MATHEMATICAL ANALYSIS AND NUMERICAL MATHEMATICS VOLUME: 17 |NUMBER: 3 |2019 |SEPTEMBER
Taylo Se ies Based In eg a ion in Elec ic
Ci cui s Simula ions
Vacla SATEK 1,2, Pe VEIGEND1, Gab iela NECASOVA1
1Depa men o In elligen Sys ems, Facul y o In o ma ion Technology, B no Uni e si y o Technology,
Boze echo a 2, 612 66 B no, Czech Republic
2IT4Inno a ions Na ional Supe compu ing Cen e , VSB–Technical Uni e si y o Os a a,
S uden ska 6231/1B, 708 00 Os a a, Czech Republic
[email p o ec ed], [email p o ec ed], i [email p o ec ed], inecaso [email p o ec ed]
DOI: 10.15598/aeee. 17i3.3369
Abs ac . This pape deals wi h he ex emely p e-
cise, s able and as solu ion o he o dina y di e en-
ial equa ions. The solu ion o hese is pe o med using
a me hod based on he Taylo se ies - The Mode n Tay-
lo Se ies Me hod. The pape in es iga es wo p oblems
o demons a e he posi i e p ope ies o he me hod:
linea p oblem - he beha io o signal ansmission
on he eleg aph line and a non-linea p oblem - he
Van de Pol oscilla o . Bo h p oblems we e analyzed
and sol ed using newly implemen ed MATLAB Mode n
Taylo Se ies Me hod sol e s. The esul s we e hen
compa ed o he s a e-o - he-a MATLAB sol e s.
Keywo ds
Ini ial alue p oblems, MATLAB, o dina y di -
e en ial equa ions, Taylo se ies me hod, ele-
g aph line, Van de Pol oscilla o .
1. In oduc ion
The pape deals wi h he solu ion o echnical Ini-
ial Value P oblems (IVPs) ep esen ing he p oblems
which a ise om common echnical p ac ice (especially
om elec ical and mechanical enginee ing). Ini ial
alue p oblems a e ep esen ed by he sys em o O -
dina y Di e en ial Equa ions (ODEs).
The bes -known and he mos accu a e me hod
o calcula ing a new alue o he nume ical solu ion
o ODE [1]:
y0= ( , y), y( 0) = y0,(1)
is o cons uc he Taylo se ies in he o m:
yi+1 =yi+h· ( i, yi) + h2
2! · 0( i, yi) + . . .
+hn
n!· [n−1]( i, yi),
(2)
whe e his he size o in eg a ion s ep, yi=y( i)is he
p e ious alue and yi+ 1 = y( i+h)is he nex alue
o he unc ion y( ).
The Taylo se ies can be e y e ec i ely imple-
men ed as he a iable-o de , a iable-s ep-size nu-
me ical me hod [2] - Mode n Taylo Se ies Me hod
(MTSM). The me hod is based on a ecu en calcu-
la ion o he e ms o he Taylo se ies o each in e-
g a ion s ep. The e o e, he complica ed calcula ion
o highe o de de i a i es does no need o be pe -
o med, he alue o highe de i a i e is calcula ed nu-
me ically om he p e ious one [3]. Equa ion (2) can
hen be ew i en in he o m:
yi+1 =DY0+DY1+DY2+···+DYn,(3)
whe e DYideno es he e ms o he Taylo se ies. The-
o e ically, i is possible o compu e he solu ion o ho-
mogeneous linea di e en ial equa ions wi h cons an
coe icien s wi h a bi a y o de and wi h a bi a y ac-
cu acy. Le us deno e as ORD he unc ion which
changes du ing he compu a ion and de ines he num-
be o e ms o he Taylo se ies used in he cu en
in eg a ion s ep (ORDi+1 =n).
The impo an p ope y o he me hod is au oma ic
o de con ol, i.e. using as many e ms o he Taylo
se ies as he de ined accu acy equi es. The e o e i is
common ha he numbe o e ms o he Taylo se ies
a ies o di e en cons an in eg a ion s ep sizes.
