

T

0

T

0

-Re@d H t +sLD ‡ ‰

T

Out[384]=

0

AbsAHH-d L M p He K + rL v + L Hp Hc K + qL r + b K M He q - c rLL v +

M Hc K L H-f p + a rL + q H- a e K L + p He K - f L + rLLL wL Hv y h1 + w z h2 LL ë I2 L Hb e K M + p rL v2 -

2 L M Hb c K + a e K + Hd + f L pL v w + 2 M Ha c K L + p qL w2 ME „ t

juga ada untuk d>0, s(-e,e),

In[385]:= In[396]:= Out[396]=

akibatnya untuk suatu fungsi yang kontinu bagian demi akibatnya bagian demi fungsi kontinu suatu untuk yang

In[386]:=

bagian a HtL, Abs @∑t Exp@-d Ht +sLD Hh1 v y + h2 w zL U D < a HtL

In[386]:=

untuk setiap s H-e, eL, t œ @0, ¶L , dan ‡ a HtL „ t < ¶, ¶

0

In[386]:=

sehingga

Teorema 7 dapat digunakan.

61

Lampiran 1 (Titik Tetap)

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