"T 1 (P1S) - T 1 (P1E) + .... + T 1 (P4S) - T 0 (P4E) = 0.53 * ð /4"

"T 3 (P1S) - T 3 (P1E) + .... + T 3 (P4S) - T 3 (P4E) = 0"

"T 5 (P1S) - T 5 (P1E) + .... + T 5 (P4S) - T 5 (P4E) = 0"

".... .... ...."

"T 15 (P1S) - T 15 (P1E) + .... + T 15 (P4S) - T 15 (P4E) = 0"

"P1 start: 17.9125 P1 end: 21.4007"

"P2 start: 36.1121 P2 end: 42.7902"

"P3 start: 54.8818 P3 end: 64.1028"

"P4 start: 74.4503 P4 end: 85.1345"

"A single transition."

STEP(t-x)

f1(t,x):=STEP(t-x)-STEP(t-2+x)-STEP(t-2-x)+STEP(t-4+x)

FOURIER(y,t,t1,t2,n)

FOURIER(f1(t,x),t,0,4,15)

x:epsilonReal (0, 1)

;Simp(#14)
4*SIN(15*pi*t/2)*COS(15*pi*x/2)/(15*pi)+4*SIN(13*pi*t/2)*COS(13*pi*x/2)/(13*p~
i)+4*SIN(11*pi*t/2)*COS(11*pi*x/2)/(11*pi)+4*SIN(9*pi*t/2)*COS(9*pi*x/2)/(9*p~
i)+4*SIN(7*pi*t/2)*COS(7*pi*x/2)/(7*pi)+4*SIN(5*pi*t/2)*COS(5*pi*x/2)/(5*pi)+~
4*SIN(3*pi*t/2)*COS(3*pi*x/2)/(3*pi)+4*SIN(pi*t/2)*COS(pi*x/2)/pi

f2(x,n):=COS(n*pi*x/2)

vx:=[x1,x2,x3,x4,x5,x6,x7,x8]

SUM((-1)^(i+1)*f2(vx SUB i,n),i,1,DIM(vx))

va:=VECTOR(SUM((-1)^(i+1)*f2(vx SUB i,n),i,1,DIM(vx)),n,1,15,2)

vb:=va=[0.53*pi/4,0,0,0,0,0,0,0]

cx=COS(pi*x/2)

cvx:=[cx1,cx2,cx3,cx4,cx5,cx6,cx7,cx8]

CHEBYCHEV_T(n,x)

CHEBYCHEV_T(n,COS(pi*x/2))=COS(n*pi*x/2)

cva:=VECTOR(SUM((-1)^(i+1)*CHEBYCHEV_T(n,cvx SUB i),i,1,DIM(cvx)),n,1,15,2)

cvb:=cva=[0.53*pi/4,0,0,0,0,0,0,0]