The MTSM has been implemen ed in MATLAB [4],
in C/C++ languages (FOS and TKSL/C so wa e [2]).
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Addi ionally, he me hod can be e ec i ely imple-
men ed di ec ly in ha dwa e [5].
Se e al o he implemen a ions o he Taylo se ies
me hod in a a iable o de and a iable s ep con-
ex we e p esen ed by di e en au ho s. TIDES so -
wa e [6] and TAYLOR [7], which includes de ailed de-
sc ip ion o a a iable s ep size e sion. O he imple-
men a ions based on Taylo se ies include ATOMF [8],
COSY INFINITY [9], and DAETS [10]. The a i-
able s epsize a iable-o de scheme is also desc ibed
in [11], [12] and [13], whe e simula ions on a pa -
allel compu e a e shown. The app oach based
on an app oxima e o mula ion o he Taylo me hods
can be ound in [14].
This pape amends and ex ends he pape p esen ed
a he 2018 Mode n Ma hema ical Me hods in Engi-
nee ing con e ence [15]. The solu ion o linea ODEs
now uses ixed s ep and ixed o de ins ead o a i-
able s ep a iable o de app oach shown in he pape .
The MTSM algo i hm o nonlinea quad a ic ODEs
is imp o ed wi h ecu en calcula ion o e ms o he
Taylo se ies (highe de i a i es a e no needed o cal-
cula ion). Due o hese changes, he compu a ion ime
o MTSM was imp o ed in compa ison o he pape [15]
p esen ed a he con e ence.
The pape is di ided in o se e al sec ions. In Sec. 2.
he e ec i e nume ical solu ion o a sys em o linea
ODEs using highe o de MTSM is shown and he Tele-
g aph equa ion is analyzed. The Sec. 3. p esen s he
solu ion o quad a ic nonlinea ODEs and he nonlin-
ea Van de Pol oscilla o is discussed. All algo i hms
o MTSM a e e icien ly implemen ed in MATLAB
so wa e [4] using ec o iza ion. Finally, he MTSM
algo i hms a e compa ed wi h MATLAB sol e s [16].
2. Solu ion o Linea ODEs
Equa ion (2) o linea sys ems o ODEs in he o m
~y 0=A~y +~
bcan be ew i en as:
~yi+1 =~yi+hA~yi+~
b+h2
2! AA~yi+~
b+. . .
+hn
n!A(n−1) A~yi+~
b,
(4)
whe e Ais he cons an Jacobian ma ix and ~
bis he
cons an igh -hand side o he sys em.
Mo eo e , Eq. (4) can be ew i en in he o m
Eq. (3) whe e e ms o he Taylo se ies can be com-
pu ed ecu en ly:
~
DY0=~yi,~
DY1=hA~yi+~
b,
~
DYl=h
lADYl−1, l = 2, . . . , n.
(5)
The ecu en calcula ion o Taylo se ies e ms is
use ul in e o con ol. I is s opped when:
n
X
j=n−s op || ~
DYj|| ≤ eps, (6)
whe e eps means e o pe s ep and s op deno es he
numbe o successi e e ms o he Taylo se ies, which
ha e me he s opping c i e ion Eq. (6). In his pape ,
he s op is se o 3.
Fo he solu ion o linea ODEs wi h cons an in-
eg a ion s ep size h, he cons an numbe o Taylo
se ies e ms (ORD =n) which sa is ies Eq. (6) is ob-
ained.
The Eq. (6) can be also ew i en in he o m:
~yi+1 =Ay~yi+Ab~
b, (7)
whe e he ma ices Ayand Aba e in he o m:
Ay=
n
X
j=0
hj
j!Aj,Ab=
n
X
j=1
hj
j!Aj−1.(8)
The ixed in eg a ion s ep size can be app oxima ed
using:
h < n
eps ·n
||An||,(9)
he e o e, cons an ma ices Ayand Aba e p ecalcu-
la ed only once a he beginning o he solu ion.
2.1. Teleg aph Equa ion
The eleg aph line is ep esen ed by he elec ic ci cui
depic ed in Fig. 1 [17]. The beha io o he ci cui is
desc ibed by he sys em o ODEs:
u0
Cj=1
Cj
(ij−ij+1),
i0
j=uLj
Lj
, j = 1, . . . , S,
(10)
whe e Sdeno es he numbe o RLC segmen s o he
eleg aph line. The solu ion o he sys em o ODEs
Eq. (10) leads o he linea IVP in he o m:
~y 0=A~y +~
b, ~y(0) = ~y0,(11)
whe e Ais a ma ix o cons an s (R,L, and Cpa am-
e e s o he ci cui ), ~y is a ec o o a iables ( ol ages
and cu en s), ~
bis a ec o o cons an s and ~y0is a ec-
o o ini ial condi ions. The block s uc u e o ma ix
Aand ec o s ~y and ~
bcan be ound in [17].
Fo he simula ion expe imen s in his pape , he
capaci ances and induc ances a e he same, C1=C2=
··· =CS= 1 pF and L1=L2=··· =LS= 10 nH
(homogeneous lossless eleg aph line).
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u0
R1L1
i1
C1
uC1
Segmen 1
i2iS−1
CS−1
uCS−1
LS
iS
CS
uCS
Segmen S
iS+1
R2
Fig. 1: Model o he eleg aph line - se ies o Ssegmen s.
Mo eo e , he ansmission line is adjus ed i
R1=R2=√L·C−1= 100 Ω. The angula eloci y is
se ω= 3·109 ad·s−1. The inpu ol age u0should be
gene ally cons an (DC ci cui ) o ha monic (AC ci -
cui ) signal. In he case o DC ci cui , he inpu ol age
u0is hidden in cons an igh -hand side ~
b. In he case
o AC ci cui , he inpu ol age u0=U0sin(ω )can
be compu ed using auxilia y sys em o coupled linea
ODEs:
u0
0=ωx, u0(0) = 0,
x0=−ωu0, x(0) = U0.(12)
The AC ci cui wi h inpu ol age u0= sin(ω )is
used. The p opaga ion cons an pe uni leng h o one
segmen o simple model o ansmission line in Fig. 1
can be calcula ed as LC =√LC. The o al delay o
he inpu signal can be compu ed as delay =S LC.
The delay o he ou pu ol age uC100 o 100 segmen s
is shown in Fig. 2. The ime o simula ion was se
a max = 2 delay o all expe imen s.
0 0.5 1 1.5 2
(s) 10-8
-1
-0.5
0
0.5
1
Vol age (V)
uC1
uC100
Fig. 2: Delay o he signal on he ansmission line wi h
S= 100 segmen s.
The MATLAB sol e , using he explici MTSM wi h
a cons an o de and cons an s ep size scheme Eq. (7)
o linea sys ems o ODEs Eq. (11) has been imple-
men ed. This algo i hm was es ed on a se o examples
o a eleg aph line wi h di e en numbe o segmen s S.
The maximum simula ion ime max is dependen on
he numbe o segmen s. The numbe o in eg a ion
s eps ises wi h he maximum simula ion ime, see
Tab. 3. Mo eo e , i is essen ial o e i y he s abil-
i y o me hods.
The MTSM was compa ed wi h ec o ized MAT-
LAB explici ode sol e s. Bo h ela i e and absolu e
ole ances o all sol e s we e se o eps = 1010. All
codes a e implemen ed in MATLAB 2015a and compu-
a ions we e pa ially pe o med on SALOMON supe -
compu e a IT4Inno a ions Na ional Supe compu ing
Cen e , VSB–Technical Uni e si y o Os a a [18].
Compa isons o MTSM and MATLAB ode
sol e s [16] a e in Tab. 1 and Tab. 2 o MTSM wi h
o de 30 and 60, espec i ely. Each un ime is aken
as a median alue o 100 compu a ions. Ra ios o
compu a ion imes a io =ode/expTay 1indica e
signi ican ly as e compu a ion using MTSM in all
cases.