cvb

;Simp(#28)
[cx1-cx2+cx3-cx4+cx5-cx6+cx7-cx8=53*pi/400,4*cx1^3-3*cx1-4*cx2^3+3*cx2+4*cx3^~
3-3*cx3-4*cx4^3+3*cx4+4*cx5^3-3*cx5-4*cx6^3+3*cx6+4*cx7^3-3*cx7-cx8*(4*cx8^2-~
3)=0,16*cx1^5-20*cx1^3+5*cx1-16*cx2^5+20*cx2^3-5*cx2+16*cx3^5-20*cx3^3+5*cx3-~
16*cx4^5+20*cx4^3-5*cx4+16*cx5^5-20*cx5^3+5*cx5-16*cx6^5+20*cx6^3-5*cx6+16*cx~
7^5-20*cx7^3+5*cx7-cx8*(16*cx8^4-20*cx8^2+5)=0,64*cx1^7-112*cx1^5+56*cx1^3-7*~
cx1-64*cx2^7+112*cx2^5-56*cx2^3+7*cx2+64*cx3^7-112*cx3^5+56*cx3^3-7*cx3-64*cx~
4^7+112*cx4^5-56*cx4^3+7*cx4+64*cx5^7-112*cx5^5+56*cx5^3-7*cx5-64*cx6^7+112*c~
x6^5-56*cx6^3+7*cx6+64*cx7^7-112*cx7^5+56*cx7^3-7*cx7-cx8*(64*cx8^6-112*cx8^4~
+56*cx8^2-7)=0,256*cx1^9-576*cx1^7+432*cx1^5-120*cx1^3+9*cx1-256*cx2^9+576*cx~
2^7-432*cx2^5+120*cx2^3-9*cx2+256*cx3^9-576*cx3^7+432*cx3^5-120*cx3^3+9*cx3-2~
56*cx4^9+576*cx4^7-432*cx4^5+120*cx4^3-9*cx4+256*cx5^9-576*cx5^7+432*cx5^5-12~
0*cx5^3+9*cx5-256*cx6^9+576*cx6^7-432*cx6^5+120*cx6^3-9*cx6+256*cx7^9-576*cx7~
^7+432*cx7^5-120*cx7^3+9*cx7-cx8*(256*cx8^8-576*cx8^6+432*cx8^4-120*cx8^2+9)=~
0,1024*cx1^11-2816*cx1^9+2816*cx1^7-1232*cx1^5+220*cx1^3-11*cx1-1024*cx2^11+2~
816*cx2^9-2816*cx2^7+1232*cx2^5-220*cx2^3+11*cx2+1024*cx3^11-2816*cx3^9+2816*~
cx3^7-1232*cx3^5+220*cx3^3-11*cx3-1024*cx4^11+2816*cx4^9-2816*cx4^7+1232*cx4^~
5-220*cx4^3+11*cx4+1024*cx5^11-2816*cx5^9+2816*cx5^7-1232*cx5^5+220*cx5^3-11*~
cx5-1024*cx6^11+2816*cx6^9-2816*cx6^7+1232*cx6^5-220*cx6^3+11*cx6+1024*cx7^11~
-2816*cx7^9+2816*cx7^7-1232*cx7^5+220*cx7^3-11*cx7-cx8*(1024*cx8^10-2816*cx8^~
8+2816*cx8^6-1232*cx8^4+220*cx8^2-11)=0,4096*cx1^13-13312*cx1^11+16640*cx1^9-~
9984*cx1^7+2912*cx1^5-364*cx1^3+13*cx1-4096*cx2^13+13312*cx2^11-16640*cx2^9+9~
984*cx2^7-2912*cx2^5+364*cx2^3-13*cx2+4096*cx3^13-13312*cx3^11+16640*cx3^9-99~
84*cx3^7+2912*cx3^5-364*cx3^3+13*cx3-4096*cx4^13+13312*cx4^11-16640*cx4^9+998~
4*cx4^7-2912*cx4^5+364*cx4^3-13*cx4+4096*cx5^13-13312*cx5^11+16640*cx5^9-9984~
*cx5^7+2912*cx5^5-364*cx5^3+13*cx5-4096*cx6^13+13312*cx6^11-16640*cx6^9+9984*~
cx6^7-2912*cx6^5+364*cx6^3-13*cx6+4096*cx7^13-13312*cx7^11+16640*cx7^9-9984*c~
x7^7+2912*cx7^5-364*cx7^3+13*cx7-cx8*(4096*cx8^12-13312*cx8^10+16640*cx8^8-99~
84*cx8^6+2912*cx8^4-364*cx8^2+13)=0,16384*cx1^15-61440*cx1^13+92160*cx1^11-70~
400*cx1^9+28800*cx1^7-6048*cx1^5+560*cx1^3-15*cx1-16384*cx2^15+61440*cx2^13-9~
2160*cx2^11+70400*cx2^9-28800*cx2^7+6048*cx2^5-560*cx2^3+15*cx2+16384*cx3^15-~
61440*cx3^13+92160*cx3^11-70400*cx3^9+28800*cx3^7-6048*cx3^5+560*cx3^3-15*cx3~
-16384*cx4^15+61440*cx4^13-92160*cx4^11+70400*cx4^9-28800*cx4^7+6048*cx4^5-56~
0*cx4^3+15*cx4+16384*cx5^15-61440*cx5^13+92160*cx5^11-70400*cx5^9+28800*cx5^7~
-6048*cx5^5+560*cx5^3-15*cx5-16384*cx6^15+61440*cx6^13-92160*cx6^11+70400*cx6~
^9-28800*cx6^7+6048*cx6^5-560*cx6^3+15*cx6+16384*cx7^15-61440*cx7^13+92160*cx~
7^11-70400*cx7^9+28800*cx7^7-6048*cx7^5+560*cx7^3-15*cx7-cx8*(16384*cx8^14-61~
440*cx8^12+92160*cx8^10-70400*cx8^8+28800*cx8^6-6048*cx8^4+560*cx8^2-15)=0]

cv2:=(cvb SUB 2+3*cvb SUB 1)/4

;Simp(#30')
cv2:=cx1^3-cx2^3+cx3^3-cx4^3+cx5^3-cx6^3+cx7^3-cx8^3=159*pi/1600

cv3:=(cvb SUB 3+20*cv2-5*cvb SUB 1)/16

;Simp(#32')
cv3:=cx1^5-cx2^5+cx3^5-cx4^5+cx5^5-cx6^5+cx7^5-cx8^5=53*pi/640

cv4:=(cvb SUB 4+112*cv3-56*cv2+7*cvb SUB 1)/64

;Simp(#34')
cv4:=cx1^7-cx2^7+cx3^7-cx4^7+cx5^7-cx6^7+cx7^7-cx8^7=371*pi/5120

cv5:=(cvb SUB 5+576*cv4-432*cv3+120*cv2-9*cvb SUB 1)/256

;Simp(#36')
cv5:=cx1^9-cx2^9+cx3^9-cx4^9+cx5^9-cx6^9+cx7^9-cx8^9=3339*pi/51200

cv6:=(cvb SUB 6+2816*cv5-2816*cv4+1232*cv3-220*cv2+11*cvb SUB 1)/1024

;Simp(#38')
cv6:=cx1^11-cx2^11+cx3^11-cx4^11+cx5^11-cx6^11+cx7^11-cx8^11=12243*pi/204800

cv7:=(cvb SUB 7+13312*cv6-16640*cv5+9984*cv4-2912*cv3+364*cv2-13*cvb SUB 1)/4~
096

;Simp(#40')
cv7:=cx1^13-cx2^13+cx3^13-cx4^13+cx5^13-cx6^13+cx7^13-cx8^13=22737*pi/409600

cv8:=(cvb SUB 8+61440*cv7-92160*cv6+70400*cv5-28800*cv4+6048*cv3-560*cv2+15*c~
vb SUB 1)/16384

;Simp(#42')
cv8:=cx1^15-cx2^15+cx3^15-cx4^15+cx5^15-cx6^15+cx7^15-cx8^15=68211*pi/1310720

cvc:=[cvb SUB 1,cv2,cv3,cv4,cv5,cv6,cv7,cv8]

cvc

[cx1:epsilonReal (0, 1),cx2:epsilonReal (0, 1),cx3:epsilonReal (0, 1),cx4:eps~
ilonReal (0, 1),cx5:epsilonReal (0, 1),cx6:epsilonReal (0, 1),cx7:epsilonReal~
 (0, 1),cx8:epsilonReal (0, 1)]

cvd:=[cx1-cx2+cx3-cx4+cx5-cx6+cx7-cx8-53*pi/400,cx1^3-cx2^3+cx3^3-cx4^3+cx5^3~
-cx6^3+cx7^3-cx8^3-159*pi/1600,cx1^5-cx2^5+cx3^5-cx4^5+cx5^5-cx6^5+cx7^5-cx8^~
5-53*pi/640,cx1^7-cx2^7+cx3^7-cx4^7+cx5^7-cx6^7+cx7^7-cx8^7-371*pi/5120,cx1^9~
-cx2^9+cx3^9-cx4^9+cx5^9-cx6^9+cx7^9-cx8^9-3339*pi/51200,cx1^11-cx2^11+cx3^11~
-cx4^11+cx5^11-cx6^11+cx7^11-cx8^11-12243*pi/204800,cx1^13-cx2^13+cx3^13-cx4^~
13+cx5^13-cx6^13+cx7^13-cx8^13-22737*pi/409600,cx1^15-cx2^15+cx3^15-cx4^15+cx~
5^15-cx6^15+cx7^15-cx8^15-68211*pi/1310720]