Tab. 1: Time o solu ions: explici Taylo expTay (ORD = 30)
and MATLAB explici ode sol e s compa ison.
Sode23 ode45 ode113 expTay
a io a io a io (s)
200 2919.2 439.3 148.1 0.0054
600 1101.4 215.7 61.3 0.052
1000 934.7 163.7 47.9 0.14
1400 1015.9 180.5 46.2 0.26
1800 989.9 161.8 41.7 0.42
Tab. 2: Time o solu ions: explici Taylo expTay (ORD = 60)
and MATLAB explici ode sol e s compa ison.
Sode23 ode45 ode113 expTay
a io a io a io (s)
200 4573.5 693.3 234.9 0.0035
600 1639.3 311.8 91.1 0.035
1000 1267.2 221.2 65.9 0.1
1400 1505.6 273.5 68.3 0.17
1800 1536.4 248.3 64.4 0.28
The numbe o in eg a ion s eps is depic ed in Tab. 3,
whe e abb e ia ions Tay30 and Tay60 mean expTay
o ORD = 30 and ORD = 60, espec i ely.
Tab. 3: Nume ical solu ion: numbe o in eg a ion s eps.
Sode23 ode45 ode113 Tay30 Tay60
200 95627 26548 4297 147 55
600 267984 71908 12866 440 165
1000 439362 114316 21436 733 275
1400 610765 155172 30005 1026 385
1800 782287 194984 38574 1319 495
Mo e compa isons o MTSM nume ical solu ions o
linea sys ems o ODEs can be ound in [19] and [20].
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3. Solu ion o Nonlinea
(Quad a ic) ODEs
In his sec ion, he e ec i e solu ion o a special case
o nonlinea quad a ic sys ems o ODEs is desc ibed.
The nonlinea quad a ic sys em o ODEs is any
i s o de ODE ha is quad a ic in he unknown unc-
ion. Fo such sys em, Taylo se ies based nume ical
me hod can be implemen ed in a e y e ec i e way.
Equa ion (1) o nonlinea -quad a ic sys ems
o ODEs can be ew i en as:
~y 0=A~y +B~yjk +C~y 2+~
b, ~y(0) = ~y0,(13)
whe e A∈ <ne×ne is he ma ix o linea pa o he
sys em, B∈ <ne×ne(ne−1)/2is he ma ix o mixed
quad a ic e m, C∈ <ne×ne is he ma ix o pu e
quad a ic e m, ~
b∈ <ne is he igh -hand side o he
o ces incoming o he sys em, ~y0is a ec o o ini ial
condi ions, and he symbol ne s ands o he numbe
o equa ions o he sys em o ODEs. The unknown
unc ion ~y 2 ep esen s he ec o o mul iplica ions
(y1y1,y2y2,. . . , yneyne)Tand simila ly ~yjk ep-
esen s he ec o o mixed e ms mul iplica ions
(yj1yk1,yj2yk2,. . . , yjne(ne−1)/2ykne(ne−1)/2)T. The in-
dexes jand kcome om combina o ics C(ne, 2). Fo
simpli ica ion, he ma ices A,B,Cand he ec o ~
b
a e cons an .
Highe de i a i es o Eq. (3) can be e ec i ely com-
pu ed in MATLAB so wa e [4] using ma ix- ec o
mul iplica ion, e.g. highe de i a i e ~y[p] o pu e
quad a ic e m (wi h ma ix C) can be exp essed as:
~y[p]=C p−2
X
i=0
~y[p−1−i].∗~y[i]p−1
i+~y. ∗~y[p−1]!,
(14)
whe e he ope a ion ‘.*’ s ands o he elemen -
byelemen mul iplica ion, i.e. ~y[p1].∗~y[p2]is a ec-
o (y[p1]
1y[p2]
1,. . . ,y[p1]
ne y[p2]
ne )T. The binomial coe icien s
p−1
ican be e ec i ely p ecalcula ed using Pascal i-
angle, o mo e in o ma ion, see pascal unc ion in
MATLAB so wa e.