NEWTONS(u,x,x0,n)

pd:=[17.9125,21.4007,36.1121,42.7902,54.8818,64.1028,74.4503,85.1345]

cx0:=VECTOR(COS(pi*x/2/90),x,pd)

;Approx(#50')
cx0:=[0.951527326421722077941838885457765820854687001928379737060755189,0.931~
05135798216697879971313810501975950091365491566825872618611,0.807865436543013~
8239101858899365710452007656275767774398576080038,0.7338460669502759040852284~
044246420292639065975688539129257492744,0.57526510822491812888631855017665615~
2938322073295257401944302904,0.4367578271599814437015934658423473531839335265~
367063023178860916,0.26807415608582170486654350603266753098852862403692046186~
28904897,0.08481696973504412823268302326550332891577910077225182636004473282]

cve:=cva-[0.53*pi/4,0,0,0,0,0,0,0]

cvx1:=(NEWTONS(cvd,cvx,cx0,10)) SUB 10

;Approx(#53')
cvx1:=[0.9515289584148314902917575948045316897040838580529112548672665896,0.9~
310528512956324875373389082888800469259066328815894808893214427,0.80786631683~
88302258767160099945570096450573434396894965679880449,0.733846983486380758238~
0674954217458723248608047072105962795450151,0.5752666360094235900993150486793~
590530789209979641099037018125631,0.43675807753960117547218876917802864601357~
81193337230870402703674,0.268074205647632831844749294769009538040296273873278~
2745642996313,0.0848171779884561127686425270742685930478879709902642442130498~
4532]

cp1:=VECTOR(180*ACOS(x)/pi,x,cvx1)

;Approx(#55')
cp1:=[17.91219597540949342428994226619440633502995309105527969786705607,21.40~
046551443607894614916464369409646659369518266812952249339406,36.1120144212546~
7597968815339510520183640504290101414195457026981,42.790122696205216102379014~
83545318413472018005453085872868587145,54.88169298387477401389200008871297076~
387030761461002391465298161,64.1027840528797605598334631784930444193801199510~
3175091196627531,74.450297052431294044106020445765002274868208089649794813791~
39215,85.13448802480628688608711194467029449730282687282910603099429378]

cvf:=VECTOR([n,SUM((-1)^(i+1)*CHEBYCHEV_T(n,cvx1 SUB i),i,1,DIM(cvx1))],n,1,3~
1,2)

;Approx(#57')
cvf:=[[1,0.4162610266006476040963002482845341321561249454172015212791801583],~
[3,-2.607973645025239114399394935970104396072359895912710648985767217*10^(-64~
)],[5,-4.974980277904127508336022837204297878649156363291994430698635132*10^(~
-64)],[7,-7.167510580477017336637105617005379425922211043428449348925773957*1~
0^(-64)],[9,-8.55421135596763025873032338755991327375392203971305637104348344~
2*10^(-64)],[11,-4.6950612573761910883915194834591428172051428406653981654775~
11250*10^(-64)],[13,1.0221380207128593029590718188653233001057074410672649940~
19933322*10^(-62)],[15,-1.074920388460489294988813234697666051764008551992638~
926547840428*10^(-61)],[17,-5.51152210442895196355188071596938944711577536347~
8436113231813338],[19,4.57823216659180774227020188749173615985438546546004946~
4758632096],[21,1.57275347010821629973366472263833663850851897824274964341920~
0475],[23,0.1820997568781743146385410744381143332476223147800061940891041693]~
,[25,0.01052569694032809474314705646014383614079443742237866331648139622],[27~
,0.001135326674488682474093905233846822365445042192876925413043516007],[29,-0~
.01006311927186493965140709807253086601220190999824787453805761339],[31,0.136~
5921891343238002785211503831155218370443061102872206983067014]]