Mo eo e , he highe de i a i es o he e ms B~yjk,
C~y 2, in Eq. (13) can be included in a ecu en calcu-
la ion o Taylo se ies e ms ~
DYBand ~
DYC:
~
DYB0=~
DYC0=~
0,
~
DYB1=h(B~yjk),~
DYC1=h(C~y 2
i),
~
DYBl=h
lBPl
m=1 ~
DYj,l−m.∗~
DYk,m−1,
~
DYCl=h
lCPl
m=1 ~
DYl−m.∗~
DYm−1,
~
DYl=~
DYAl−1+~
DYBl−1+~
DYCl−1,
l= 2, . . . , n,
(15)
whe e he linea e m DYAl−1is compu ed using
Eq. (5).
3.1. Van de Pol Oscilla o
The Van de Pol oscilla o is a nonconse a i e oscil-
la o wi h nonlinea damping. Ene gy is dissipa ed
a high ampli udes and gene a ed a low ampli udes.
As a esul , he e a e oscilla ions a ound a s a e a
which ene gy gene a ion and dissipa ion balance.
Due o he unique na u e o he Van de Pol oscilla-
o , i has become he co ne s one o s udying sys ems
wi h limi cycle oscilla ions. In ac , he Van de Pol
equa ion has become a s aple model o oscilla o y p o-
cesses no only in physics, bu also in biology, sociology,
and e en economics.
F om ma hema ical poin o iew, he Van de Pol
equa ion is an o dina y di e en ial equa ion:
y00 +µ(y2−1)y0+y=b, (16)
whe e he pa ame e µ ep esen s he nonlinea i y and
he s eng h o damping [21]. Righ -hand side unc-
ion b ep esen s he o cing unc ion. When µ= 0,
he equa ion becomes y00 +y= 0, which is a sim-
ple ha monic oscilla o . Fo µ > 0 he sys em en e s
a limi cycle: nea he o igin (y=y0= 0), he sys em
is uns able; a om he o igin, he sys em is damped,
see Fig. 3.
-2 0 2
-5
0
5
=0.1
=1
=5
y ( ad)
y' ( ad s )
.
-1
Fig. 3: Limi cycles o un o ced Van de Pol oscilla o .
The beha io o he sys em o di e en pa ame e s
µin he ime domain is shown in Fig. 4.
The second o de ODE Eq. (16) can be ans o med
(using subs i u ions y1=y0and y2=y2) in o he
auxilia y sys em o ODEs:
y0=y1,
y0
1=µ(1 −y2)y1−y+b,
y0
2= 2yy1,
(17)
wi h ini ial condi ions y(0),y1(0) = y0(0),y(0) = y2(0)
and ze o igh -hand side b= 0.
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5 10 15 20
(s)
-2
-1
0
1
2
3
Posi ion ( ad), eloci y ( ad s )
.
-1
y ( ad)
y' ( ad s )
-1
.
(a) µ= 0.1.
0 5 10 15 20
(s)
-3
-2
-1
0
1
2
3
Posi ion ( ad), eloci y ( ad s )
.
-1
y ( ad)
y' ( ad s )
-1
.
(b) µ= 1.
0 5 10 15 20
(s)
-5
0
5
Posi ion ( ad), eloci y ( ad s )
.
-1
y ( ad)
y' ( ad s )
-1
.
(c) µ= 5.
Fig. 4: Beha io o Van de Pol oscilla o in he ime domain.
The sys em Eq. (17) is equi alen wi h Eq. (13),
whe e:
~y =

y
y1
y2
,~
b=

0
0
0
,A=0,
B=

0 0 0
0 0 −µ
2 0 0 
,
C=

0 1 0
−1µ0
0 0 0 
.
(18)
0 5 10 15 20
(s)
5
10
15
20
ORD
(a) µ= 0.1.
(b) µ= 1.
0 5 10 15 20
(s)
10
20
30
40
50
60
ORD
(c) µ= 5.
Fig. 5: The MTSM solu ion: ORD unc ion.
The MATLAB sol e o explici a iable s ep size
and a iable o de MTSM o nonlina quad a ic sys-
ems o ODEs Eq. (11) has been implemen ed. This
algo i hm was es ed on a se o examples o Van de
Pol sys ems Eq. (17) wi h di e en alues o he pa-
ame e µ. The MTSM was again compa ed wi h
ec o ized MATLAB explici ode sol e s. Bo h el-
a i e and absolu e ole ances o all sol e s we e se o
eps = 10−10. Resul s o he compa isons o he MTSM
wi h MATLAB ode sol e s a e shown in Tab. 4. Ra ios
o compu a ion imes a io =ode =expTay >1
indica e as e compu a ion o he MTSM in all
cases. The numbe o in eg a ion s eps is shown in
Tab. 5. The MTSM O de (ORD) is shown in Fig. 5.
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As can be seen, he ORD is changing apidly especially
o p oblems in ol ing highe non-linea i ies (la ge
pa ame e µ). Beha io o he unc ion ORD demon-
s a es ha he me hod changes used in eg a ion o -
de dynamically du ing calcula ion wi h espec o he
alue o pa ame e eps.
Tab. 4: Time o solu ions: explici Taylo expTay and MAT-
LAB explici ode sol e s compa ison.
µode23 ode45 ode113 expTay
a io a io a io (s)
0.1156.4 8.43 1.9 0.063
1193.9 8.9 4.12 0.088
5108.3 8.2 3.1 0.15
10 58.1 5.9 2.1 0.2
Tab. 5: Nume ical solu ion: numbe o in eg a ion s eps.
µode23 ode45 ode113 expTay
0.1283858 22280 1969 100
1354201 36660 3986 250
5337565 57344 5002 1000
10 241783 54700 4581 2000
3.2. Fo ced Van de Pol Oscilla o
The o ced Van de Pol oscilla o equa ion adds o he
Eq. (16) nonze o igh -hand side (e.g. some ha monic
signal b( ) = Fsin(ω )):
y00 −µ(1 −y2)y0+y=Fsin(ω ),(19)
whe e Fis he ampli ude and ωis angula eloci y o
he wa e signal ( ad·s−1). The limi cycle o µ= 5,
ω= 1 and F= 2 is depic ed in Fig. 7.
The beha io o such sys em in he ime domain is
shown in Fig. 8. In elec ic ci cui s, he wa e unc-
ion can be ep esen ed as some AC powe supply
b( ) = u0=U0sin(ω ), see Fig. 6.
Equa ion (19) is equi alen o he sys em o i e
ODEs Eq. (17) wi h gene a ing sys em o igh -hand
side sine unc ion Eq. (12). This au onomous sys em o
ODEs can be again ep esen ed in ma ix- ec o o m
Eq. (13).
Resul s o nume ical solu ions o o ced Van de Pol
Eq. (19) wi h di e en pa ame e µand se pa ame e s
ω= 1,F= 2 a e in Tab. 6. Ra io o compu a ion imes
a io =ode =expTay >1indica es again as e
compu a ion o he MTSM in all cases. The numbe
o in eg a ion s eps o all me hods can be ound in
Tab. 7.
Take a no e, ha he pa ame e ωin Eq. (19) has
also signi ican impac on nume ical compu a ions, see
Tab. 8 and Tab. 9. The las line o Tab. 9 ( o ω > 104)
indica es ha he compu a ion ime o ode23 sol e is
mo e han 50 hou s.
i
R
ES
C
L
M
Ug
ia
Fig. 6: Fo ced Van de Pol Oscilla o .
-2 0 2
-5
0
5
=5
y ( ad)
y' ( ad s )
.
-1
Fig. 7: Limi cycle o o ced Van de Pol oscilla o .
0 10 20 30 40 50
(s)
-10
-5
0
5
10 y ( ad)
Posi ion ( ad), eloci y ( ad s )
.
-1
y' ( ad s )
-1
.
Fig. 8: Beha io o he o ced Van de Pol oscilla o in he ime
domain.
Mo e compa isons o MTSM nume ical solu ions o
nonlinea sys ems o ODEs can be ound in [22].
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Tab. 6: Fo ced Van de Pol oscilla o : ime o nume ical solu-
ion.
µode23 ode45 ode113 expTay
a io a io a io (s)
0.1250.6 9.4 3.9 0.076
1190 10 3.8 0.12
5130.4 8.5 3.1 0.17
10 18.7 6.7 2.3 0.22
Tab. 7: Fo ced Van de Pol oscilla o : numbe o in eg a ion
s eps.
µode23 ode45 ode113 expTay
0.1382029 32184 3347 100
1449441 52732 4779 400
5464230 67424 5727 1000
10 385322 68740 5665 2000
Tab. 8: Fo ced Van de Pol oscilla o (µ= 5,F= 2): ime o
nume ical solu ion.
ωode23 ode45 ode113 expTay
( ad·s−1) a io a io a io (s)
101434.9 21.8 4.2 0.21
1021025.8 51.3 8.5 0.9
1031526.3 77.9 10.7 10.5
104>1500 88 12.7 121.4
Tab. 9: Fo ced Van de Pol oscilla o (µ= 5,F= 2): numbe
o in eg a ion s eps.
ω( ad·s−1) ode23 ode45 ode113 expTay
10119 ·10521 ·10411 ·10410 ·102
10219 ·10621 ·10594 ·10320 ·102
10319 ·10721 ·10694 ·10420 ·103
104-21 ·10794 ·10520 ·104
4. Conclusion
This a icle deal wi h he nume ical solu ion o lin-
ea and nonlinea sys ems o ODEs coming om elec-
ical ci cui s simula ions. The model o he ele-
g aph line was chosen as he example o he linea
p oblem, he Van de Pol oscilla o as he example
o nonlinea one. All calcula ions we e pe o med us-
ing MATLAB so wa e. The MTSM sol e o non-
linea quad a ic sys ems o ODEs was success ully im-
plemen ed. The MTSM is able o sol e hese p oblems
as e and mo e e icien ly han he s a e-o - he-a ode
sol e s in MATLAB.
Fu u e wo k will be ocused on he solu ion o gen-
e al nonlinea p oblems, pa alleliza ion o he MTSM
algo i hm and i s ha dwa e ep esen a ion.
Acknowledgmen
This esea ch was inancially suppo ed by he Min-
is y o Educa ion, You h and Spo s om he Na-
ional P og amme o Sus ainabili y (NPU II) p ojec
“IT4Inno a ions excellence in science - LQ1602”. The
pape also includes he esul s o he in e nal BUT FIT
p ojec FIT-S-17-4014.
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Abou Au ho s
Vacla SATEK was bo n in Os a a, Czech Re-
public. He ecei ed his M.Sc. om B no Uni e si y
o Technology, Facul y o In o ma ion Technology in
2006 and Ph.D. in 2012 a he same ins i u ion. His
esea ch in e es s include nume ical solu ion o o di-
na y and pa ial di e en ial equa ions - s i sys ems,
compu a ional luid dynamics, con ac p oblems e c.
Pe VEIGEND was bo n in B no, Czech Re-
public. He ecei ed his M.Sc. om B no Uni e si y
o Technology, Facul y o In o ma ion Technology
in 2014. His esea ch in e es s include nume ical
calcula ions and sys em modeling and simula ions.
Gab iela NECASOVA was bo n in B no, Czech
Republic. She ecei ed his M.Sc. om B no Uni e si y
o Technology, Facul y o In o ma ion Technology in
2014. He esea ch in e es s include nume ical solu ion
o di e en ial equa ions and pa allel me hods.
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